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Expansion of (1-4*x)^(13/2).
+10
6
1, -26, 286, -1716, 6006, -12012, 12012, -3432, -858, -572, -572, -728, -1092, -1848, -3432, -6864, -14586, -32604, -76076, -184184, -460460, -1184040, -3121560, -8414640, -23140260, -64792728, -184410072, -532740208, -1560167752, -4626704368, -13880113104
FORMULA
a(n) = (-2)^n * Product_{i=0..n-1} (13-2*i) / n! for n>0. - R. J. Mathar, Feb 19 2008 D-finite with recurrence: n*a(n) - 2*(2*n-13)*a(n-1) = 0 for n>0. - Bruno Berselli, Jul 02 2018 a(n) ~ -135135 * 2^(2*n - 7) / (sqrt(Pi) * n^(15/2)). - Vaclav Kotesovec, Jul 02 2018 a(n) = (-4)^n*binomial(13/2, n).
Sum_{n>=0} 1/a(n) = 960/1001 - 10*Pi/(3^8*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 244659776/234609375 - 12*log(phi)/(5^7*sqrt(5)), where phi is the golden ratio ( A001622). (End)
MAPLE
f := k -> -135135*(2*k)!/((2*k-1)*(2*k-3)*(2*k-5)*(2*k-7)*(2*k-9)*(2*k-11)*(-13+2*k)*(k!)^2):
MATHEMATICA
CoefficientList[Series[(1-4*x)^(13/2), {x, 0, 50}], x] (* Amiram Eldar, Mar 25 2022 *)
PROG
(PARI) my(x = 'x + O('x^40)); Vec((1-4*x)^(13/2)) \\ Michel Marcus, Jul 02 2018
Expansion of (1-4*x)^(15/2).
+10
4
1, -30, 390, -2860, 12870, -36036, 60060, -51480, 12870, 2860, 1716, 1560, 1820, 2520, 3960, 6864, 12870, 25740, 54340, 120120, 276276, 657800, 1614600, 4071600, 10518300, 27768312, 74760840, 204900080, 570793080, 1613966640, 4626704368, 13432367520
FORMULA
D-finite with recurrence: n*a(n) +2*(-2*n+17)*a(n-1)=0. - R. J. Mathar, Jan 17 2020 a(n) = (-4)^n*binomial(15/2, n).
Sum_{n>=0} 1/a(n) = 972/1001 + 34*Pi/(3^10*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 18235778692/17595703125 - 68*log(phi)/(5^9*sqrt(5)), where phi is the golden ratio ( A001622). (End)
MATHEMATICA
CoefficientList[Series[(1-4x)^(15/2), {x, 0, 30}], x] (* Harvey P. Dale, Oct 03 2012 *)
Expansion of (1-4*x)^(17/2).
+10
4
1, -34, 510, -4420, 24310, -87516, 204204, -291720, 218790, -48620, -9724, -5304, -4420, -4760, -6120, -8976, -14586, -25740, -48620, -97240, -204204, -447304, -1016600, -2386800, -5768100, -14304888, -36312408, -94143280, -248807240, -669205680, -1829162192
FORMULA
D-finite with recurrence: n*a(n) +2*(-2*n+19)*a(n-1)=0. - R. J. Mathar, Jan 17 2020 a(n) = (-4)^n*binomial(17/2, n).
Sum_{n>=0} 1/a(n) = 49600/51051 - 38*Pi/(3^11*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 1542987607648/1495634765625 - 76*log(phi)/(5^10*sqrt(5)), where phi is the golden ratio ( A001622). (End)
MATHEMATICA
CoefficientList[Series[(1 - 4 x)^(17/2), {x, 0, 33}], x] (* Vincenzo Librandi, Jan 18 2020 *)
Expansion of (1-4*x)^(19/2).
