A319743 Row sums of A174158.

1, 2, 11, 74, 602, 5452, 53559, 558602, 6106034, 69298580, 811086718, 9740402476, 119550632788, 1495039156600, 19002275811887, 244983878813514, 3198363309664658, 42225545561470084, 563083734161627910, 7576864105285884420, 102790882283750139060, 1404908982711268821720

Offset: 1

Author

Stefano Spezia, Dec 23 2018

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..830
• Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Enumeration of parallelogram polycubes, arXiv:2105.00971 [cs.DM], 2021.

Crossrefs

Cf. A000290, A007318, A174158, A174696.

Programs

GAP

List([1..20], n->Sum([1..n], m->(Binomial(n-1, m-1)*Binomial(n, m-1)/m)^2));

Magma

[(&+[(Binomial(n-1,j-1)*Binomial(n,j-1)/j)^2 : j in [1..n]]): n in [1..25]]; // G. C. Greubel, Feb 15 2021

Maple

a := n -> add(binomial(n-1, m-1)^2*binomial(n, m-1)^2/m^2, m = 1 .. n): seq(a(n), n = 1 .. 20)

Mathematica

Table[HypergeometricPFQ[{1-n,1-n,-n,-n},{1,2,2},1],{n,1,20}]

PARI

a(n) = sum(m=1, n, (binomial(n-1, m-1)*binomial(n, m-1)/m)^2);

Sage

[hypergeometric([-n, -n, -n+1, -n+1], [1, 2, 2], 1).simplify_hypergeometric() for n in (1..25)] # G. C. Greubel, Feb 15 2021

Formula

a(n) = Sum_{m=1..n} (binomial(n - 1, m - 1)*binomial(n, m - 1)/m)^2.

a(n) = Sum_{m=1..n} A000290(A007318(n - 1, m - 1)*A007318(n, m - 1)/m).

a(n) = 4F3([1 - n, 1 - n, - n, - n], [1, 2, 2], 1), where F is the generalized hypergeometric function.

From Vaclav Kotesovec, Dec 24 2018: (Start)

Recurrence: n*(n+1)^3*(5*n^2 - 10*n + 4)*a(n) = 2*n*(2*n - 1)*(15*n^4 - 30*n^3 + 7*n^2 + 8*n - 8)*a(n-1) + 4*(n-2)^2*(4*n - 5)*(4*n - 3)*(5*n^2 - 1)*a(n-2).

a(n) ~ 2^(4*n + 1/2) / (Pi^(3/2) * n^(7/2)).

(End)