A359926 a(n) = coefficient of x^n*y^(n+1)/n! in (1/4) * log( Sum_{n>=0} (n + 2*y)^(2*n) * x^n/n! ).

1, 8, 168, 6016, 309760, 20957184, 1762991104, 177690607616, 20895204704256, 2810343286374400, 425698411965054976, 71735043897868419072, 13313460758336789020672, 2698754565131159025483776, 593332971403056575938560000, 140634107346363806457259884544

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Formula

a(n) ~ c * d^n * n! / n^(5/2), where d = 4*(1 + sqrt(2)) * exp(2 - sqrt(2)) = 17.347603772617734513447467379678826546908822081006190652539615... and c = sqrt((2 - sqrt(2))/Pi)/4 = 0.107953003168342979028946547859477378793474... - Vaclav Kotesovec, Feb 13 2023, updated Mar 17 2024

Example

E.g.f. A(x) = x + 8*x^2/2! + 168*x^3/3! + 6016*x^4/4! + 309760*x^5/5! + 20957184*x^6/6! + 1762991104*x^7/7! + 177690607616*x^8/8! + 20895204704256*x^9/9! + 2810343286374400*x^10/10! + ...
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! where a(n) equals the coefficient of y^(n+1)*x^n/n! in the series given by
(1/4) * log( Sum_{n>=0} (n + 2*y)^(2*n) * x^n/n! ) = (y^2 + y + 1/4)*x + (8*y^3 + 18*y^2 + 14*y + 15/4)*x^2/2! + (168*y^4 + 632*y^3 + 933*y^2 + 639*y + 683/4)*x^3/3! + (6016*y^5 + 33088*y^4 + 76480*y^3 + 92680*y^2 + 58720*y + 31019/2)*x^4/4! + (309760*y^6 + 2304384*y^5 + 7521360*y^4 + 13763280*y^3 + 14855385*y^2 + 8940045*y + 9342629/4)*x^5/5! + (20957184*y^7 + 200377344*y^6 + 865825536*y^5 + 2188392960*y^4 + 3486312960*y^3 + 3490688496*y^2 + 2027376336*y + 525120804)*x^6/6! + ...
Exponentiation yields the e.g.f. of A266485:
exp(A(x)) = 1 + x + 9*x^2/2! + 193*x^3/3! + 6929*x^4/4! + 356001*x^5/5! + 24004825*x^6/6! + 2012327521*x^7/7! + 202156421409*x^8/8! + 23701550853313*x^9/9! + ... + A266485(n)*x^n/n! + ...
which equals
lim_{N->oo} [ Sum_{n>=0} (N + 2*n)^(2*n) * (x/N)^n/n! ]^(1/N).
RELATED SEQUENCES.
a(n) is divisible by n where a(n)/n begins:
[1, 4, 56, 1504, 61952, 3492864, 251855872, 22211325952, 2321689411584, ...].

Prog

(PARI) {a(n) = (1/4) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + 2*y)^(2*m)*x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A266485, A359928, A319834, A318634.

Sequence in context: A253974 A254459 A254129 * A334780 A084941 A139564

Adjacent sequences: A359923 A359924 A359925 * A359927 A359928 A359929

Keyword

nonn

Author

Paul D. Hanna, Jan 20 2023

Status

approved