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Search: a296044 -id:a296044
Displaying 1-9 of 9 results found.
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A296043
a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k-1))/(1 + x^(2*k)))^n.
+10
6
1, 1, -1, -5, -1, 31, 65, -90, -641, -644, 3329, 11386, -1471, -87021, -164634, 317935, 1881471, 1418719, -11370760, -33937951, 17468929, 294971868, 468897758, -1304743033, -6275603903, -2804572819, 42665919997, 109181454826, -106020803386, -1063546684834, -1362993953395
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..500
MATHEMATICA
Table[SeriesCoefficient[Product[((1 + x^(2 k - 1))/(1 + x^(2 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[(x^(1/8) EllipticTheta[2, 0, x^(1/2)]/EllipticTheta[2, 0, x])^n, {x, 0, n}], {n, 0, 30}]
CROSSREFS
Main diagonal of A296067.
Cf. A029838, A029839, A029840, A029841, A029842, A029843, A029844, A029845, A296044, A296045.
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Dec 03 2017
STATUS
approved
A296045
a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^n.
+10
5
1, 1, 3, 13, 55, 231, 981, 4222, 18351, 80320, 353453, 1562364, 6932185, 30856541, 137725710, 616190583, 2762605791, 12408541299, 55825435656, 251523510045, 1134741006825, 5125453110196, 23175983361270, 104899547541255, 475228898015025, 2154737528486881, 9777332125043577
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..500
FORMULA
a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 - x^(4*k)))^n.
a(n) ~ c * d^n / sqrt(n), where d = 4.62579056836776492108784045382518984897... (see A192540) and c = 0.255113338880004277664416308115912337... - Vaclav Kotesovec, Dec 05 2017
MATHEMATICA
Table[SeriesCoefficient[Product[((1 + x^(2 k - 1))/(1 - x^(2 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^(4 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[(2 (-x)^(1/8)/EllipticTheta[2, 0, Sqrt[-x]])^n, {x, 0, n}], {n, 0, 26}]
Table[(-1)^n * 2^n * SeriesCoefficient[1/(QPochhammer[-1, x]*QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 07 2020 *)
(* Calculation of constants {d, c}: *) Chop[{1/r, 4/Sqrt[Pi*(77/2 - 4*s*(-r*s)^(7/8) * Derivative[0, 0, 2][EllipticTheta][2, 0, Sqrt[-r*s]])]} /. FindRoot[{s == (2*(-r*s)^(1/8))/EllipticTheta[2, 0, Sqrt[-r*s]], 7*I*r + 2*(-r*s)^(7/8)*Sqrt[r*s] * Derivative[0, 0, 1][EllipticTheta][2, 0, Sqrt[-r*s]] == 0}, {r, 1/5}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)
CROSSREFS
Cf. A006950, A106337, A192540, A273225, A273226, A273228, A295832, A296043, A296044.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 03 2017
STATUS
approved
A296163
a(n) = [x^n] Product_{k>=1} ((1 - x^(5*k))/(1 - x^k))^n.
+10
4
1, 1, 5, 22, 105, 501, 2456, 12160, 60801, 306130, 1550255, 7887034, 40281720, 206405967, 1060602800, 5463059772, 28199365873, 145832364580, 755420838614, 3918935839970, 20357605331355, 105878815699042, 551273881133750, 2873161931172668, 14988243880188600
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..24.
Eric Weisstein's World of Mathematics, Partition Function b_k
FORMULA
a(n) = [x^n] Product_{k>=1} (1 + x^k + x^(2*k) + x^(3*k) + x^(4*k))^n.
a(n) ~ c * d^n / sqrt(n), where d = 5.3271035802753567624196808294779171420899175782347488197... and c = 0.2712048688090020853684153670711011713396954... - Vaclav Kotesovec, May 13 2018
MATHEMATICA
Table[SeriesCoefficient[Product[((1 - x^(5 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[Product[(1 + x^k + x^(2 k) + x^(3 k) + x^(4 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]
(* Calculation of constant d: *) With[{k = 5}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)
CROSSREFS
Cf. A008485, A035959, A121591, A263002, A270913, A285928, A296044, A296162.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 06 2017
STATUS
approved
A296068
Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2*j))/(1 - x^(2*j-1)))^k.
