A316705 E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} (2*n+1-2*k) + (2*k-1)*x.

1, 1, 5, 45, 513, 7745, 142485, 3095421, 77642145, 2207145825, 70130493765, 2463100122285, 94752421655265, 3962161404127329, 178943401595685909, 8680576995359894205, 450150904632193002945, 24850264116962803786305, 1455015398837011003805445, 90062955077484708745769133, 5876178416626462668682616385, 403059737428052979318873127425

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

E.g.f.: A(x) = sqrt( (1 + 2*x*W(x))*(1 + 2*x^2*W(x)) ) such that W(x) satisfies: W(x)/(1 + 2*x*W(x)) = ( W(x)/(1 + 2*x^2*W(x)) )^x.

a(n)/n! ~ c * d^n / sqrt(n), where d = 3.346513389529679772056152566067040813392... and c = 0.34882587166136471331152567... - Vaclav Kotesovec, Jul 18 2018

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 45*x^3/3! + 513*x^4/4! + 7745*x^5/5! + 142485*x^6/6! + 3095421*x^7/7! + 77642145*x^8/8! + 2207145825*x^9/9! + 70130493765*x^10/10! + ...
such that
A(x) = 1 + (1 + x)*x + (3 + x)*(1 + 3*x)*x^2/2! + (5 + x)*(3 + 3*x)*(1 + 5*x)*x^3/3! + (7 + x)*(5 + 3*x)*(3 + 5*x)*(1 + 7*x)*x^4/4! + (9 + x)*(7 + 3*x)*(5 + 5*x)*(3 + 7*x)*(1 + 9*x)*x^5/5! + ...
Also,
A(x) = sqrt( (1 + 2*x*W(x))*(1 + 2*x^2*W(x)) )
where
W(x) = 1 + (1 + x)*(2*x) + (2 + x)*(1 + 2*x)*(2*x)^2/2! + (3 + x)*(2 + 2*x)*(1 + 3*x)*(2*x)^3/3! + (4 + x)*(3 + 2*x)*(2 + 3*x)*(1 + 4*x)*(2*x)^4/4! + (5 + x)*(4 + 2*x)*(3 + 3*x)*(2 + 4*x)*(1 + 5*x)*(2*x)^5/5! + ...
Explicitly,
W(x) = 1 + 2*x + 12*x^2/2! + 108*x^3/3! + 1312*x^4/4! + 20320*x^5/5! + 381408*x^6/6! + 8420160*x^7/7! + 213813248*x^8/8! + ... + A316704(n)*x^n/n! + ...
where W(x) satisfies
W(x)/(1 + 2*x*W(x)) = ( W(x)/(1 + 2*x^2*W(x)) )^x.

Prog

(PARI) {a(n) = my(A); A = sum(m=0, n, x^m/m! * prod(k=1, m, 2*m+1-2*k + (2*k-1)*x +x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A316704, A316370.

Sequence in context: A233834 A188267 A133305 * A248586 A275576 A365564

Adjacent sequences: A316702 A316703 A316704 * A316706 A316707 A316708

Keyword

nonn

Author

Paul D. Hanna, Jul 16 2018

Status

approved