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

%I #29 Jun 22 2022 11:18:08

%S 0,1,-1,-1,8,-14,-46,314,48,-7244,13484,233804,-1002112,-10692824,

%T 79853864,661278344,-7758425472,-52788251504,928828095344,

%U 5226295555184,-135732896444032,-617858881186784,23892314499643424,83518025870961824

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

%C 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]

%H Vincenzo Librandi, <a href="/A009453/b009453.txt">Table of n, a(n) for n = 0..200</a>

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

%F 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

%t 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 *)

%t With[{nn = 30}, CoefficientList[Series[Sin[Log[1 + Tanh[x]]], {x, 0, nn}], x] Range[0, nn]!] (* _Vincenzo Librandi_, Feb 09 2014 *)

%o (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 */

%Y Cf. A009453, A102573.

%K sign,easy

%O 0,5

%A _R. H. Hardin_

%E Extended with signs by _Olivier GĂ©rard_, Mar 15 1997