cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A386665 G.f. satisfies: A(x) = Sum_{n>=0} ( (1+x)^n - A(x)^(-1/2) )^n / ( 2 - (1+x)^n * A(x)^(-1/2) )^(n+1).

Original entry on oeis.org

1, 1, 8, 90, 1336, 24406, 530234, 13410942, 388841734, 12762735148, 469004980720, 19105730068460, 855146084504046, 41724450644602328, 2204075802189470532, 125300401263988607716, 7626356269363721248332, 494723229572772238087966, 34070289390944902842701094, 2482276670026891882801017812
Offset: 0

Views

Author

Paul D. Hanna, Aug 29 2025

Keywords

Comments

It appears that lim_{n->oo} ( a(n+1)/a(n) )/(n+1) exists and is near 4.

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 90*x^3 + 1336*x^4 + 24406*x^5 + 530234*x^6 + 13410942*x^7 + 388841734*x^8 + 12762735148*x^9 + 469004980720*x^10 + ...
RELATED SERIES.
A(x)^(1/2) = 1 + 2*(x/4) + 62*(x/4)^2 + 2756*(x/4)^3 + 163574*(x/4)^4 + 11997852*(x/4)^5 + 1047984172*(x/4)^6 + 106571791752*(x/4)^7 + 12417003030694*(x/4)^8 + ...
A(x)^(-1/2) = 1 - 2*(x/4) - 58*(x/4)^2 - 2516*(x/4)^3 - 149434*(x/4)^4 - 11055996*(x/4)^5 - 976190180*(x/4)^6 - 100318703592*(x/4)^7 - 11796814729146*(x/4)^8 - ...

Links

Crossrefs

Cf. A322735, A317350, A322737.

Programs

  • PARI
    {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( (1+x)^m - Ser(A)^(-1/2) )^m / (2 - (1+x)^m*Ser(A)^(-1/2))^(m+1) ) ) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n along with B(x) = A(x)^(1/2) satisfies the following formulas.
(1) A(x) = Sum_{n>=0} ( (1+x)^n - 1/B(x) )^n / ( 2 - (1+x)^n/B(x) )^(n+1).
(2) A(x) = Sum_{n>=0} ( (1+x)^n + 1/B(x) )^n / ( 2 + (1+x)^n/B(x) )^(n+1).
(3) B(x) = Sum_{n>=0} ( (1+x)^n*B(x) - 1 )^n / ( 2*B(x) - (1+x)^n )^(n+1).
(4) B(x) = Sum_{n>=0} ( (1+x)^n*B(x) + 1 )^n / ( 2*B(x) + (1+x)^n )^(n+1).

A322736 G.f. satisfies: A(x) = Sum_{n>=0} 2^n * ( (1+x)^n - A(x)^(1/2) )^n / ( 3 - 2*(1+x)^n * A(x)^(1/2) )^(n+1).

Original entry on oeis.org

1, 2, 8, 96, 2956, 114992, 5244896, 277303392, 16680895688, 1124043943848, 83860544099264, 6863636560150656, 611673708807594944, 58982083391411043456, 6120766911879901270784, 680339106407429897733760, 80661483112436517009089168, 10162784535291704640507410016, 1356175692780348173552997926272, 191103836643650458447321745220736, 28358934286111202643351952170366400, 4420810085328675478052952299755080000
Offset: 0

Views

Author

Paul D. Hanna, Jan 25 2019

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 96*x^3 + 2956*x^4 + 114992*x^5 + 5244896*x^6 + 277303392*x^7 + 16680895688*x^8 + 1124043943848*x^9 + ...
such that A(x) and B = A(x)^(1/2) satisfy
A(x) = 1/(3 - 2*B)  +  2*((1+x) - B)/(3 - 2*(1+x)*B)^2  +  2^2*((1+x)^2 - B)^2/(3 - 2*(1+x)^2*B)^3  +  2^3*((1+x)^3 - B)^3/(3 - 2*(1+x)^3*B)^4  +  2^4*((1+x)^4 - B)^4/(3 - 2*(1+x)^4*B)^5  +  2^5*((1+x)^5 - B)^5/(3 - 2*(1+x)^5*B)^6 + ...
also,
A(x) = 1/(3 + 2*B)  +  2*((1+x) + B)/(3 + 2*(1+x)*B)^2  +  2^2*((1+x)^2 + B)^2/(3 + 2*(1+x)^2*B)^3  +  2^3*((1+x)^3 + B)^3/(3 + 2*(1+x)^3*B)^4  +  2^4*((1+x)^4 + B)^4/(3 + 2*(1+x)^4*B)^5  +  2^5*((1+x)^5 + B)^5/(3 + 2*(1+x)^5*B)^6 + ...

Crossrefs

Cf. A317350, A322735.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( (1+x)^m - Ser(A)^(1/2) )^m *2^m / (3 - 2*(1+x)^m*Ser(A)^(1/2) )^(m+1) ) ) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) and B(x) = A(x)^(1/2) satisfy:
(1) A(x) = Sum_{n>=0} 2^n * ( (1+x)^n - B(x) )^n / ( 3 - 2*(1+x)^n * B(x) )^(n+1),
(2) A(x) = Sum_{n>=0} 2^n * ( (1+x)^n + B(x) )^n / ( 3 + 2*(1+x)^n * B(x) )^(n+1).
Showing 1-2 of 2 results.

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