The On-Line Encyclopedia of Integer Sequences (OEIS)

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A363112

Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(2*n-1).
(history; published version)

#9 by Paul D. Hanna at Tue May 16 04:22:05 EDT 2023

STATUS

editing

approved

#8 by Paul D. Hanna at Tue May 16 04:22:03 EDT 2023

FORMULA

(2) -1 = Sum_{n=-oo..+oo} x^(2*n^2) / (1 - 2*x^n*A(x)*x^n)^(2*n+1).

STATUS

approved

editing

#7 by OEIS Server at Mon May 15 08:47:01 EDT 2023

LINKS

Paul D. Hanna, <a href="/A363112/b363112_1.txt">Table of n, a(n) for n = 0..300</a>

#6 by Michael De Vlieger at Mon May 15 08:47:01 EDT 2023

STATUS

reviewed

approved

Discussion

Mon May 15

08:47

OEIS Server: Installed first b-file as b363112.txt.

#5 by Joerg Arndt at Mon May 15 00:33:33 EDT 2023

STATUS

proposed

reviewed

#4 by Paul D. Hanna at Sun May 14 21:36:55 EDT 2023

STATUS

editing

proposed

#3 by Paul D. Hanna at Sun May 14 21:36:30 EDT 2023

LINKS

Paul D. Hanna, <a href="/A363112/b363112_1.txt">Table of n, a(n) for n = 0..300</a>

#2 by Paul D. Hanna at Sun May 14 21:35:08 EDT 2023

NAME

allocated for Paul D. Hanna

Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(2*n-1).

DATA

1, 1, 6, 51, 470, 4716, 49350, 534115, 5929892, 67175779, 773473709, 9025907984, 106511693025, 1268898400188, 15240421643846, 184348620664449, 2243749948233175, 27459089491691552, 337685454820968084, 4170918486201555250, 51719670553572755173, 643610071084847351183

OFFSET

0,3

FORMULA

Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(2*n-1).

(2) -1 = Sum_{n=-oo..+oo} x^(2*n^2) / (1 - 2*x^n*A(x))^(2*n+1).

EXAMPLE

G.f.: A(x) = 1 + x + 6*x^2 + 51*x^3 + 470*x^4 + 4716*x^5 + 49350*x^6 + 534115*x^7 + 5929892*x^8 + 67175779*x^9 + 773473709*x^10 + ...

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);

A[#A] = polcoeff(-1 + sum(m=-#A, #A, x^m * (2*Ser(A) - x^m)^(2*m-1) ), #A-1)/2); A[n+1]}

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

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);

A[#A] = polcoeff(1 - sum(m=-#A, #A, x^(2*m^2)/(1 - 2*Ser(A)*x^m)^(2*m+1) ), #A-1)/2); A[n+1]}

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

CROSSREFS

Cf. A357227, A363113, A363114.

Cf. A361772.

KEYWORD

allocated

nonn

AUTHOR

Paul D. Hanna, May 14 2023

STATUS

approved

editing

#1 by Paul D. Hanna at Sun May 14 19:33:16 EDT 2023

NAME

allocated for Paul D. Hanna

KEYWORD

allocated

STATUS

approved