A325295 E.g.f. A(x) satisfies: Sum_{n>=0} x^n * A(x)^(n*(n+1)/2) = Sum_{n>=0} x^n * exp(n^2*x).

1, 1, 3, 22, 293, 6396, 192247, 7487810, 359234409, 20571901624, 1375291211051, 105546986373918, 9179888213024509, 895591657896191348, 97191443720659321503, 11651115500493779746906, 1533793016323134419714513, 220607836736797491462201456, 34514721658321036998293295955, 5850764317548636197562977724758, 1070825456345525996396825473813221

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 293*x^4/4! + 6396*x^5/5! + 192247*x^6/6! + 7487810*x^7/7! + 359234409*x^8/8! + 20571901624*x^9/9! + 1375291211051*x^10/10! + ...
such that the following series are equal:
B(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^6 + x^4*A(x)^10 + x^5*A(x)^15 + x^6*A(x)^21 + x^7*A(x)^28 + x^8*A(x)^36 + x^9*A(x)^45 + ...
B(x) = 1 + x*exp(x) + x^2*exp(4*x) + x^3*exp(9*x) + x^4*exp(16*x) + x^5*exp(25*x) + x^6*exp(36*x) + x^7*exp(49*x) + x^8*exp(64*x) + x^9*exp(81*x) + ...
where
B(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 436*x^4/4! + 8185*x^5/5! + 206046*x^6/6! + 6622945*x^7/7! + 263313688*x^8/8! + ... + A193421(n)*x^n/n! + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = -polcoeff( sum(m=0, #A, x^m*( Ser(A)^(m*(m+1)/2) - exp(m^2*x +x*O(x^#A))) ), #A) ); n!*A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A193421.

Sequence in context: A108991 A247659 A244468 * A298693 A326430 A119390

Adjacent sequences: A325292 A325293 A325294 * A325296 A325297 A325298

Keyword

nonn

Author

Paul D. Hanna, Apr 26 2019

Status

approved