A370335 -id:A370335 - cp's OEIS frontend

A370336 Expansion of [ Sum_{n>=0} 5^n * (24^n + 1)/3 x^(n*(n+1)/2) ]^(1/3).

1, 5, -25, 300, -3000, 34375, -426750, 5539375, -73968750, 1010175000, -14043011250, 198006675000, -2824523453125, 40684553625000, -590871274218750, 8642318714253125, -127185323309250000, 1881843237600000000, -27976771190059687500, 417688301999460937500, -6259735680122821875000

Offset: 0

Paul D. Hanna, Feb 23 2024

sign

Self-convolution cube equals A370335.

			G.f.: A(x) = 1 + 5*x - 25*x^2 + 300*x^3 - 3000*x^4 + 34375*x^5 - 426750*x^6 + 5539375*x^7 - 73968750*x^8 + 1010175000*x^9 - 14043011250*x^10 + ...
RELATED SERIES.
The cube of the g.f. A(x) yields the g.f. of A370335 starting as
A(x)^3 = 1 + 15*x + 275*x^3 + 5375*x^6 + 106875*x^10 + 2134375*x^15 + 42671875*x^21 + 853359375*x^28 + 17066796875*x^36 + ... + 5^n*(2*4^n + 1)/3 * x^(n*(n+1)/2) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..630

Cf. A370335, A370016.

{a(n) = my(A);
A = sum(m=0, sqrtint(2*n+1), 5^m*(2*4^m + 1)/3 * x^(m*(m+1)/2) +x*O(x^n))^(1/3);
polcoeff(H=A, n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ (-1)^(n+1) * c * d^n / n^(4/3), where d = 16.061038491618401040959460250524051290971925631740259277535... and c = 0.25648790376068702946627569573532916303248367815529074... - 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

A370336 Expansion of [ Sum_{n>=0} 5^n * (24^n + 1)/3 x^(n*(n+1)/2) ]^(1/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

cp's OEIS Frontend

A370336 Expansion of [ Sum_{n>=0} 5^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) ]^(1/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

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

A370336 Expansion of [ Sum_{n>=0} 5^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).