A182520 G.f.: x = Sum_{n>=1} a(n)*x^n * Sum_{k=0..n} (k+1)*binomial(n,k)*(-x)^k.

1, 2, 8, 42, 264, 1916, 15744, 144546, 1467544, 16335972, 197916768, 2593286692, 36547123728, 551308006392, 8863973173120, 151328667184530, 2734113089546040, 52120618871199060, 1045503182929422240, 22014065919151444140, 485475449631284066160

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..300

Formula

G.f.: C(x) + Sum_{n>=2} (n+1)!/3! * C(x)^n, where C(x) = (1-sqrt(1-4*x))/2 is a g.f. of the Catalan numbers A000108.

Recurrence: (n-1)*n*a(n) = (n-1)*(n^2 + 9*n - 19)*a(n-1) - 2*(n-2)*(4*n^2 + 2*n - 31)*a(n-2) + 4*(n-2)*(2*n-7)*(2*n-3)*a(n-3). - Vaclav Kotesovec, Jul 05 2014

a(n) ~ exp(1)/6 * n * n!. - Vaclav Kotesovec, Jul 05 2014

Example

G.f.: A(x) = x + 2*x^2 + 8*x^3 + 42*x^4 + 264*x^5 + 1916*x^6 + 15744*x^7 +...
such that
A(x) = C(x) + C(x)^2 + 4*C(x)^3 + 20*C(x)^4 + 120*C(x)^5 +...+ (n+1)!/3!*C(x)^n +...
where
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 +...+ A000108(n)*x^(n+1) +...
By definition, the terms a(n) satisfy:
x = a(1)*x*(1-2*x) + a(2)*x^2*(1-4*x+3*x^2) + a(3)*x^3*(1-6*x+9*x^2-4*x^3) + a(4)*x^4*(1-8*x+18*x^2-16*x^3+5*x^4) +...

Prog

(PARI) {a(n)=if(n<1, 0, polcoeff(x-sum(m=1, n-1, a(m)*x^m*sum(k=0, m, (k+1)*binomial(m, k)*(-x)^k)+x*O(x^n)), n))}
(PARI) {a(n)=local(C=serreverse(x-x^2+x^2*O(x^n))); polcoeff(C+sum(m=2, n, (m+1)!/3!*C^m), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A000108.

Sequence in context: A229285 A339460 A005315 * A121635 A002874 A324961

Adjacent sequences: A182517 A182518 A182519 * A182521 A182522 A182523

Keyword

nonn

Author

Paul D. Hanna, May 03 2012

Status

approved