A009453 Expansion of e.g.f. sin(log(1 + tanh(x))).

0, 1, -1, -1, 8, -14, -46, 314, 48, -7244, 13484, 233804, -1002112, -10692824, 79853864, 661278344, -7758425472, -52788251504, 928828095344, 5226295555184, -135732896444032, -617858881186784, 23892314499643424, 83518025870961824

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Comments

Related to A102573: letting T(q,r) be the coefficient of n^(r+1) in the polynomial 2^(q-n)/n times Sum_{k=0..n} binomial(n,k)*k^q, then A009453(x) equals (-1)^(x+1) times the real part of Sum_{k=0..x-1} T(x,k)*i^k, where i is the imaginary unit. See Mathematica code below. [John M. Campbell, Nov 17 2011]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A009453, A102573.

Programs

Mathematica

poly[q_] := 2^(q - n)/n*FunctionExpand[Sum[Binomial[n, k]*k^q, {k, 0, n}]]; T[q_, r_] := First[Take[CoefficientList[poly[q], n], {r + 1, r + 1}]]; Table[Re[Sum[T[x, k]*I^k, {k, 0, x - 1}]]*(-1)^(x + 1), {x, 0, 23}] (* John M. Campbell, Nov 17 2011 *)
With[{nn = 30}, CoefficientList[Series[Sin[Log[1 + Tanh[x]]], {x, 0, nn}], x] Range[0, nn]!] (* Vincenzo Librandi, Feb 09 2014 *)

Maxima

a(n):=sum(sum((stirling1(r,2*m+1)*sum(binomial(k-1,r-1)*k!*2^(n-k)*stirling2(n,k)*(-1)^(r+k+m),k,r,n))/r!,r,2*m+1,n),m,0,(n-1)/2); /* Vladimir Kruchinin, Jun 21 2011 */

Formula

E.g.f. sin(log(1 + tanh(x))).

a(n) = Sum_{m=0..(n-1)/2} (Sum_{r=2*m+1..n} (stirling1(r,2*m+1) * Sum_{k=r..n} binomial(k-1,r-1)*k!*2^(n-k)*stirling2(n,k)*(-1)^(r+k+m)/r!)). - Vladimir Kruchinin, Jun 21 2011

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997