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A121545 Coefficients of Taylor series expansion of the operad Prim L. 1
0, 1, 1, 4, 17, 81, 412, 2192, 12049, 67891, 390041, 2276176, 13455356, 80402284, 484865032, 2947107384, 18036248337, 111046920567, 687345582787, 4274642610932, 26697307240777, 167377288848977 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Olivier Gérard and Vincenzo Librandi, Table of n, a(n) for n = 0..200 (first 51 terms from Olivier Gérard)

Philippe Leroux, An equivalence of categories motivated by weighted directed graphs, arXiv:math-ph/0709.3453, 2007-2008.

FORMULA

G.f.: sin^2( (1/3)*arcsin(sqrt(27*x/4)) ) / ( 3/4 + sin^2( (1/3)*arcsin(sqrt(27*x/4)) )).

G.f.: x*G(x)^2 / (1 + x*G(x)^2), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 03 2012

As to a signed variant for n>0: (1, -1, 4, -17,...), a(n) = upper left term of M^n, M = the following infinite square production matrix:

1, 1, 0, 0, 0, 0,...

-2, -2, 1, 0, 0, 0,...

3, 3, -2, 1, 0, 0,...

-4, -4, 3, -2, 1, 0,...

5, 5, -4, 3, -2, 1,...

-6, -6, 5, -4, 3, -2,...

...

(each column is (1, -2, 3, -4, 5,...) prepended with (0, 0, 1, 2, 3,...) zeros by columns).

- Gary W. Adamson, Jul 13 2011

Recurrence: 32*n*(2*n-1)*a(n) = 16*(11*n^2 - n - 15)*a(n-1) + 6*(278*n^2 - 1351*n + 1670)*a(n-2) + 45*(3*n-8)*(3*n-7)*a(n-3). - Vaclav Kotesovec, Nov 19 2012

a(n) ~ 3^(3*n+1/2)/(2^(2*n+4)*n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Nov 19 2012

MATHEMATICA

CoefficientList[Series[Sin[1/3*ArcSin[Sqrt[27*x/4]]]^2/(3/4 + Sin[1/3*ArcSin[Sqrt[27*x/4]]]^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 19 2012 *)

PROG

(PARI) {a(n)=local(G=1); if(n<1, 0, for(i=1, n, G=1+x*G^3+O(x^(n+1))); polcoeff(x*G^2/(1+x*G^2), n))} \\ Paul D. Hanna, Nov 03 2012

(PARI) x='x+O('x^22); concat(0, Vec(serreverse(x*(2*x-1)^2/(1-x)^3))) \\ Gheorghe Coserea, Aug 18 2017

CROSSREFS

Cf. A006013, A001764.

Sequence in context: A204326 A151250 A174810 * A078845 A230126 A181517

Adjacent sequences:  A121542 A121543 A121544 * A121546 A121547 A121548

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Oct 07 2007

EXTENSIONS

More terms from Olivier Gérard, Oct 11 2007

STATUS

approved

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Last modified October 19 03:54 EDT 2019. Contains 328211 sequences. (Running on oeis4.)