A286180 -id:A286180 - cp's OEIS frontend

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

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

N. J. A. Sloane

nonn

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

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

Seiichi Manyama, Table of n, a(n) for n = 0..10000
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Case k=12, Theorem 7.

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.

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

More terms from Seiichi Manyama, May 05 2017

A014809 Expansion of Jacobi theta constant (theta_2/2)^24.

Original entry on oeis.org

1, 24, 276, 2048, 11178, 48576, 177400, 565248, 1612875, 4200352, 10131156, 22892544, 48897678, 99448320, 193740408, 363315200, 658523925, 1157743824, 1980143600, 3303168000, 5386270686, 8602175744, 13477895856, 20748607488, 31425764410, 46883528256, 68969957700

Offset: 0

N. J. A. Sloane

nonn

Number of ways of writing n as the sum of 24 triangular numbers from A000217.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
James G. Huard and Kenneth S. Williams, Sums of sixteen and twenty-four triangular numbers, Rocky Mountain J. Math. Volume 35, Number 3 (2005), 857-868.
Ken Ono, Sinai Robins and Patrick T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Case k=24, Theorem 8; alternative link.

Column k=24 of A286180.
Cf. A000217, A000594, A096963.
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.

a[n_] := Module[{e = IntegerExponent[n+3, 2]}, (2^(11*e) * DivisorSigma[11, (n+3)/2^e] - RamanujanTau[n+3] - 2072 * If[OddQ[n], RamanujanTau[(n+3)/2], 0]) / 176896]; Array[a, 27, 0] (* Amiram Eldar, Jan 11 2025 *)

From Wolfdieter Lang, Jan 13 2017: (Start)
G.f.: 24th power of the g.f. for A010054.
a(n) = (A096963(n+3) - tau(n+3) - 2072*tau((n+3)/2))/176896, with Ramanujan's tau function given in A000594, and tau(n) is put to 0 if n is not integer. See the Ono et al. link, case k=24, Theorem 8. (End)
a(n) = 1/72 * Sum_{a, b, x, y > 0, a*x + b*y = n + 3, x == y == 1 mod 2 and a > b} (a*b)^3*(a^2 - b^2)^2. - Seiichi Manyama, May 05 2017
a(0) = 1, a(n) = (24/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 24*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017

More terms from Seiichi Manyama, May 05 2017

A106337 Number of ways of writing n as the sum of n triangular numbers.

Original entry on oeis.org

1, 1, 1, 4, 13, 31, 82, 253, 757, 2173, 6341, 18888, 56266, 167324, 499773, 1499059, 4503557, 13546893, 40824379, 123233868, 372472353, 1127080252, 3414310032, 10353722919, 31425764410, 95463814056, 290222666436, 882954212908, 2688037654049, 8188468874808

Offset: 0

Paul D. Hanna, Apr 29 2005

nonn

Number of compositions of n into n triangular numbers with 0's allowed. a(3) = 4: [1,1,1], [0,0,3], [0,3,0], [3,0,0]. - Alois P. Heinz, Jul 31 2017

The radius of convergence is equal to A106335. - Vaclav Kotesovec, Nov 15 2017

			G106336(x) = exp(x + 1/2*x^2 + 4/3*x^3 + 13/4*x^4 + 31/5*x^5 +...).
G106336(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 +...+ A106336(n)*x^n +...
G106336(x) = 1 + x*G106336(x) + (x*G106336(x))^3 + (x*G106336(x))^6 +...

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A000217, A007294, A023361, A106333, A106334, A106335, A106336, A296045.

Main diagonal of A286180.

b:= proc(n) option remember; expand(`if`(n=0, 1,
      add(`if`(issqr(8*j+1), x*b(n-j), 0), j=1..n)))
    end:
a:= n-> (p-> add(coeff(p, x, i)*binomial(n, i), i=0..n))(b(n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 31 2017

QP = QPochhammer; a[0] = 1; a[n_] := SeriesCoefficient[(QP[-1, x]*QP[x^2]/2 )^n, {x, 0, n}]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jun 04 2017 *)

{a(n)=local(X); if(n<1,1,X=x+x*O(x^n); polcoeff(eta(X^2)^(2*n)/eta(X)^n,n))}

Log.g.f.: Sum_{n>=1} a(n)/n*x^n = log(G106336(x)), where G106336(x) is the g.f. of A106336 and satisfies: Sum_{n>=0} (x*G106336(x))^(n*(n+1)/2) = G106336(x).

a(n) = [x^n] Product_{j=1..n} (1+x^j-x^(2*j)-x^(3*j))^n. - Alois P. Heinz, Aug 01 2017

a(0) changed to 1 by Alois P. Heinz, Jul 31 2017

A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j(j+1)(j+2)/6))^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 1, 0, 1, 5, 6, 1, 2, 0, 0, 1, 6, 10, 4, 3, 2, 0, 0, 1, 7, 15, 10, 5, 6, 0, 0, 0, 1, 8, 21, 20, 10, 12, 3, 0, 0, 0, 1, 9, 28, 35, 21, 21, 12, 0, 1, 0, 0, 1, 10, 36, 56, 42, 36, 30, 4, 3, 0, 1, 0, 1, 11, 45, 84, 78, 63, 61, 20, 6, 3, 2, 0, 0, 1, 12, 55, 120, 135, 112, 112, 60, 15, 12, 3, 2, 0, 0

Offset: 0

Ilya Gutkovskiy, Jul 31 2017

A(n,k) is the number of ways of writing n as a sum of k tetrahedral (or triangular pyramidal) numbers (A000292).

