A370045 -id:A370045 - cp's OEIS frontend

A370044 Expansion of [ Sum_{n>=0} (-4)^n * (24^n + 1)/3 x^(n*(n+1)/2) ]^(-1/3).

1, 4, 32, 240, 2048, 17920, 163904, 1526784, 14473216, 138743808, 1342326528, 13078851584, 128177979392, 1262257356800, 12481163427840, 123845494105088, 1232601926811648, 12300407336042496, 123037059803447296, 1233275751577944064, 12385053557486911488, 124585853452251328512

Offset: 0

Paul D. Hanna, Feb 24 2024

nonn

			G.f.: A(x) = 1 + 4*x + 32*x^2 + 240*x^3 + 2048*x^4 + 17920*x^5 + 163904*x^6 + 1526784*x^7 + 14473216*x^8 + 138743808*x^9 + 1342326528*x^10 + ...
RELATED SERIES.
The cube of 1/A(x) equals the g.f. A370018 which starts as
1/A(x)^3 = 1 - 12*x + 176*x^3 - 2752*x^6 + 43776*x^10 - 699392*x^15 + 11186176*x^21 + ... + (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) + ...
and 1/A(x) equals the g.f. of A370019, which begins
1/A(x) = 1 - 4*x - 16*x^2 - 48*x^3 - 384*x^4 - 2816*x^5 - 24384*x^6 - 206336*x^7 - 1815552*x^8 - 16189440*x^9 + ... + A370019(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..1035

Cf. A370019, A370018, A370045.

{a(n) = my(A);
A = sum(m=0, sqrtint(2*n+1), (-4)^m * (1 + 2*4^m)/3 * x^(m*(m+1)/2) +x*O(x^n))^(-1/3);
polcoeff(H=A, n)}
for(n=0, 25, print1(a(n), ", "))

a(n) ~ c * d^n / n^(2/3), where d = 10.3933629985595735031515117628403087010816839988881759248638104... and c = 0.42093748110527419326289922348630166534660617909266766696... - Vaclav Kotesovec, Feb 24 2024

A370018 Expansion of Sum_{n>=0} (-4)^n * (24^n + 1)/3 x^(n*(n+1)/2).

Original entry on oeis.org

1, -12, 0, 176, 0, 0, -2752, 0, 0, 0, 43776, 0, 0, 0, 0, -699392, 0, 0, 0, 0, 0, 11186176, 0, 0, 0, 0, 0, 0, -178962432, 0, 0, 0, 0, 0, 0, 0, 2863333376, 0, 0, 0, 0, 0, 0, 0, 0, -45813071872, 0, 0, 0, 0, 0, 0, 0, 0, 0, 733008101376, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11728125427712

Offset: 0

Paul D. Hanna, Feb 23 2024

sign

Equals the self-convolution cube of A370019.

			G.f.: A(x) = 1 - 12*x + 176*x^3 - 2752*x^6 + 43776*x^10 - 699392*x^15 + 11186176*x^21 - 178962432*x^28 + 2863333376*x^36 + ... + (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) + ...
RELATED SERIES.
The cube root of g.f. A(x) is an integer series starting as
A(x)^(1/3) = 1 - 4*x - 16*x^2 - 48*x^3 - 384*x^4 - 2816*x^5 - 24384*x^6 - 206336*x^7 - 1815552*x^8 - 16189440*x^9 + ... + A370019(n)*x^n + ...
Also,
A(x)^(1/6) = 1 - 2*x - 10*x^2 - 44*x^3 - 330*x^4 - 2508*x^5 - 21476*x^6 - 185720*x^7 - 1658778*x^8 - 15042060*x^9 - 138464620*x^10 + ...
The expansion of 1/A(x) begins
1/A(x) = 1 + 12*x + 144*x^2 + 1552*x^3 + 16512*x^4 + 172800*x^5 + 1803200*x^6 + 18765312*x^7 + 195167232*x^8 + 2028914688*x^9 + ... + A370045(n)*x^n + ...
Further,
1/A(x)^(1/3) = 1 + 4*x + 32*x^2 + 240*x^3 + 2048*x^4 + 17920*x^5 + 163904*x^6 + 1526784*x^7 + 14473216*x^8 + 138743808*x^9 + ... + A370044(n)*x^n + ...

Cf. A370019, A370045, A370044, A370015, A370335.

{a(n) = my(A);
A = sum(m=0, sqrtint(2*n+1), (-4)^m * (1 + 2*4^m)/3 * x^(m*(m+1)/2) +x*O(x^n));
polcoeff(H=A, n)}
for(n=0, 66, print1(a(n), ", "))

cp's OEIS Frontend

A370044 Expansion of [ Sum_{n>=0} (-4)^n * (24^n + 1)/3 x^(n*(n+1)/2) ]^(-1/3).

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A370018 Expansion of Sum_{n>=0} (-4)^n * (24^n + 1)/3 x^(n*(n+1)/2).

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cp's OEIS Frontend

A370044 Expansion of [ Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) ]^(-1/3).

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Author

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PARI

Formula

A370018 Expansion of Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A370044 Expansion of [ Sum_{n>=0} (-4)^n * (24^n + 1)/3 x^(n*(n+1)/2) ]^(-1/3).

A370018 Expansion of Sum_{n>=0} (-4)^n * (24^n + 1)/3 x^(n*(n+1)/2).