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

0,3

Comments

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

Links

Table of n, a(n) for n=0..16.

Example

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 +...

Prog

(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)}

Crossrefs

Cf. A186236, A168590, A027907.

Sequence in context: A304982 A039719 A332653 * A014466 A304981 A284860

Adjacent sequences: A198200 A198201 A198202 * A198204 A198205 A198206

Keyword

nonn

Author

Paul D. Hanna, Oct 22 2011

Status

approved