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Search: a214775 -id:a214775
Displaying 1-7 of 7 results found. page 1
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A215002 Number of all solid standard Young tableaux of shape [[n,k],[n-k]] for 0<=k<=n. +10
7
1, 2, 10, 60, 398, 2764, 19796, 144536, 1070294, 8007052, 60380940, 458185992, 3494554380, 26764583096, 205711091880, 1585822364592, 12256625999718, 94942581080204, 736895626109148, 5729374337686696, 44615143884080996, 347905737091032552, 2716349710039969688 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau
FORMULA
See Maple programs.
a(n) ~ sqrt((5*sqrt(5))/2-11/2) * 8^n / sqrt(Pi*n). - Vaclav Kotesovec, Jul 16 2014
MAPLE
b:= proc(x, y, z) option remember; `if`(z>y, b(x, z, y),
`if`({x, y, z}={0}, 1, `if`(x>y and x>z, b(x-1, y, z), 0)+
`if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0)))
end:
a:= n-> add(b(n, k, n-k), k=0..n):
seq(a(n), n=0..25);
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1, 2, 10, 60][n+1],
((1640*n^8 -1180*n^7 -7114*n^6 +5615*n^5 +20240*n^4 -35170*n^3
+20379*n^2 -4050*n) *a(n-1) +(-7640*n^8 +14560*n^7 +47374*n^6
-140900*n^5 -37160*n^4 +601810*n^3 -944154*n^2 +580680*n -113400)
*a(n-2) +(-28800*n^8 +181440*n^7 -138240*n^6 -874800*n^5 +670680*n^4
+3165480*n^3 -3646440*n^2 -12960*n -453600) *a(n-3) +(207360*n^8
-1451520*n^7 +912384*n^6 +11767680*n^5 -15720480*n^4 -42042240*n^3
+92516256*n^2 -50388480*n +16329600) *a(n-4)) / (n* (n+1) *(2*n-1)
*(2*n+1) *(20*n^4-47*n^2-33*n+90)))
end:
seq(a(n), n=0..25);
MATHEMATICA
b[x_, y_, z_] := b[x, y, z] = If[z>y, b[x, z, y], If[Union[{x, y, z}] == {0}, 1, If[x>y && x>z, b[x-1, y, z], 0] + If[y>0, b[x, y-1, z], 0] + If[z>0, b[x, y, z-1], 0]]]; T[n_, k_] := b[n, k, n-k]; a[n_] := Sum[T[n, k], {k, 0, n}]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)
CROSSREFS
Row sums of A214775.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 30 2012
STATUS
approved
A214801 Number of solid standard Young tableaux of shape [[2*n,n],[n]]. +10
6
1, 6, 174, 7020, 325590, 16290708, 854630476, 46305862488, 2568272967270, 144984584562180, 8298621602461476, 480298712286979560, 28052543639835133692, 1650956086756046986440, 97790578929910135588440, 5824509559447044190027952, 348581174512709008160833158 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..555
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau
FORMULA
Recurrence: (n-1)*n^2*(2*n-1)*(2*n+1)*(4*n-1)*(4*n+1)*(392*n^4 - 2044*n^3 + 4216*n^2 - 3944*n + 1377)*a(n) = 2*(n-1)*(1859648*n^10 - 13670048*n^9 + 43255264*n^8 - 75152192*n^7 + 75863336*n^6 - 41825576*n^5 + 7317576*n^4 + 5067372*n^3 - 3441344*n^2 + 785094*n - 59535)*a(n-1) - 4*(2*n-3)*(4*n-7)*(4*n-5)*(1310848*n^8 - 7998592*n^7 + 19695952*n^6 - 24269488*n^5 + 15125236*n^4 - 3514192*n^3 - 1066614*n^2 + 533457*n - 45927)*a(n-2) + 5184*n*(2*n-5)*(2*n-3)*(4*n-11)*(4*n-9)*(4*n-7)*(4*n-5)*(392*n^4 - 476*n^3 + 436*n^2 - 76*n - 3)*a(n-3). - Vaclav Kotesovec, Aug 31 2014
a(n) ~ sqrt((5*sqrt(5)-11)/4) * 64^n / (Pi*n). - Vaclav Kotesovec, Aug 31 2014
MAPLE
b:= proc(x, y, z) option remember; `if`(z>y, b(x, z, y),
`if`({x, y, z}={0}, 1, `if`(x>y and x>z, b(x-1, y, z), 0)+
`if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0)))
end:
a:= n-> b(2*n, n, n):
seq(a(n), n=0..20);
MATHEMATICA
b[x_, y_, z_] := b[x, y, z] = If[z>y, b[x, z, y], If[Union[{x, y, z}] == {0}, 1, If[x>y && x>z, b[x-1, y, z], 0] + If[y>0, b[x, y-1, z], 0] + If[z>0, b[x, y, z-1], 0]]]; a[n_] := b[2n, n, n]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)
CROSSREFS
Central row elements of A214775.
