A201954 E.g.f. satisfies: A(sinh(x)) = x*cosh(A(x)).

1, 0, 2, 0, 24, 0, 552, 0, 15936, 0, 427008, 0, 35567232, 0, 7352654592, 0, -1064016089088, 0, -285531029213184, 0, 515396596477280256, 0, -177833505179096580096, 0, -286336846462160593059840, 0, 573948928392687509387673600, 0, -211350462543692217044652785664

Offset

1,3

Links

Table of n, a(n) for n=1..29.

Formula

a(n)=T(n,1), T(n,m)=sum(k=1..n-m, T(n-m,k)*sum(i=0..m, (m-2*i)^k*binomial(m,i))/(k!*2^m)-(2^(-m-k+1)*sum(i=0..m+k-1, (-1)^i*(m+k-2*i-1)^n*binomial(m+k-1,i)))/n!*T(k+m-1,m))), n>m, with T(n,n)=1.

E.g.f. satisfies: A(x) = log(sqrt(1+x^2) + x) * cosh( A( log(sqrt(1+x^2) + x) ) ). - Paul D. Hanna, Dec 06 2011

Example

A(x) = x +x^3/3 +x^5/5 +23*x^7/210 + 83*x^9/1890 +...

Prog

(Maxima)
array(B, 100, 100);
fillarray (B, makelist (-1, i, 1, 10000));
T(n, m):=if B[n, m]=-1 then B[n, m]:(if n=m then 1 else sum(T(n-m, k)*sum((m-2*i)^k*binomial(m, i), i, 0, m)/(k!*2^m)-(2^(-m-k+1)*sum((-1)^i*(m+k-2*i-1)^n*binomial(m+k-1, i), i, 0, m+k-1))/n!*T(k+m-1, m), k, 1, n-m)) else B[n, m];
makelist(n!*T(n, 1), n, 1, 30);
(PARI) /* Using A(x) = arcsinh(x)*cosh(A(arcsinh(x))), Paul D. Hanna, Dec 06 2011 */
{a(n)=local(A=1+x, X=x+x*O(x^n)); for(i=1, n+1, A=log(sqrt(1+X^2)+x)*cosh(subst(A, x, log(sqrt(1+X^2)+x)))); n!*polcoeff(A, n)}

Crossrefs

Sequence in context: A138551 A133490 A051728 * A005359 A008842 A347898

Adjacent sequences: A201951 A201952 A201953 * A201955 A201956 A201957

Keyword

sign

Author

Vladimir Kruchinin, Dec 06 2011

Status

approved