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A162649 E.g.f. satisfies: A(x) = exp( x*cosh(x*A(x)) ). 2
1, 1, 1, 4, 37, 276, 2221, 26888, 397097, 6055696, 103023481, 2047621632, 44856857101, 1051415079872, 26792169643877, 743266588537216, 22085066412427729, 698048232173484288, 23495515539312273265 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

FORMULA

a(n) = Sum_{k=0..n} C(n,k)*(n-k+1)^(k-1)*Sum_{j=0..k} C(k,j)*(2j-k)^(n-k)/2^k.

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then

a(n,m) = Sum_{k=0..n} C(n,k)*m*(n-k+m)^(k-1)*Sum_{j=0..k} C(k,j)*(2j-k)^(n-k)/2^k.

a(n) ~ sqrt((s*(r*s+sqrt(1+r^4*s^2))) / (r*(1+r*s*sqrt(1+r^4*s^2)))) * n^(n-1) / (exp(n)*r^n), where r = 0.49285491893166753586122556276745..., s = 2.549795671338977846249694869195317... are roots of the system of equations r*cosh(r*s) = log(s), r^2*sinh(r*s) = 1/s. - Vaclav Kotesovec, Jul 14 2014

EXAMPLE

E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 37*x^4/4! + 276*x^5/5! +...

log(A(x)) = x*cosh(A(x)) where

cosh(A(x)) = 1 + x^2/2! + 6*x^3/3! + 25*x^4/4! + 160*x^5/5! + 1921*x^6/6! +...

PROG

(PARI) {a(n, m=1)=sum(k=0, n, binomial(n, k)*m*(n-k+m)^(k-1)*sum(j=0, k, binomial(k, j)*(2*j-k)^(n-k)/2^k))}

CROSSREFS

Sequence in context: A335773 A201865 A025542 * A221059 A197966 A199690

Adjacent sequences:  A162646 A162647 A162648 * A162650 A162651 A162652

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 08 2009

STATUS

approved

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Last modified December 3 18:19 EST 2021. Contains 349467 sequences. (Running on oeis4.)