A319834 a(n) = coefficient of x^n*y^(n+1)/n! in log( Sum_{n>=0} (n^2 + n*y + y^2)^n * x^n/n! ).

1, 2, 15, 184, 3325, 79056, 2345539, 83505920, 3472829721, 165321395200, 8868765212791, 529513463098368, 34831327847918485, 2503184803456354304, 195151614670701520875, 16405316791445973139456, 1479333355684885588136881, 142443466217414911148359680, 14587416733382035646737882591, 1583199811285962289889116160000

Offset

1,2

Comments

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! equals the logarithm of the e.g.f. of A319147.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..100

Formula

a(n) ~ c * d^n * n! / n^(5/2), where d = 6.1601834100761946... (same as for A319147) and c = 0.193396776821391327... - Vaclav Kotesovec, Mar 19 2024

Example

E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 184*x^4/4! + 3325*x^5/5! + 79056*x^6/6! + 2345539*x^7/7! + 83505920*x^8/8! + 3472829721*x^9/9! + ...
Exponentiation yields the e.g.f. of A319147:
exp(A(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4776*x^5/5! + 111967*x^6/6! + 3280264*x^7/7! + 115550073*x^8/8! +...+ A319147(n)*x^n/n! + ...
which equals
Limit_{N->oo} [ Sum_{n>=0} (N^2 + N*n + n^2)^n * (x/N)^n/n! ]^(1/N).
RELATED SEQUENCES.
a(n) is divisible by n where a(n)/n begins:
[1, 1, 5, 46, 665, 13176, 335077, 10438240, 385869969, 16532139520, ...].

Prog

(PARI) {a(n) = n! * polcoeff( polcoeff( log( sum(m=0, 2*n, (m^2 + m*y + y^2)^m *x^m/m! ) +x*O(x^(2*n)) ), n, x), n+1, y)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A319147, A318634.

Sequence in context: A210655 A052857 A053492 * A268420 A208402 A098343

Adjacent sequences: A319831 A319832 A319833 * A319835 A319836 A319837

Keyword

nonn

Author

Paul D. Hanna, Sep 30 2018

Status

approved