			Square array begins:
1,  1,  1,  1,   1,   1,  ...
0,  1,  2,  3,   4,   5,  ...
0,  0,  1,  3,   6,  10,  ...
0,  0,  0,  1,   4,  10,  ...
0,  1,  2,  3,   5,  10,  ...
0,  0,  2,  6,  12,  21,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers

Cf. A000292, A104246, A286180, A290054, A290430.
Cf. A000007 (column 0), A023533 (column 1), A282172 (column 5).
Main diagonal gives A303170.
Similar to, but different from, A045847.

Table[Function[k, SeriesCoefficient[Sum[x^(i (i + 1) (i + 2)/6), {i, 0, n}]^k, {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

G.f. of column k: (Sum_{j>=0} x^(j*(j+1)*(j+2)/6))^k.

A014805 Expansion of Jacobi theta constant (theta_2/2)^16.

Original entry on oeis.org

1, 16, 120, 576, 2060, 6048, 15424, 35200, 73518, 143280, 263584, 461376, 775160, 1256928, 1973760, 3017088, 4503557, 6572880, 9411984, 13249280, 18340932, 25034976, 33739520, 44879616, 59057510, 76949920, 99212352, 126838080, 160884264, 202296960, 252645376

Offset: 0

N. J. A. Sloane

Number of ways of writing n as the sum of 16 triangular numbers from A000217. - Seiichi Manyama, May 05 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000
J. G. Huard and K. S. Williams, Sums of sixteen and twenty-four triangular numbers, Rocky Mountain J. Math. Volume 35, Number 3 (2005), 857-868.

Column k=16 of A286180.

a(n) = 1/192 * Sum_{a, b, x, y > 0, a*x + b*y = 2*n + 4, a == b == x == y == 1 mod 2 and a > b} a*b*(a^2 - b^2)^2. - Seiichi Manyama, May 05 2017

a(0) = 1, a(n) = (16/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017

More terms from Seiichi Manyama, May 05 2017

A014806 Expansion of Jacobi theta constant (theta_2/2)^20.

Original entry on oeis.org

1, 20, 190, 1160, 5225, 18924, 58350, 158840, 391020, 886540, 1877676, 3753640, 7140485, 13014240, 22846170, 38794448, 63969485, 102744780, 161143180, 247386480, 372472353, 550858280, 801535160, 1148976360, 1624208445, 2266848372, 3126467670, 4264095520

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Column k=20 of A286180.

Cf. A002129, A010054.

a(0) = 1, a(n) = (20/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017

More terms from Seiichi Manyama, May 06 2017

A287990 Expansion of Jacobi theta constant (theta_2/2)^36.

Original entry on oeis.org

1, 36, 630, 7176, 60165, 398412, 2184078, 10255320, 42321942, 156590980, 527649912, 1639560888, 4745867595, 12904341336, 33190117110, 81222775680, 190066236318, 427113304920, 925107172122, 1937505253320, 3934709716500, 7767340567380, 14937197788890

Offset: 0

Seiichi Manyama, Jun 04 2017

nonn

Number of ways of writing n as the sum of 36 triangular numbers.

			5*1 + 3*1 + 1*3 = 7*1 + 3*1 + 1*1 = 2 + 9. So a(1) = (5*3*1*((25-9)*(25-1)*(9-1))^2 + 7*3*1*((49-9)*(49-1)*(9-1))^2) / 141557760 = 36.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018

Column k=36 of A286180.

Cf. A008438 (k=4*1^2), A014805 (k=4*2^2), this sequence (k=4*3^2).

a:= proc(n) option remember; `if`(n=0, 1, -add(a(n-j)*add(
      36*d*(-1)^d, d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 23 2018

A002129[n_] := DivisorSum[n, -(-1)^#*#&];
a[n_] := a[n] = If[n == 0, 1, (36/n)*Sum[A002129[k]*a[n-k], {k, 1, n}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 17 2022 *)

a(0) = 1, a(n) = (36/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0.
a(n) = 1/141557760 * Sum_{a, b, c, x, y, z > 0, a*x + b*y + c*z = 2*n + 9, a == b == c == x == y == z == 1 mod 2 and a > b > c} a*b*c*((a^2 - b^2)*(a^2 - c^2)*(b^2 - c^2))^2.
Euler transform of [36, -36, 36, -36, 36, -36, ...]. - Simon Plouffe, Jun 23 2018

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

Seiichi Manyama, Jun 04 2017

nonn

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

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

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Column k=48 of A286180.

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

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 *)

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

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

cp's OEIS Frontend

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

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A014809 Expansion of Jacobi theta constant (theta_2/2)^24.

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A106337 Number of ways of writing n as the sum of n triangular numbers.

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A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j*(j+1)*(j+2)/6))^k.

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A014805 Expansion of Jacobi theta constant (theta_2/2)^16.

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A014806 Expansion of Jacobi theta constant (theta_2/2)^20.

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A287990 Expansion of Jacobi theta constant (theta_2/2)^36.

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

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A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j(j+1)(j+2)/6))^k.