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.

Previous Showing 11-15 of 15 results.

A361979 Expansion of 1 / Sum_{k>=0} x^(k*(2*k - 1)).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 0, 1, -2, 3, -4, 5, -5, 4, -2, -2, 7, -13, 19, -24, 27, -25, 17, -2, -20, 48, -80, 110, -132, 137, -116, 62, 30, -158, 314, -479, 622, -704, 680, -507, 150, 405, -1135, 1973, -2797, 3432, -3662, 3250, -1983, -280, 3540, -7592, 11977, -15953
Offset: 0

Views

Author

Ilya Gutkovskiy, May 25 2023

Keywords

Crossrefs

Cf. A000384, A006950, A106507, A308806, A317665, A322798, A363149, A363275.

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[1/Sum[x^(k (2 k - 1)), {k, 0, nmax}], {x, 0, nmax}], x]

A363149 Expansion of 1 / Sum_{k>=0} x^(k*(5*k - 3)/2).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, -2, 3, -4, 5, -6, 7, -8, 10, -13, 17, -22, 27, -33, 40, -49, 61, -77, 98, -123, 153, -189, 233, -288, 358, -448, 561, -701, 872, -1082, 1342, -1666, 2073, -2584, 3223, -4016, 4997, -6212, 7720, -9598, 11942, -14869, 18517, -23053, 28687
Offset: 0

Views

Author

Ilya Gutkovskiy, May 25 2023

Keywords

Crossrefs

Cf. A000566, A106507, A308806, A317665, A322799, A361979, A363275.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/Sum[x^(k (5 k - 3)/2), {k, 0, nmax}], {x, 0, nmax}], x]

A363275 Expansion of 1 / Sum_{k>=0} x^(k*(3*k - 2)).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, -1, 0, 1, -2, 3, -4, 5, -6, 7, -7, 6, -4, 1, 3, -9, 16, -24, 32, -39, 44, -46, 44, -35, 18, 8, -43, 86, -135, 187, -238, 280, -304, 300, -259, 171, -28, -174, 435, -746, 1088, -1431, 1736, -1952, 2017, -1864, 1425, -641, -527, 2086, -4002
Offset: 0

Views

Author

Ilya Gutkovskiy, May 25 2023

Keywords

Crossrefs

Cf. A000567, A106507, A195848, A308806, A317665, A322800, A361979, A363149.

Programs

  • Mathematica
    nmax = 56; CoefficientList[Series[1/Sum[x^(k (3 k - 2)), {k, 0, nmax}], {x, 0, nmax}], x]

A208061 G.f. 1/sum(k>=0, (-1)^k * x^(k*(k+1)/2)).

Original entry on oeis.org

1, 1, 1, 0, -1, -2, -1, 1, 4, 5, 2, -5, -12, -13, -3, 17, 34, 32, -1, -54, -93, -72, 28, 169, 248, 152, -147, -510, -646, -282, 582, 1484, 1627, 375, -2045, -4195, -3927, 110, 6716, 11544, 9002, -3458, -20996, -30921, -19123, 17974, 63154, 80435, 35553, -71525, -183969
Offset: 0

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Author

Franklin T. Adams-Watters, Feb 23 2012

Keywords

Examples

			G.f. = 1 + x + x^2 - x^4 - 2*x^5 - x^6 + x^7 + 4*x^8 + 5*x^9 + 2*x^10 - 5*x^11 + ...

Crossrefs

Cf. A006950, A023361, A106507, A197870.

Programs

  • PARI
    al(n)=Vec(1/(sum(k=0,sqrtint(2*n),(-1)^k*x^(k*(k+1)\2))+x*O(x^n)))

Formula

G.f.: 1 / (1 - x*(1 - x^2*(1 - x^3*(1 - x^4*(1 - ...))))). - Michael Somos, Mar 03 2014
Convolution inverse of A197870. - Michael Somos, Mar 03 2014

A258386 Expansion of Product_{k>=1} 1/(1-x^k)^(k+(-1)^k).

Original entry on oeis.org

1, 0, 3, 2, 11, 10, 35, 40, 107, 138, 310, 432, 871, 1262, 2355, 3504, 6186, 9318, 15799, 23934, 39351, 59672, 95772, 144970, 228258, 344244, 533552, 800952, 1225164, 1829530, 2767227, 4109504, 6155310, 9089834, 13497964, 19822252, 29208812, 42660456
Offset: 0

Views

Author

Vaclav Kotesovec, May 28 2015

Keywords

Links

Crossrefs

Cf. A000219, A106507.

Programs

  • Mathematica
    nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k+(-1)^k),{k,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ (2*Zeta(3))^(13/36) / (sqrt(3) * Pi * n^(31/36)) * exp(Zeta'(-1) + 3*Zeta(3)^(1/3) * (n/2)^(2/3)), where Zeta(3) = A002117, Zeta'(-1) = A084448 = 1/12 - log(A074962). - Vaclav Kotesovec, May 28 2015
Previous Showing 11-15 of 15 results.

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