A156325 E.g.f.: A(x) = exp( Sum_{n>=1} n(n+1)/2 * a(n-1)*x^n/n! ) = Sum_{n>=0} a(n)*x^n/n! with a(0)=1.

1, 1, 4, 34, 482, 10056, 286372, 10591372, 491169996, 27826318000, 1887581200256, 150885500428224, 14028718134958936, 1500672248541122944, 182987661921689610000, 25231215606822797450176

Offset

0,3

Links

Table of n, a(n) for n=0..15.

Formula

a(n) = Sum_{k=1..n} k(k+1)/2 * C(n-1,k-1)*a(k-1)*a(n-k) for n>0, with a(0)=1.

E.g.f. satisifies: A(x) = exp( d/dx x^2*A(x)/2 ). - Paul D. Hanna, Dec 17 2017

Example

E.g.f: A(x) = 1 + x + 4*x^2/2! + 34*x^3/3! + 482*x^4/4! + 10056*x^5/5! +...
log(A(x)) = x + 3*1*x^2/2! + 6*4*x^3/3! + 10*34*x^4/4! + 15*482*x^5/5! +...
such that log(A(x))  =  x*A(x) + x^2*A'(x)/2  =  d/dx x^2*A(x)/2.

Prog

(PARI) {a(n) = if(n==0, 1, n!*polcoeff(exp(sum(k=1, n, k*(k+1)/2*a(k-1)*x^k/k!)+x*O(x^n)), n))}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n) = if(n==0, 1, sum(k=1, n, k*(k+1)/2*binomial(n-1, k-1)*a(k-1)*a(n-k)))}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = exp(deriv(x^2*A/2 +x^2*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 17 2017

Crossrefs

Cf. A156326, A156327.

Sequence in context: A208831 A294475 A198976 * A248654 A336495 A111169

Adjacent sequences: A156322 A156323 A156324 * A156326 A156327 A156328

Keyword

nonn

Author

Paul D. Hanna, Feb 08 2009

Status

approved