A008439 -id:A008439 - cp's OEIS frontend

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

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

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

N. J. A. Sloane, Jun 01 2013

nonn

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.

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.

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

A340950 Number of ways to write n as an ordered sum of 5 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 5, 0, 10, 5, 10, 20, 5, 35, 11, 40, 30, 35, 55, 30, 90, 25, 100, 60, 80, 120, 60, 140, 90, 161, 100, 165, 135, 165, 210, 140, 220, 180, 265, 170, 295, 200, 285, 330, 205, 365, 260, 395, 295, 391, 350, 355, 480, 340, 455, 490, 415, 480, 515, 445, 600, 510, 565, 550, 680, 545, 555

Offset: 5

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..10000

Cf. A000217, A008439, A010054, A053603, A053604, A319815, A340949, A340951, A340952, A340953, A340954, A340955.

b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..67);  # Alois P. Heinz, Jan 31 2021

nmax = 67; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^5, {x, 0, nmax}], x] // Drop[#, 5] &

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^5, where theta_2() is the Jacobi theta function.

A286180 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k in powers of x.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 1, 0, 1, 4, 3, 2, 0, 0, 1, 5, 6, 4, 2, 0, 0, 1, 6, 10, 8, 6, 0, 1, 0, 1, 7, 15, 15, 13, 3, 3, 0, 0, 1, 8, 21, 26, 25, 12, 6, 2, 0, 0, 1, 9, 28, 42, 45, 31, 14, 9, 0, 0, 0, 1, 10, 36, 64, 77, 66, 35, 24, 3, 2, 1, 0, 1, 11, 45

Offset: 0

Seiichi Manyama, May 07 2017

A(n, k) is the number of ways of writing n as the sum of k triangular numbers.

			Square array begins:
   1, 1, 1, 1,  1,  1, ...
   0, 1, 2, 3,  4,  5, ...
   0, 0, 1, 3,  6, 10, ...
   0, 1, 2, 4,  8, 15, ...
   0, 0, 2, 6, 13, 25, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-12 give A000007, A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787.

Main diagonal gives A106337.

Table[Function[k, SeriesCoefficient[Product[(1 + x^i) (1 - x^(2 i)), {i, Infinity}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten (* Michael De Vlieger, May 07 2017 *)

G.f. of column k: (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k.

A282172 Expansion of (Sum_{k>=0} x^(k(k+1)(k+2)/6))^5.

Original entry on oeis.org

1, 5, 10, 10, 10, 21, 30, 20, 15, 30, 35, 30, 40, 40, 35, 60, 65, 25, 30, 60, 46, 50, 80, 50, 55, 120, 95, 20, 60, 90, 60, 80, 100, 40, 80, 145, 85, 30, 90, 85, 105, 155, 100, 40, 155, 170, 90, 80, 100, 90, 171, 145, 40, 60, 140, 110, 125, 130, 80, 140, 250, 170, 70, 110, 140, 160, 190, 140, 90, 180, 220, 170, 95, 70, 110, 215

Offset: 0

Ilya Gutkovskiy, Feb 07 2017

nonn

Number of ways to write n as an ordered sum of 5 tetrahedral (or triangular pyramidal) numbers (A000292).

a(n) > 0 for all n ("Pollock's Conjecture").

			a(4) = 10 because we have:
[4, 0, 0, 0, 0]
[0, 4, 0, 0, 0]
[0, 0, 4, 0, 0]
[0, 0, 0, 4, 0]
[0, 0, 0, 0, 4]
[1, 1, 1, 1, 0]
[1, 1, 1, 0, 1]
[1, 1, 0, 1, 1]
[1, 0, 1, 1, 1]
[0, 1, 1, 1, 1]

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Pollock's Conjecture
Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers

Cf. A000292, A000797, A008439, A104246.

nmax = 75; CoefficientList[Series[(Sum[x^(k (k + 1) (k + 2)/6), {k, 0, nmax}])^5, {x, 0, nmax}], x]

G.f.: (Sum_{k>=0} x^(k*(k+1)*(k+2)/6))^5.

A341022 Number of partitions of n into 5 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 1, 1, 3, 0, 2, 2, 0, 1, 3, 2, 3, 2, 1, 2, 3, 2, 0, 5, 1, 3, 3, 2, 3, 2, 4, 2, 5, 2, 2, 6, 1, 4, 6, 1, 5, 6, 3, 4, 4, 4, 4, 5, 3, 5, 7, 6, 4, 8, 2, 5, 7, 3, 7, 7, 7, 5, 7, 6, 4, 12, 5, 4, 10, 3, 11, 9, 5, 9, 8, 5

Offset: 35

Ilya Gutkovskiy, Feb 02 2021

nonn

Cf. A000217, A008439, A010054, A307597, A307598, A319815, A340950, A341021, A341023, A341024, A341025, A341026, A341027.