+10
3
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 9, 10, 4, 0, 1, 5, 14, 22, 18, 6, 0, 1, 6, 20, 40, 48, 32, 9, 0, 1, 7, 27, 65, 101, 99, 55, 12, 0, 1, 8, 35, 98, 185, 236, 194, 90, 16, 0, 1, 9, 44, 140, 309, 481, 518, 363, 144, 22, 0, 1, 10, 54, 192, 483, 882, 1165, 1080, 657, 226, 29, 0, 1, 11, 65, 255, 718, 1498, 2330, 2665, 2162, 1155, 346, 38, 0
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
Table of n, a(n) for n=0..90.
FORMULA
G.f. of column k: Product_{j>=1} ((1 + x^(2*j))/(1 - x^(2*j-1)))^k.
G.f. of column k: Product_{j>=1} ((1 - x^(4*j))/(1 - x^j))^k.
G.f. of column k: 2^(-k/2)*(theta_2(0,x)/(x^(1/8)*theta_2(Pi/4,sqrt(x))))^k, where theta_() is the Jacobi theta function.
EXAMPLE
G.f. of column k: A_k(x) = 1 + k*x + (1/2)*k*(k + 3)*x^2 + (1/6)*k*(k^2 + 9*k + 8)*x^3 + (1/24)*k*(k^3 + 18*k^2 + 59*k + 18)*x^4 + (1/120)*k*(k^4 + 30*k^3 + 215*k^2 + 330*k + 144)*x^5 + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 2, 5, 9, 14, 20, ...
0, 3, 10, 22, 40, 65, ...
0, 4, 18, 48, 101, 185, ...
0, 6, 32, 99, 236, 481, ...
MATHEMATICA
Table[Function[k, SeriesCoefficient[Product[((1 + x^(2 i))/(1 - x^(2 i - 1)))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Product[((1 - x^(4 i))/(1 - x^i))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[2^(-k/2) (EllipticTheta[2, 0, x]/(x^(1/8) EllipticTheta[2, Pi/4, Sqrt[x]]))^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
CROSSREFS
Columns k=0..8 give A000007, A001935, A001936, A001937, A093160, A001939, A001940, A001941, A092877.
Main diagonal gives A296044.
Antidiagonal sums give A302020.
Cf. A296067.
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Dec 04 2017
STATUS
approved
A304625
a(n) = [x^n] Product_{k>=1} ((1 - x^(n*k))/(1 - x^k))^n.
+10
3
1, 0, 3, 19, 101, 501, 2486, 12398, 62329, 315436, 1605330, 8207552, 42124368, 216903051, 1119974861, 5796944342, 30068145889, 156250892593, 813310723907, 4239676354631, 22130265931880, 115654632452514, 605081974091853, 3168828466966365, 16610409114771876, 87141919856550506
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Number of partitions of n into 2 or more parts of n kinds. - Ilya Gutkovskiy, May 16 2018
LINKS
Table of n, a(n) for n=0..25.
Index entries for sequences related to partitions
FORMULA
a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.3527013334866426877724... and c = 0.268015212710733315686... - Vaclav Kotesovec, May 16 2018
MATHEMATICA
Table[SeriesCoefficient[Product[((1 - x^(n k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[1/(1 - x^k)^n, {k, 1, n - 1}], {x, 0, n}], {n, 0, 25}]
CROSSREFS
Cf. A000065, A008485, A022567, A093160, A270913, A285927, A285928, A286653, A296044, A296162, A296163, A304626.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 15 2018
STATUS
approved
A296162
a(n) = [x^n] Product_{k>=1} ((1 - x^(3*k))/(1 - x^k))^n.
+10
2
1, 1, 5, 19, 89, 406, 1913, 9073, 43505, 209971, 1019390, 4971781, 24343037, 119579006, 589050663, 2908727839, 14393759457, 71360342129, 354372852011, 1762416036422, 8776797353574, 43761058841638, 218431753457637, 1091385769314272, 5458068218285909, 27319061357620056
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..25.
Eric Weisstein's World of Mathematics, Partition Function b_k
FORMULA
a(n) = [x^n] Product_{k>=1} (1 + x^k + x^(2*k))^n.
a(n) ~ c * d^n / sqrt(n), where d = 5.1069752682291604222843644516987970799026747758649349... and c = 0.271879273312907861082536692355942116774864... - Vaclav Kotesovec, May 13 2018
MATHEMATICA
Table[SeriesCoefficient[Product[((1 - x^(3 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[(1 + x^k + x^(2 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
(* Calculation of constant d: *) With[{k = 3}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)
CROSSREFS
Cf. A000726, A008485, A121596, A128758, A270913, A285927, A296044, A296163.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 06 2017
STATUS
approved
A304628
a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(4*k)))^n.
