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 11-14 of 14 results.

A014787 Expansion of Jacobi theta constant (theta_2/2)^12.

Original entry on oeis.org

1, 12, 66, 232, 627, 1452, 2982, 5544, 9669, 16016, 25158, 38160, 56266, 80124, 111816, 153528, 205260, 270876, 353870, 452496, 574299, 724044, 895884, 1103520, 1353330, 1633500, 1966482, 2360072, 2792703, 3299340, 3892922, 4533936, 5273841, 6134448
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Number of ways of writing n as the sum of 12 triangular numbers from A000217.

Examples

			a(2) = (A001160(7) - A000735(3))/256 = (16808 - (-88))/256 = 66. - _Wolfdieter Lang_, Jan 13 2017

Links

Crossrefs

Column k=12 of A286180.
Cf. A000217, A000735, A001160.
Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809.

Formula

From Wolfdieter Lang, Jan 13 2017: (Start)
G.f.: 12th power of g.f. for A010054.
a(n) = (A001160(2*n+3) - A000735(n+1))/256. See the Ono et al. link, case k=12, Theorem 7. (End)
a(0) = 1, a(n) = (12/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 12*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017

Extensions

More terms from Seiichi Manyama, May 05 2017

A226255 Number of ways of writing n as the sum of 11 triangular numbers.

Original entry on oeis.org

1, 11, 55, 176, 440, 957, 1848, 3245, 5412, 8580, 12892, 18888, 26895, 36916, 50160, 66935, 86658, 111870, 142582, 177320, 221100, 272690, 329065, 399102, 480040, 566808, 672969, 793760, 920326, 1074040, 1248412, 1425974, 1640595, 1882145, 2123385, 2418339, 2743928, 3062895, 3453978, 3880855
Offset: 0

Views

Author

N. J. A. Sloane, Jun 01 2013

Keywords

Links

Crossrefs

Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809.

Formula

G.f. is 11th power of g.f. for A010054.
a(0) = 1, a(n) = (11/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 11*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017

A169976 Expansion of (psi(x)^24 + psi(-x)^24) / 2 in powers of x^2 where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 276, 11178, 177400, 1612875, 10131156, 48897678, 193740408, 658523925, 1980143600, 5386270686, 13477895856, 31425764410, 68969957700, 143635113000, 285718115112, 545796171084, 1005775268868, 1794713445350, 3111031518000
Offset: 0

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Author

Michael Somos, Aug 15 2010

Keywords

Examples

			1 + 276*x + 11178*x^2 + 177400*x^3 + 1612875*x^4 + 10131156*x^5 + ...
q^3 + 276*q^5 + 11178*q^7 + 177400*q^9 + 1612875*q^11 + 10131156*q^13 + ...

Links

Crossrefs

Cf. A000594, A013959, A014809.

Programs

  • Mathematica
    QP:= Pochhammer; a[n_]:= SeriesCoefficient[(QP[q, q])^24*(QP[-q^(1/2), q^(1/2)]^24 + QP[q^(1/2), -q^(1/2)]^24)/2, {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Apr 04 2018 *)
a[n_] := (DivisorSigma[11, 2*n+3] - RamanujanTau[2*n+3]) / 176896; Array[a, 20, 0] (* Amiram Eldar, Jan 11 2025 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, n *= 2; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^24, n))}
  • PARI
    {a(n) = if( n<0, 0, n = 2*n + 3; (sigma(n, 11) - polcoeff( x * eta(x + x * O(x^n))^24, n)) / 176896 )}

Formula

a(n) = (A013959(2*n + 3) - A000594(2*n + 3)) / 176896 = A014809(2*n).

A287991 Expansion of Jacobi theta constant (theta_2/2)^48.

Original entry on oeis.org

1, 48, 1128, 17344, 196836, 1764192, 13051008, 82244736, 452197434, 2210431056, 9753024192, 39328459968, 146436844568, 507826976160, 1652238451200, 5074887938688, 14794635174459, 41126600601168, 109456398969568, 279899944411776, 689873759134308
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2017

Keywords

Comments

Number of ways of writing n as the sum of 48 triangular numbers.

Examples

			4*1 + 2*1 + 1*1 = 1 + 6. So a(1) = (4*2*1)^3*((16-1)*(16-4)*(4-1))^2 / 3110400 = 48.

Links

Crossrefs

Column k=48 of A286180.
Cf. A007331 (k=4*1*2), A014809 (k=4*2*3), this sequence (k=4*3*4).

Programs

  • Mathematica
    a002129[n_]:=-Sum[(-1)^d*d, {d, Divisors[n]}]; a[n_]:=a[n]=If[n==0, 1, 48 Sum[a002129[k] a[n - k], {k, n}]/n]; Table[a[n], {n, 0, 100}] (* Indranil Ghosh, Aug 02 2017 *)
  • Python
    from sympy import divisors
from sympy.core.cache import cacheit
def a002129(n): return -sum((-1)**d*d for d in divisors(n))
@cacheit
def a(n): return 1 if n==0 else 48*sum(a002129(k)*a(n - k) for k in range(1, n + 1))//n
print([a(n) for n in range(101)]) # Indranil Ghosh, Aug 02 2017

Formula

a(0) = 1, a(n) = (48/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0.
a(n) = 1/3110400 * Sum_{a, b, c, x, y, z > 0, a*x + b*y + c*z = n + 6, x == y == z == 1 mod 2 and a > b > c} (a*b*c)^3*((a^2 - b^2)*(a^2 - c^2)*(b^2 - c^2))^2.
G.f.: exp(48*Sum_{k>=1} (x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Aug 02 2017
Previous Showing 11-14 of 14 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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