Column k=2 of A176129.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 28 2012
STATUS
approved
A214955 Number of solid standard Young tableaux of shape [[n,n-1],[1]]. +10
2
1, 6, 25, 98, 378, 1452, 5577, 21450, 82654, 319124, 1234506, 4784276, 18572500, 72209880, 281150505, 1096087770, 4278278070, 16717354500, 65388738030, 256000696380, 1003116947820, 3933750236520, 15437682614250, 60625494924228, 238235373671148, 936735006679752 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) is odd if and only if n = 2^i-1 for i in {1, 2, 3, ...} = A000027.
Form an array with m(1,n) = n*(n+1)/2, m(n,1) = n*(n-1)+1, and m(i,j) = m(i,j-1) + m(i-1,j); A000217 in the top row, A002061 in the first column, A086514 in the second column. Then on the diagonal m(n,n) = a(n). - J. M. Bergot, May 02 2013
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
S. B. Ekhad and D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229 [math.CO], 2012.
Wikipedia, Young tableau.
FORMULA
a(n) = 2*(2*n-1)^2/((n+1)*(2*n-3)) * a(n-1) for n>1; a(1) = 1.
a(n) = (2*n-1) * C(2*n,n)/(n+1) = A060747(n) * A000108(n).
a(n) = [x^n] x*(1 + 2*x)/(1 - x)^(n+2). - Ilya Gutkovskiy, Oct 12 2017
Sum_{n>=1} 1/a(n) = 1/6 + G + 13*Pi/(36*sqrt(3)) - Pi*log(2+sqrt(3))/8, where G is Catalan's constant (A006752). - Amiram Eldar, Mar 06 2022
From Stefano Spezia, Mar 29 2023: (Start)
O.g.f.: 1 + (3 - 3*sqrt(1 - 4*x) - 8*x)/(2*x*sqrt(1 - 4*x)).
E.g.f.: 1 + exp(2*x)*(3*I_1(2*x) - I_0(2*x)), where I_n(x) is the modified Bessel function of the first kind.
a(n) ~ 2^(1+2*n)/sqrt(n*Pi). (End)
MAPLE
a:= proc(n) option remember;
`if`(n<2, n, 2*(2*n-1)^2*a(n-1)/((n+1)*(2*n-3)))
end:
seq(a(n), n=1..30);
MATHEMATICA
a[n_]:= a[n] = If[n<2, n, 2*(2*n-1)^2*a[n-1]/((n+1)*(2*n-3))]; Array[a, 30] (* Jean-François Alcover, Aug 14 2017, translated from Maple *)
PROG
(PARI) a(n) = (2*n-1) * binomial(2*n, n)/(n+1); \\ Michel Marcus, Mar 06 2022
CROSSREFS
Column k=1 of A214775.
