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A255839 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..3*n} binomial(3*n,k)^2 * x^k] / A(x)^n * x^n/n ). 2
1, 1, 9, 18, 64, 172, 477, 1368, 3681, 10485, 28701, 80829, 225090, 632160, 1778553, 5010948, 14181849, 40161357, 114151716, 324873027, 926918784, 2649218580, 7585705665, 21758756931, 62508649059, 179859399129, 518234494662, 1495275239115, 4319808231645, 12495043092609, 36183457564425 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare the definition of this sequence to G(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} binomial(2*n,k)^2 * x^k] / G(x)^n * x^n/n ), which is satisfied by the rational function: G(x) = (1+x^2)^2*(1+x^3)/((1-x)*(1-x^2)).

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..100

EXAMPLE

G.f.: A(x) = 1 + x + 9*x^2 + 18*x^3 + 64*x^4 + 172*x^5 + 477*x^6 +...

where

log(A(x)) = (1 + 3^2*x + 3^2*x^2 + x^3)/A(x) * x +

(1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)/A(x)^2 * x^/2 +

(1 + 9^2*x + 36^2*x^2 + 84^2*x^3 + 126^2*x^4 + 126^2*x^5 + 84^2*x^6 + 36^2*x^7 + 9^2*x^8 + x^9)/A(x)^3 * x^3/3 +

(1 + 12^2*x + 66^2*x^2 + 220^2*x^3 + 495^2*x^4 + 792^2*x^5 + 924^2*x^6 + 792^2*x^7 + 495^2*x^8 + 220^2*x^9 + 66^2*x^10 + 12^2*x^11 + x^12)/A(x)^4 * x^4/4 +...

which involves the squares of the coefficients in (1 + x)^(3*n).

PROG

(PARI) /* By Definition: */

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, min(3*m, n-m), binomial(3*m, k)^2 * x^k) / (A +x*O(x^n))^m * x^m/m)+x*O(x^n))); polcoeff(A, n)}

for(n=0, 40, print1(a(n), ", "))

CROSSREFS

Cf. A248876, A200537, A251687, A251688, A005721.

Sequence in context: A107313 A232921 A295473 * A166640 A140149 A197345

Adjacent sequences:  A255836 A255837 A255838 * A255840 A255841 A255842

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 07 2015

STATUS

approved

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Last modified September 24 21:51 EDT 2022. Contains 356949 sequences. (Running on oeis4.)