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A186236
G.f.: exp( Sum_{n>=0} [ Sum_{k=0..2*n} A027907(n,k)^2 * x^k ]* x^n/n ), where A027907 is the triangle of trinomial coefficients.
6
1, 1, 2, 5, 13, 34, 93, 262, 753, 2198, 6502, 19449, 58724, 178739, 547836, 1689407, 5237939, 16318137, 51056027, 160363129, 505456920, 1598263936, 5068483189, 16116397411, 51371962474, 164123564499, 525447953073, 1685534207788, 5416719384326, 17437073203711
OFFSET
0,3
COMMENTS
Trinomial coefficients satisfy: Sum_{k=0..2*n} A027907(n,k)*x^k = (1+x+x^2)^n.
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 34*x^5 + 93*x^6 +...
The logarithm begins:
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 31*x^4/4 + 91*x^5/5 + 282*x^6/6 + 890*x^7/7 + 2831*x^8/8 + 9055*x^9/9 + 29133*x^10/10 +...
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 + 3^2*x + 6^2*x^2 + 7^2*x^3 + 6^2*x^4 + 3*x^5 + x^6)*x^3/3
+ (1 + 4^2*x + 10^2*x^2 + 16^2*x^3 + 19^2*x^4 + 16^2*x^5 + 10^2*x^6 + 4^2*x^7 + x^8)*x^4/4
+ (1 + 5^2*x + 15^2*x^2 + 30^2*x^3 + 45^2*x^4 + 51^2*x^5 + 45^2*x^6 + 30^2*x^7 + 15^2*x^8 + 5^2*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)^2 *x^k) *x^m/m)+x*O(x^n)), n)}
CROSSREFS
Cf. A180718 (variant).
Sequence in context: A062465 A261237 A318229 * A064780 A368599 A148289
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 19 2011
STATUS
approved