Cf. A000027, A000108, A006752, A060747, A215002.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 30 2012
STATUS
approved
A215298 Number of solid standard Young tableaux of shape [[n,n-2],[2]]. +10
2
2, 25, 174, 962, 4804, 22689, 103510, 461318, 2021916, 8752042, 37520972, 159633060, 674969224, 2839400945, 11893509990, 49637986590, 206519808300, 856904298030, 3547095101220, 14652264350940, 60412895258040, 248675669866650, 1022088942267900, 4195255959533052 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
LINKS
Alois P. Heinz, Table of n, a(n) for n = 2..500
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau
FORMULA
See Maple program.
a(n) ~ 3 * 2^(2*n-1) * sqrt(n) / sqrt(Pi). - Vaclav Kotesovec, Sep 02 2014
a(n) = 12*(1 - 4*n + 10*n^2 - 8*n^3 + 2*n^4) * (2*n-4)! / ((n-2)! * (n+1)!). - Vaclav Kotesovec, Sep 02 2014
MAPLE
a:= proc(n) option remember; `if`(n=2, 2,
2*(4*n^5 -26*n^4 +60*n^3 -58*n^2 +22*n -5)*a(n-1)/
(2*n^5 -14*n^4 +30*n^3 -10*n^2 -31*n +25))
end:
seq(a(n), n=2..30);
MATHEMATICA
Table[12*(1 - 4*n + 10*n^2 - 8*n^3 + 2*n^4) * (2*n-4)! / ((n-2)! * (n+1)!), {n, 2, 20}] (* Vaclav Kotesovec, Sep 02 2014 *)
CROSSREFS
Column k=2 of A214775.
Cf. A215002.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 07 2012
STATUS
approved
A215299 Number of solid standard Young tableaux of shape [[n,n-3],[3]]. +10
2
5, 98, 962, 7020, 43573, 245962, 1305238, 6633172, 32649890, 156817044, 738717796, 3425580376, 15679951989, 70992594650, 318450985230, 1417072222020, 6261985407990, 27502477286460, 120137081521500, 522256720264680, 2260525598620770, 9746264904755652 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,1
LINKS
Alois P. Heinz, Table of n, a(n) for n = 3..500
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau
FORMULA
See Maple program.
For n > 3, a(n) = (8*(45 - 180*n + 580*n^2 - 756*n^3 + 484*n^4 - 144*n^5 + 16*n^6) * (2*n-6)!) / (3 * (n-3)! * (n+1)!). - Vaclav Kotesovec, Sep 02 2014
MAPLE
a:= proc(n) option remember; `if`(n<5, [0$2, 5, 98][n],
2*(32*n^7 -400*n^6 +1976*n^5 -4900*n^4 +6452*n^3 -4420*n^2
+1350*n-315)*a(n-1) / (16*n^7 -224*n^6 +1204*n^5 -3008*n^4
+2980*n^3 +1072*n^2 -4155*n +2205))
end:
seq(a(n), n=3..30);
MATHEMATICA
Flatten[{5, Table[(8*(45 - 180*n + 580*n^2 - 756*n^3 + 484*n^4 - 144*n^5 + 16*n^6) * (2*n-6)!) / (3*(n-3)!*(n+1)!), {n, 4, 20}]}] (* Vaclav Kotesovec, Sep 02 2014 *)
CROSSREFS
Column k=3 of A214775.
Cf. A215002.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 07 2012
STATUS
approved
A215300 Number of solid standard Young tableaux of shape [[n,n-4],[4]]. +10
2
14, 378, 4804, 43573, 325590, 2149454, 13054108, 74688594, 408634828, 2159302420, 11097147528, 55747502501, 274790652518, 1332928973766, 6377276361900, 30149660760870, 141057202034340, 653892592144620, 3006490865152440, 13722387184879650, 62220533305358076 (list; graph; refs; listen; history; text; internal format)
OFFSET
4,1
LINKS
Alois P. Heinz, Table of n, a(n) for n = 4..500
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau
FORMULA
See Maple program.
