A198203 G.f.: exp( Sum_{n>=0} [ Sum_{k=0..2*n} A027907(n,k)^n * x^k ]* x^n/n ), where A027907 is the triangle of trinomial coefficients.

1, 1, 2, 5, 19, 160, 3418, 179705, 19488053, 4590422901, 2738580784946, 3583015072969210, 9255051219746866753, 56916338252385095986978, 871826913772059843867743765, 26753845554560439025697319191184, 1695956186616651065722319776300825712

Offset: 0

Author

Paul D. Hanna, Oct 22 2011

Keywords

• nonn

Comments

Trinomial coefficients satisfy: Sum_{k=0..2*n} A027907(n,k)*x^k = (1+x+x^2)^n.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 19*x^4 + 160*x^5 + 3418*x^6 +...
The logarithm begins:
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 55*x^4/4 + 691*x^5/5 + 19440*x^6/6 + 1232750*x^7/7 + 154436735*x^8/8 + 41136723397*x^9/9 +...
which equals the sum of the series:
log(A(x)) = (1 + x + x^2)*x
+ (1 + 2^2*x + 3^2*x^2 + 2^2*x^3 + x^4)*x^2/2
+ (1 + 3^3*x + 6^3*x^2 + 7^3*x^3 + 6^3*x^4 + 3^3*x^5 + x^6)*x^3/3
+ (1 + 4^4*x + 10^4*x^2 + 16^4*x^3 + 19^4*x^4 + 16^4*x^5 + 10^4*x^6 + 4^4*x^7 + x^8)*x^4/4
+ (1 + 5^5*x + 15^5*x^2 + 30^5*x^3 + 45^5*x^4 + 51^5*x^5 + 45^5*x^6 + 30^5*x^7 + 15^5*x^8 + 5^5*x^9 + x^10)*x^5/5 +...
		

Crossrefs

Cf. A186236, A168590, A027907.

Programs

PARI

{A027907(n, k)=polcoeff((1+x+x^2)^n, k)}
{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, 2*m, A027907(m, k)^m *x^k) *x^m/m)+x*O(x^n)), n)}