A132745 Row sums of (A008550 formatted as a triangular array).

1, 2, 3, 5, 11, 32, 114, 467, 2130, 10642, 57629, 335381, 2082582, 13716502, 95352529, 696790819, 5334094259, 42649956716, 355261078352, 3075741331481, 27620835538407, 256810928552476, 2468108094076860, 24481671811988907, 250296546308500181, 2634309876797453868, 28509045368598994348

Offset: 0

Author

Philippe Deléham, Nov 21 2007

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A001263, A008550, A243631.

Programs

Magma

A243631:= func< n,k | n eq 0 select 1 else (&+[ Binomial(n,j)^2*k^j*(n-j)/(n*(j+1)): j in [0..n-1]]) >;
A132745:= func< n | (&+[A243631(k,n-k): k in [0..n]]) >;
[A132745(n): n in [0..30]]; // G. C. Greubel, Feb 16 2021

Mathematica

Table[Sum[Hypergeometric2F1[1-k, -k, 2, n-k], {k,0,n}], {n,0,30}] (* G. C. Greubel, Feb 16 2021 *)

Sage

def A243631(n,k): return 1 if n==0 else sum( binomial(n,j)^2*k^j*(n-j)/(n*(j+1)) for j in [0..n-1])
def A132745(n): return sum( A243631(k, n-k) for k in [0..n] )
[A132745(n) for n in [0..30]] # G. C. Greubel, Feb 16 2021

Formula

From G. C. Greubel, Feb 16 2021: (Start)

a(n) = Sum_{k=0..n} Hypergeometric2F1([1-n+k, k-n], [2], k).

a(n) = Sum_{k=0..n} Hypergeometric2F1([1-k, -k], [2], n-k).

a(n) = 1 + Sum_{k=1..n} Sum_{j=0..k-1} binomial(k,j)^2 * ((k-j)*(n-k)^j/(k*(j+1))).

a(n) = 1 + Sum_{k=1..n} Sum_{j=0..k-1} A001263(k, k-j) * (n-k)^j. (End)

Extensions

Terms a(11) onward added by G. C. Greubel, Feb 16 2021