+10
2
1, 1, 3, 13, 47, 181, 729, 2948, 12031, 49540, 205153, 853546, 3565505, 14943839, 62810786, 264650683, 1117486463, 4727486583, 20032950744, 85017558081, 361289789377, 1537198394570, 6547611493822, 27917246924099, 119141276756545, 508884954441331, 2175284934712217, 9305217981192748
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..500
FORMULA
a(n) = [x^n] Product_{k>=1} ((1 - x^(8*k-4))/(1 - x^(2*k-1)))^n.
a(n) ~ c * d^n / sqrt(n), where d = 4.3582188263213968630940316689... and c = 0.266443662680498334500839... - Vaclav Kotesovec, May 18 2018
MATHEMATICA
Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(4 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]
Table[SeriesCoefficient[Product[((1 - x^(8 k - 4))/(1 - x^(2 k - 1)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]
(* Calculation of constants {d, c}: *) With[{k = 4}, {1/r, Sqrt[QPochhammer[-1, (r*s)^k] / (2*Pi*(r^2*s*Derivative[0, 2][QPochhammer][-1, r*s] - k^2*(r*s)^(2*k) * Derivative[0, 2][QPochhammer][-1, (r*s)^k] - k*(1 + k)*(r*s)^k * Derivative[0, 1][QPochhammer][-1, (r*s)^k]))]} /. FindRoot[{s == QPochhammer[-1, r*s]/QPochhammer[-1, (r*s)^k], QPochhammer[-1, (r*s)^k] + k*(r*s)^k*Derivative[0, 1][QPochhammer][-1, (r*s)^k] == r*Derivative[0, 1][QPochhammer][-1, r*s]}, {r, 1/4}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)
CROSSREFS
Cf. A070048, A255526, A285290, A296044, A296164, A304629.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 15 2018
STATUS
approved
A319456
a(n) = [x^n] Product_{k>=1} ((1 - x^k)*(1 - x^(2*k)))^n.
+10
1
1, -1, -3, 14, -11, -81, 282, -57, -2043, 5405, 2417, -46476, 94522, 110512, -943407, 1505289, 2807589, -16888311, 23645199, 46006542, -265972791, 472882620, 187884672, -3981273597, 14234579226, -19187383356, -78662039004, 502118911904, -847583768679, -2627514175002
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..29.
FORMULA
a(n) = [x^n] Product_{k>=1} (1 - x^(2*k))^(2*n)/(1 + x^k)^n.
a(n) = [x^n] exp(n*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k).
MATHEMATICA
Table[SeriesCoefficient[Product[((1 - x^k) (1 - x^(2 k)))^n , {k, 1, n}], {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[(QPochhammer[x] QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[Exp[n Sum[(DivisorSigma[1, 2 k] - 4 DivisorSigma[1, k]) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 29}]
CROSSREFS
Cf. A002171, A002288, A008705, A030204, A030211, A066535, A296043, A296044, A319457.
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Sep 19 2018
STATUS
approved
A319457
a(n) = [x^n] Product_{k>=1} 1/((1 - x^k)*(1 - x^(2*k)))^n.
+10
1
1, 1, 7, 31, 175, 931, 5209, 29114, 165087, 940828, 5396777, 31090962, 179832625, 1043516371, 6072302726, 35420582431, 207051636799, 1212583329959, 7113193757656, 41788933655049, 245831162935825, 1447891754747672, 8537111315442222, 50387162650271055, 297664212003582753
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..24.
FORMULA
a(n) = [x^n] Product_{k>=1} (1 + x^k)^n/(1 - x^(2*k))^(2*n).
a(n) = [x^n] exp(n*Sum_{k>=1} (4*sigma(k) - sigma(2*k))*x^k/k).
MATHEMATICA
Table[SeriesCoefficient[Product[1/((1 - x^k) (1 - x^(2 k)))^n , {k, 1, n}], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[1/(QPochhammer[x] QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[Exp[n Sum[(4 DivisorSigma[1, k] - DivisorSigma[1, 2 k]) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 24}]
CROSSREFS
Cf. A002513, A008485, A270919, A296043, A296044, A319455, A319456.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 19 2018
STATUS
approved
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