A012123 E.g.f.: exp(arcsin(tanh(x))).

1, 1, 1, 0, -3, -4, 21, 80, -263, -2224, 4841, 88960, -99723, -4942144, -199939, 366928640, 501445617, -35219691264, -101818966319, 4251941253120, 19731909099757, -631113275843584, -4192563651606299, 113005305852006400, 1009030667701246697

Offset

0,5

Links

Table of n, a(n) for n=0..24.

Formula

a(n) = (-i)^n * Z(n,i), where i = sqrt(-1) and Z(n,x) denotes the n-th zigzag polynomial as described in A147309. Alternative form of the egf: {sec(i*x) - tan(i*x)}^i. - Peter Bala, Jan 26 2011

a(n)=sum(m=1..n, sum(r=m..n, (sum(k=r..n, (-1)^((3*k)/2)*(sum(i=0..k, (2^i*stirling1(m+i,m)* binomial(m+k-1,m+i-1))/(m+i)!))*binomial((r-2)/2,(r-m-k)/2)))*((-1)^(r-m)+1)*sum(k=0..r-m, binomial(k-1,r-1)*k!*2^(n-k)*stirling2(n,k)*(-1)^(r+k))))/2, n>0, a(0)=1. - Vladimir Kruchinin, Jun 09 2011

a(n) = sum(i=0..n-1, binomial(n-1,i)*euler(i)*a(n-i-1)), a(0)=1. - Vladimir Kruchinin, Feb 26 2015

Example

exp(arcsin(tanh(x))) = 1 + x + 1/2!*x^2 - 3/4!*x^4 - 4/5!*x^5 + 21/6!*x^6 ...

Mathematica

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, i]*EulerE[i]*a[n-i-1], {i, 0, n-1}]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, May 22 2017, after Vladimir Kruchinin *)

Prog

(Maxima)
a(n):=sum(sum((sum((-1)^((3*k)/2)*(sum((2^i*stirling1(m+i, m)* binomial(m+k-1, m+i-1))/(m+i)!, i, 0, k))*binomial((r-2)/2, (r-m-k)/2), k, 0, r-m))*((-1)^(r-m)+1)*sum(binomial(k-1, r-1)*k!*2^(n-k)*stirling2(n, k)*(-1)^(r+k), k, r, n), r, m, n), m, 1, n)/2; /* Vladimir Kruchinin, Jun 09 2011 */
(Maxima)
a(n):=if n=0 then 1 else sum(binomial(n-1, i)*euler(i)*a(n-i-1), i, 0, n-1); /* Vladimir Kruchinin, Feb 26 2015 */
(PARI) x='x+O('x^66); Vec(serlaplace(exp(asin(tanh(x))))) \\ Joerg Arndt, Feb 26 2015

Crossrefs

Sequence in context: A094632 A081698 A182096 * A012255 A012247 A057791

Adjacent sequences: A012120 A012121 A012122 * A012124 A012125 A012126

Keyword

sign

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Status

approved