For n > 4, a(n) = (2*(n-5))!/(3*(n-5)!*(n+1)!)*(160*(-567 + 2394*n - 8862*n^2 + 15592*n^3 - 15484*n^4 + 9152*n^5 - 3292*n^6 + 704*n^7 - 82*n^8 + 4*n^9)). - Vaclav Kotesovec, Sep 02 2014
MAPLE
a:= proc(n) option remember; `if`(n<6, [0$3, 14, 378, 4804][n],
((-460961029024*n^4 +54902186125572*n^3 -347074341314956*n^2
+421934757637074*n +6838164520124) *a(n-1) +(104238656896016*n^4
-2317913124589048*n^3 +16535317231755832*n^2 -44274446438628908*n
+29901662719961532)*a(n-2) +(-391233321452352*n^4
+7447800734464704*n^3 -48294258553516272*n^2 +122447135865649584*n
-105955729051546080)*a(n-3)) / (286655151052*n^4 -1210962058579*n^3
+4322649356693*n^2 -24951473774234*n -30771740340558))
end:
seq(a(n), n=4..30);
MATHEMATICA
Flatten[{14, Table[(2*(n-5))!/(3*(n-5)!*(n+1)!)*(160*(-567+2394*n-8862*n^2+15592*n^3-15484*n^4+9152*n^5-3292*n^6+704*n^7-82*n^8+4*n^9)), {n, 5, 20}]}] (* Vaclav Kotesovec, Sep 02 2014 *)
CROSSREFS
Column k=4 of A214775.
Cf. A215002.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 07 2012
STATUS
approved
A215301 Number of solid standard Young tableaux of shape [[n,n-5],[5]]. +10
2
42, 1452, 22689, 245962, 2149454, 16290708, 111709178, 711996820, 4292788212, 24777783256, 138077129921, 747501664986, 3949741123174, 20444004524804, 103955714523390, 520494659493180, 2570907398453580, 12546842041060200, 60579487688891610, 289692893191143876 (list; graph; refs; listen; history; text; internal format)
OFFSET
5,1
LINKS
Alois P. Heinz, Table of n, a(n) for n = 5..500
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau
FORMULA
See Maple program.
For n > 6, a(n) = (2*(n-6))! / (5 * (n-6)! * (n+1)!) * 64 * (-51975 + 217350*n - 873908*n^2 + 1738396*n^3 - 2038350*n^4 + 1500940*n^5 - 724004*n^6 + 231788*n^7 - 48750*n^8 + 6460*n^9 - 488*n^10 + 16*n^11). - Vaclav Kotesovec, Sep 02 2014
MAPLE
a:=proc(n) option remember; `if`(n<7, [0$4, 42, 1452, 22689][n],
((-940984202308081409937789248*n^7+36378423601372783158274124928*n^6
-540987251973268278464961515672*n^5+4140452478540141056223108638628*n^4
-17643038551017281385645661643624*n^3+40489345935054116443261823323140*n^2
-39934974057986427003556989745680*n-247683783218781902433156798480)*a(n-1)
+(5038765510419498883689330496*n^7-154613008671019208064714735488*n^6
+1939670093038831522623368803072*n^5-12888788321486668402366527701360*n^4
+48941495657518683977159471709724*n^3-105016281014420890409086708155812*n^2
+113403222542936117699329884355248*n-47046838608769352958257951122560)
*a(n-2))/(79676793824198327746135844*n^7-1949805875384464242394656236*n^6
+20900166698905174940775960603*n^5-125515785015357799830976856812*n^4
+431332553464051479008795376439*n^3-723271251684163430971195319466*n^2
+59211568171613916060478086240*n+1362260807703300463382362391640))
end:
seq(a(n), n=5..30);
MATHEMATICA
Flatten[{42, 1452, Table[(2*(n-6))! / (5 * (n-6)! * (n+1)!) * 64 * (-51975 + 217350*n - 873908*n^2 + 1738396*n^3 - 2038350*n^4 + 1500940*n^5 - 724004*n^6 + 231788*n^7 - 48750*n^8 + 6460*n^9 - 488*n^10 + 16*n^11), {n, 7, 20}]}]
CROSSREFS
Column k=5 of A214775.
Cf. A215002.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 07 2012
STATUS
approved
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Last modified August 18 23:05 EDT 2024. Contains 375284 sequences. (Running on oeis4.)