A005156 -id:A005156 - cp's OEIS frontend

A138176 Hankel transform of A138175.

1, -4, 144, -43264, 106832896, -2155963622400, 354617391605760000, -474686810341051524710400, 5166329306435680284163452174336, -456900249341808231214942590547243565056

Offset: 0

Paul Barry, Mar 03 2008

When aerated, gives Hankel transform of A138175.

A. V. Razumov and Yu. G. Stroganov, On refined enumerations of some symmetry classes of alternating sign matrices, arXiv:math-ph/0312071, 2003.

sqrt(abs(a(n)))/2^n = A005156(n).

a(n) = (-1)^n*(Product_{k=1..n} (6*k-2)!*(2*k-1)!/((4*k-1)!*(4*k-2)!))^2.

A173312 Partial sums of A005130.

Original entry on oeis.org

1, 2, 4, 11, 53, 482, 7918, 226266, 11076482, 922911942, 130457184642, 31226202037017, 12642538061714517, 8652026056359367017, 10004193381504526849017, 19539080428042781631746217

Offset: 0

Jonathan Vos Post, Feb 16 2010

nonn

Partial sums of Robbins numbers. Partial sums of the number of descending plane partitions whose parts do not exceed n. Partial sums of the number of n X n alternating sign matrices (ASM's). After 2, 11, 53, when is this partial sum again prime, as it is not again prime through a(32)?

			a(17) = 1 + 1 + 2 + 7 + 42 + 429 + 7436 + 218348 + 10850216 + 911835460 + 129534272700 + 31095744852375 + 12611311859677500 + 8639383518297652500 + 9995541355448167482000 + 19529076234661277104897200 + 64427185703425689356896743840 + 358869201916137601447486156417296.

Cf. A005130, A006366, A048601, also A003827, A005156, A005158, A005160-A005164, A050204, A049503, A160707, A160708.

Table[Sum[Product[(3 k + 1)!/(j + k)!, {k, 0, j - 1}], {j, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 26 2017 *)
Accumulate[Table[Product[(3k+1)!/(n+k)!,{k,0,n-1}],{n,0,20}]] (* Harvey P. Dale, Feb 06 2019 *)

a(n) = Sum_{i=0..n} A005130(i) = Sum_{i=0..n} Product_{k=0..i-1} (3k+1)!/(i+k)!. [corrected by Vaclav Kotesovec, Oct 26 2017]

a(n) ~ Pi^(1/3) * exp(1/36) * 3^(3*n^2/2 - 7/36) / (A^(1/3) * Gamma(1/3)^(2/3) * n^(5/36) * 2^(2*n^2 - 5/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 26 2017

A156106 Expansion of F(1/3,2/3;1/2;27x/2) / F(1/3,-1/3;-1/2;27x/2).

Original entry on oeis.org

1, 3, 15, 99, 783, 6987, 67671, 694035, 7418943, 81800091, 923720679, 10630297827, 124224709455, 1470172954347, 17585028636279, 212248303720371, 2581823992868703

Offset: 0

Paul Barry, Feb 04 2009

Hankel transform is 3^n*2^(n^2)*A005156 = 6^n*4^C(n,2)*A005156 = 3^n*A002416*A005156.

I. Gessel and G. Xin, The Generating Function of Ternary Trees and Continued Fractions, arXiv:math/0505217 [math.CO], 2005.

D-finite with recurrence: 2*(2*n-1)*(n-2)*a(n) + (-72*(n-3)^2-171*n+420)*a(n-1) + (297*(n-3)^2+675*n-1674)*a(n-2) - 81*(3*n-5)*(3*n-7)*a(n-3) = 0. - Georg Fischer, Nov 30 2022

cp's OEIS Frontend

A138176 Hankel transform of A138175.

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A173312 Partial sums of A005130.

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A156106 Expansion of F(1/3,2/3;1/2;27x/2) / F(1/3,-1/3;-1/2;27x/2).

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cp's OEIS Frontend

A138176 Hankel transform of A138175.

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Author

Keywords

Comments

Links

Formula

A173312 Partial sums of A005130.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A156106 Expansion of F(1/3,2/3;1/2;27*x/2) / F(1/3,-1/3;-1/2;27*x/2).

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Author

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Formula

A156106 Expansion of F(1/3,2/3;1/2;27x/2) / F(1/3,-1/3;-1/2;27x/2).