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

1,2

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.

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)

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}]

Prog

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

Crossrefs

Cf. A000290, A007318, A174158, A174696.

Sequence in context: A207397 A365153 A346424 * A166992 A058789 A212028

Adjacent sequences: A319740 A319741 A319742 * A319744 A319745 A319746

Keyword

nonn

Author

Stefano Spezia, Dec 23 2018

Status

approved