+10
2
1, -38, 646, -6460, 41990, -184756, 554268, -1108536, 1385670, -923780, 184756, 33592, 16796, 12920, 12920, 15504, 21318, 32604, 54340, 97240, 184756, 369512, 772616, 1679600, 3779100, 8767512, 20907144, 51106352, 127765880, 326023280, 847660528, 2242198816
FORMULA
D-finite with recurrence: n*a(n) +2*(-2*n+21)*a(n-1)=0. - R. J. Mathar, Jan 17 2020 a(n) = (-4)^n*binomial(19/2, n).
Sum_{n>=0} 1/a(n) = 45052/46189 + 14*Pi/(3^11*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 6955761045148/6765966796875 - 84*log(phi)/(5^11*sqrt(5)), where phi is the golden ratio ( A001622). (End)
MATHEMATICA
CoefficientList[Series[(1-4x)^(19/2), {x, 0, 30}], x] (* Harvey P. Dale, Jul 03 2013 *)
Triangle T(n,k) = (2*k)!*(2*n)!/(k!*n!*(k+n)!) with k=0..n, read by rows.
+10
2
1, 2, 2, 6, 4, 6, 20, 10, 12, 20, 70, 28, 28, 40, 70, 252, 84, 72, 90, 140, 252, 924, 264, 198, 220, 308, 504, 924, 3432, 858, 572, 572, 728, 1092, 1848, 3432, 12870, 2860, 1716, 1560, 1820, 2520, 3960, 6864, 12870, 48620, 9724, 5304, 4420, 4760, 6120, 8976
COMMENTS
This is a companion to the triangle A068555.
REFERENCES
Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third edition), page 11.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 103.
EXAMPLE
Triangle begins:
1;
2, 2;
6, 4, 6;
20, 10, 12, 20;
70, 28, 28, 40, 70;
252, 84, 72, 90, 140, 252;
924, 264, 198, 220, 308, 504, 924;
3432, 858, 572, 572, 728, 1092, 1848, 3432;
12870, 2860, 1716, 1560, 1820, 2520, 3960, 6864, 12870;
48620, 9724, 5304, 4420, 4760, 6120, 8976, 14586, 25740, 48620;
...
Sum_{k=0..8} T(8,k) = 12870 + 2860 + 1716 + 1560 + 1820 + 2520 + 3960 + 6864 + 12870 = 2* A132310(7) + A000984(8) = 2*17085 + 12870 = 47040.
MATHEMATICA
Flatten[Table[Table[(2 k)! ((2 n)!/(k! n! (k + n)!)), {k, 0, n}], {n, 0, 9}]]
PROG
(Magma)
[Factorial(2*k)*Factorial(2*n)/(Factorial(k)*Factorial(n)*Factorial(k+n)): k in [0..n], n in [0..9]];
Expansion of (1-4*x)^(21/2).
+10
1
1, -42, 798, -9044, 67830, -352716, 1293292, -3325608, 5819814, -6466460, 3879876, -705432, -117572, -54264, -38760, -36176, -40698, -52668, -76076, -120120, -204204, -369512, -705432, -1410864, -2939300, -6348888, -14162904, -32522224, -76659528, -185040240
FORMULA
D-finite with recurrence: n*a(n) +2*(-2*n+23)*a(n-1)=0. - R. J. Mathar, Jan 17 2020 a(n) = (-4)^n*binomial(21/2, n).
Sum_{n>=0} 1/a(n) = 406240/415701 - 46*Pi/(3^13*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 728323714975904/710426513671875 - 92*log(phi)/(5^12*sqrt(5)), where phi is the golden ratio ( A001622). (End)
MATHEMATICA
CoefficientList[Series[Surd[(1-4x)^21, 2], {x, 0, 30}], x] (* Harvey P. Dale, Feb 25 2020 *)
CROSSREFS
Cf. A001622, A002420, A002421, A002422, A002423, A002424, A020923, A020925, A020927, A020929, A020931.
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