A124527 Row sums of triangle A124526.

1, 1, 2, 5, 14, 46, 162, 641, 2656, 12092, 56956, 290636, 1523088, 8559980, 49163792, 300514337, 1870652672, 12318376190, 82394305842, 580168452664, 4141242464512, 30992978322024, 234765130286990, 1858132080028884

Offset: 0

Author

Paul D. Hanna, Nov 08 2006

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A124526; A124528, A124529.

Programs

Maple

b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, b(n-1, k-1) +(k+1)*(b(n-1, k) +b(n-1, k+1))))
    end:
a:= n-> add(b(iquo(n, 2), k)*b(iquo(n+1, 2), k), k=0..n/2):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 14 2014

Mathematica

b[n_, k_] := b[n, k] = If[k < 0 || k > n, 0, If[n == 0, 1, b[n - 1, k - 1] + (k + 1) (b[n - 1, k] + b[n - 1, k + 1])]];
a[n_] := Sum[b[Quotient[n, 2], k] b[Quotient[n + 1, 2], k], {k, 0, n/2}];
a /@ Range[0, 30]
(* Second program: *)
S[n_, k_] = Sum[StirlingS2[n, i] Binomial[i, k], {i, 0, n}];
T[n_, k_] := S[Floor[n/2], k] S[Floor[(n + 1)/2], k];
a[n_] := Sum[T[n, k], {k, 0, Floor[n/2]}];
a /@ Range[0, 30] (* Jean-François Alcover, Nov 02 2020, first program after Alois P. Heinz *)

PARI

{a(n)=sum(k=0,n\2,(n\2)!*((n+1)\2)!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),n\2),k) *polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),(n+1)\2),k))}