A227317 Expansion of psi(x)^6 * phi(-x)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, -5, -10, 5, 6, 10, 40, -20, -50, 19, -52, -30, 50, -25, 74, 97, 50, -25, -140, 69, -34, -100, -50, -185, -6, 83, 310, -60, -60, 410, -128, 145, -100, -245, 250, -87, -90, -400, -410, -151, 362, 185, -50, 285, 30, 150, -240, 500, 370, -68, 222, 5, -190

Offset: 0

Michael Somos, Sep 02 2013

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			1 + 2*x - 5*x^2 - 10*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 40*x^7 - 20*x^8 + ...
q^3 + 2*q^7 - 5*q^11 - 10*q^15 + 5*q^19 + 6*q^23 + 10*q^27 + 40*q^31 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008439, A010816, A215600, A227695.

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^5 / QPochhammer[ q])^2, {q, 0, n}]

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / eta(x + A))^2, n))}

Expansion of psi(x)^5 * f(-x)^3 = psi(x)^2 * f(-x^2)^6 in powers of x where psi(), f() are Ramanujan theta functions.
Expansion of q^(-3/4) * (eta(q^2)^5 / eta(x))^2 in powers of q.
Euler transform of period 2 sequence [ 2, -8, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 128 (t / i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227695.
G.f.: (Product_{k>0} (1 - x^(2*k))^5 / (1 - x^k))^2.
Convolution of A008439 and A010816.
-8 * a(n) = A215600(2*n + 1).

A280718 Expansion of (Sum_{k>=0} x^(k(3k-1)/2))^5.

Original entry on oeis.org

1, 5, 10, 10, 5, 6, 20, 30, 20, 5, 10, 30, 35, 30, 30, 30, 25, 30, 60, 60, 25, 5, 35, 80, 70, 51, 35, 50, 80, 90, 80, 30, 35, 60, 80, 95, 90, 90, 50, 75, 140, 140, 85, 20, 70, 120, 130, 120, 95, 115, 100, 115, 140, 155, 110, 40, 80, 200, 230, 140, 81, 120, 200, 190, 180, 120, 80, 100, 160, 240, 200, 155, 120, 140, 245, 260, 230

Offset: 0

Ilya Gutkovskiy, Feb 10 2017

nonn

Number of ways to write n as an ordered sum of 5 pentagonal numbers (A000326).
a(n) > 0 for all n >= 0.
Every number is the sum of at most 5 pentagonal numbers.
Every number is the sum of at most k k-gonal numbers (Fermat's polygonal number theorem).

			a(5) = 6 because we have:
[5, 0, 0, 0, 0]
[0, 5, 0, 0, 0]
[0, 0, 5, 0, 0]
[0, 0, 0, 5, 0]
[0, 0, 0, 0, 5]
[1, 1, 1, 1, 1]

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Pentagonal Number
Index to sequences related to polygonal numbers

Cf. A000326, A003679, A008439, A038671, A280719, A282248.

nmax = 76; CoefficientList[Series[Sum[x^(k (3 k - 1)/2), {k, 0, nmax}]^5, {x, 0, nmax}], x]

G.f.: (Sum_{k>=0} x^(k*(3*k-1)/2))^5.

A296046 Expansion of Product_{k>=1} ((1 - x^(2k))/(1 - x^(2k-1)))^k.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 3, 0, 3, 0, 6, 1, 3, 5, 0, 13, -3, 15, -3, 14, 6, 11, 16, -4, 38, -16, 51, -24, 65, -14, 46, 21, 10, 80, -49, 154, -102, 216, -136, 242, -119, 198, 1, 68, 189, -153, 486, -425, 775, -672, 1024, -779, 1035, -628, 782, -97, 96, 816, -930, 2069, -2203, 3428, -3413, 4546, -4130, 4958

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

sign

Cf. A008439, A008441, A008443, A010054, A295831, A295832, A296047.

nmax = 65; CoefficientList[Series[Product[((1 - x^(2 k))/(1 - x^(2 k - 1)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 - x^(2*k))/(1 - x^(2*k-1)))^k.

cp's OEIS Frontend

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

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

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A340950 Number of ways to write n as an ordered sum of 5 nonzero triangular numbers.

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A286180 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k in powers of x.

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A282172 Expansion of (Sum_{k>=0} x^(k*(k+1)*(k+2)/6))^5.

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A341022 Number of partitions of n into 5 distinct nonzero triangular numbers.

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A227317 Expansion of psi(x)^6 * phi(-x)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

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A280718 Expansion of (Sum_{k>=0} x^(k*(3*k-1)/2))^5.

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A296046 Expansion of Product_{k>=1} ((1 - x^(2*k))/(1 - x^(2*k-1)))^k.

A282172 Expansion of (Sum_{k>=0} x^(k(k+1)(k+2)/6))^5.

A280718 Expansion of (Sum_{k>=0} x^(k(3k-1)/2))^5.

A296046 Expansion of Product_{k>=1} ((1 - x^(2k))/(1 - x^(2k-1)))^k.