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.

Showing 1-3 of 3 results.

A370019 Expansion of [ Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) ]^(1/3).

Original entry on oeis.org

1, -4, -16, -48, -384, -2816, -24384, -206336, -1815552, -16189440, -146777856, -1346648064, -12487131136, -116810932224, -1101080592384, -10447586845696, -99706199973888, -956400813293568, -9215587975397376, -89158545637244928, -865730439117078528, -8433936444598677504
Offset: 0

Views

Author

Paul D. Hanna, Feb 23 2024

Keywords

Comments

The self-convolution cube equals A370018.

Examples

			G.f.: 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 - 146777856*x^10 + ...
RELATED SERIES.
The cube of the g.f. A(x) equals the g.f. A370018 which starts as
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) + ...
The reciprocal of the g.f. A(x) equals the g.f. of A370044, which begins
1/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 + ... + A370044(n)*x^n + ...

Links

Crossrefs

Cf. A370016, A370044, A370336.

Programs

  • PARI
    {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, 20, print1(a(n), ", "))

Formula

a(n) ~ c * d^n / n^(4/3), where d = 10.39336299855957350315151176284030870108168399888... and c = -0.218294054014127126766352511836393819909572679... - Vaclav Kotesovec, Feb 24 2024

A370018 Expansion of Sum_{n>=0} (-4)^n * (2*4^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

Views

Author

Paul D. Hanna, Feb 23 2024

Keywords

Comments

Equals the self-convolution cube of A370019.

Examples

			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 + ...

Crossrefs

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

Programs

  • PARI
    {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), ", "))

A370045 Expansion of 1 / Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2).

Original entry on oeis.org

1, 12, 144, 1552, 16512, 172800, 1803200, 18765312, 195167232, 2028914688, 21089678592, 219201730560, 2278287884288, 23679245377536, 246107817345024, 2557891149933568, 26585106479751168, 276308723697205248, 2871777147680423936, 29847423508786839552, 310215112347152351232
Offset: 0

Views

Author

Paul D. Hanna, Feb 24 2024

Keywords

Examples

			G.f.: 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 + 21089678592*x^10 + ...
RELATED SERIES.
The expansion of 1/A(x) is the following series (A370018)
1/A(x) = 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) + ...
The cube root of A(x) begins
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 + ...
Also, the sixth root of A(x) is an integer series starting as
A(x)^(1/6) = 1 + 2*x + 14*x^2 + 92*x^3 + 742*x^4 + 6188*x^5 + 54956*x^6 + 498584*x^7 + 4625478*x^8 + 43493324*x^9 + 413627172*x^10 + ...

Links

Crossrefs

Cf. A370018 (1/A(x)), A370044 (A(x)^(1/3)), A370019 (A(x)^(-1/3)).

Programs

  • PARI
    {a(n) = my(A);A = 1 / 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, 25, print1(a(n), ", "))

Formula

From Vaclav Kotesovec, Feb 25 2024: (Start)
a(n) ~ c * d^n, where
d = 10.39336299855957350315151176284030870108168399888817592486381041027988779...
c = 1.433973222898078483437999597179822040398973315396494951383570608840342399...
d = 1/r, where r = 0.09621524814812982023560791941974657613430770687333255066... is the smallest positive root of the equation Sum_{k>=0} (-4)^k * (2*4^k + 1) * r^(k*(k+1)/2) = 0. (End)
Showing 1-3 of 3 results.

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