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A302535 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1). 4
1, 1, 6, 61, 846, 14746, 310016, 7665141, 218827766, 7106293246, 259169817316, 10497928495506, 467768758203676, 22739720141372196, 1197560448125948596, 67910602688355999461, 4125144974025630599846, 267199960610924528490486, 18382741943990196237909476, 1338585578875261292134492646, 102848696213697953204782043556 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

G.f. A(x) satisfies:

(1) A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1).

(2) A(x) = (1/x)*Series_Reversion( x/F(x) ), where F(x) = Sum_{n>=0} A007696(n)*x^n, the o.g.f. of the quartic factorials.

(3) A(x) = 1 + x*A(x)^2 * (A(x) + 5*x*A'(x)) / (A(x) + x*A'(x)).

(4) A(x) = 1/(1 - x*A(x)/(1 - 4*x*A(x)/(1 - 5*x*A(x)/(1 - 8*x*A(x)/(1 - 9*x*A(x)/(1 - 12*x*A(x)/(1 - 13*x*A(x)/(1 - ...)))))))), a continued fraction.

a(n) ~ sqrt(Pi) * 2^(2*n + 1/2) * n^(n - 1/4) / (Gamma(1/4) * exp(n - 1/4)). - Vaclav Kotesovec, Jun 18 2019

EXAMPLE

G.f.: A(x) = 1 + x + 6*x^2 + 61*x^3 + 846*x^4 + 14746*x^5 + 310016*x^6 + 7665141*x^7 + 218827766*x^8 + 7106293246*x^9 + 259169817316*x^10 + ...

such that

A(x) = 1 + x*A(x) + 5*x^2*A(x)^2 + 45*x^3*A(x)^3 + 585*x^4*A(x)^4 + 9945*x^5*A(x)^5 + 208845*x^6*A(x)^6 + ... + x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1) + ...

PROG

(PARI) /* Series Reversion of Quartic Factorials g.f.: */

{a(n) = polcoeff((1/x) * serreverse(x/sum(m=0, n, x^m*prod(k=1, m-1, 4*k + 1))+x^2*O(x^n)), n)}

for(n=0, 30, print1(a(n), ", "))

(PARI) /* Differential Equation: */

{a(n) = my(A=1); for(i=0, n, A = 1 + x*A^2*(A + 5*x*A')/(x*A +x^2*O(x^n))'); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

(PARI) /* Continued fraction: */

{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, A=CF; for(k=0, n, CF = 1/(1 - floor(4*floor(3*(n-k+1)/2)/3)*x*A*CF ) )); polcoeff(CF, n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A007696, A088368, A301363, A302100, A302565.

Sequence in context: A191803 A259271 A047737 * A086403 A049120 A271841

Adjacent sequences:  A302532 A302533 A302534 * A302536 A302537 A302538

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Apr 09 2018

STATUS

approved

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Last modified September 21 00:17 EDT 2019. Contains 327252 sequences. (Running on oeis4.)