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A124527 Row sums of triangle A124526.

%I A124527 #9 Nov 02 2020 09:09:38
%S A124527 1,1,2,5,14,46,162,641,2656,12092,56956,290636,1523088,8559980,
%T A124527 49163792,300514337,1870652672,12318376190,82394305842,580168452664,
%U A124527 4141242464512,30992978322024,234765130286990,1858132080028884
%N A124527 Row sums of triangle A124526.
%H A124527 Alois P. Heinz, <a href="/A124527/b124527.txt">Table of n, a(n) for n = 0..500</a>
%p A124527 b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
%p A124527       `if`(n=0, 1, b(n-1, k-1) +(k+1)*(b(n-1, k) +b(n-1, k+1))))
%p A124527     end:
%p A124527 a:= n-> add(b(iquo(n, 2), k)*b(iquo(n+1, 2), k), k=0..n/2):
%p A124527 seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 14 2014
%t A124527 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])]];
%t A124527 a[n_] := Sum[b[Quotient[n, 2], k] b[Quotient[n + 1, 2], k], {k, 0, n/2}];
%t A124527 a /@ Range[0, 30]
%t A124527 (* Second program: *)
%t A124527 S[n_, k_] = Sum[StirlingS2[n, i] Binomial[i, k], {i, 0, n}];
%t A124527 T[n_, k_] := S[Floor[n/2], k] S[Floor[(n + 1)/2], k];
%t A124527 a[n_] := Sum[T[n, k], {k, 0, Floor[n/2]}];
%t A124527 a /@ Range[0, 30] (* _Jean-François Alcover_, Nov 02 2020, first program after _Alois P. Heinz_ *)
%o A124527 (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))}
%Y A124527 Cf. A124526; A124528, A124529.
%K A124527 nonn
%O A124527 0,3
%A A124527 _Paul D. Hanna_, Nov 08 2006