A009257 Expansion of e.g.f. exp(tanh(log(1+x))).

1, 1, 0, -2, 4, 6, -74, 190, 1128, -14336, 51616, 404856, -7330520, 41023048, 250083744, -7532037344, 65189204416, 185746926720, -13324842809792, 175157684686144, -215786634724224, -35458746254274944, 711274631768613760

Offset

0,4

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = sum(m=1..n, sum(r=0..n-m, Stirling1(n,r+m)*sum(k=0..r, binomial(k+m-1,m-1)*(k+m)!*(-1)^(k)*2^(r-k)*Stirling2(r+m,k+m)))/m!), n>0, a(0)=1. [Vladimir Kruchinin, Jun 06 2011]

For n>3, a(n) = (3-2*n)*a(n-1)+(n-1)*(5-2*n)*a(n-2)+(1-n)*(n-2)*(n-3)*a(n-3)+(1-n)*(n-2)*(n-3)*(n-4)*a(n-4)/4. - Tani Akinari, Feb 21 2024

a(n) = Sum_{k=0..n} Sum_{j=0..n-k} binomial(-k,j)*binomial(k+j,n-k-j)*(n!/k!)*2^(k+j-n). - Tani Akinari, Feb 21 2024

Mathematica

With[{nn=30}, CoefficientList[Series[Exp[Tanh[Log[1+x]]], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jan 15 2015 *)

Prog

(Maxima)
a(n):=sum(sum(stirling1(n, r+m)*sum(binomial(k+m-1, m-1)*(k+m)!*(-1)^(k)*2^(r-k)*stirling2(r+m, k+m), k, 0, r), r, 0, n-m)/m!, m, 1, n); /* Vladimir Kruchinin, Jun 06 2011 */
(Maxima) a[n]:=if n<4 then (n+1)*(2-n)/2 else (3-2*n)*a[n-1]+(n-1)*(5-2*n)*a[n-2]+(1-n)*(n-2)*(n-3)*a[n-3]+(1-n)*(n-2)*(n-3)*(n-4)*a[n-4]/4;
makelist(a[n], n, 0, 50); /* Tani Akinari, Feb 21 2024 */
(Maxima) a(n):=sum(sum(binomial(-k, j)*binomial(k+j, n-k-j)*(n!/k!)*2^(k+j-n), j, 0, n-k), k, 0, n); /* Tani Akinari, Feb 21 2024 */

Crossrefs

Sequence in context: A066220 A348152 A367854 * A098757 A335709 A056012

Adjacent sequences: A009254 A009255 A009256 * A009258 A009259 A009260

Keyword

sign,easy

Author

R. H. Hardin

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Prior Mathematica program replaced by Harvey P. Dale, Jan 15 2015

Status

approved