A193439 exp( Sum_{n>=1} a(n-1)*x^n/n!^3 ) = Sum_{n>=0} a(n)/n!^3.

1, 1, 5, 68, 1936, 99336, 8326912, 1063584640, 196475565312, 50403792222720, 17382740425346304, 7847087503671023616, 4535069738055660564480, 3292828639234241171484672, 2955617286961757422869504000, 3233957295970672142211481337856

Offset

0,3

Comments

Compare to: exp(Sum_{n>=1} A006472(n)*x^n/n!^2) = Sum_{n>=0} A006472(n+1)/n!^2 where A006472(n) = n!*(n-1)!/2^(n-1).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..299

Formula

E.g.f.: A(x) = exp( Integral 1/x * Integral 1/x * Integral A(x) dx dx dx ). - Paul D. Hanna, Apr 30 2019

Example

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/(n!)^^3 starts as
A(x) = 1 + x + 5*x^2/2!^3 + 68*x^3/3!^3 + 1936*x^4/4!^3 + 99336*x^5/5!^3 + 8326912*x^6/6!^3 + 1063584640*x^7/7!^3 + 196475565312*x^8/8!^3 + 50403792222720*x^9/9!^3 + 17382740425346304*x^10/10!^3 + 7847087503671023616*x^11/11!^3 +...+ a(n)*x^n/n!^3 +...
where
log(A(x)) = x + x^2/2!^3 + 5*x^3/3!^3 + 68*x^4/4!^3 + 1936*x^5/5!^3 + 99336*x^6/6!^3 + 8326912*x^7/7!^3 +...+ a(n-1)*x^n/n!^3 +...
As a power series in x with reduced fractional coefficients, the e.g.f. begins
A(x) = 1 + x + 5/8*x^2 + 17/54*x^3 + 121/864*x^4 + 4139/72000*x^5 + 32527/1458000*x^6 + 1661851/200037600*x^7 + 85275853/28449792000*x^8 + 729221531/691329945600*x^9 + 2514864066167/6913299456000000*x^10 + 141910581301921/1150200196992000000*x^11 + 8201442668648113/198754594040217600000*x^12 + ...

Prog

(PARI) {a(n) = n!^3*polcoeff( exp(x+sum(m=2, n, a(m-1)*x^m/m!^3+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=0, n, A = exp( intformal(1/x*intformal(1/x*intformal(A +x*O(x^n)))))); (n!)^3*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Apr 30 2019

Crossrefs

Sequence in context: A093120 A264697 A319465 * A355086 A337951 A092817

Adjacent sequences: A193436 A193437 A193438 * A193440 A193441 A193442

Keyword

nonn

Author

Paul D. Hanna, Jul 25 2011

Status

approved