A008440 -id:A008440 - 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

A340951 Number of ways to write n as an ordered sum of 6 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 6, 0, 15, 6, 20, 30, 15, 66, 21, 90, 61, 90, 126, 86, 210, 90, 270, 156, 261, 320, 210, 450, 261, 516, 375, 542, 495, 570, 727, 540, 870, 650, 966, 816, 1050, 906, 1155, 1266, 1020, 1560, 1090, 1710, 1416, 1698, 1635, 1746, 2120, 1650, 2376, 1980, 2316, 2490, 2368, 2520, 2835

Offset: 6

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..10000

Cf. A000217, A008440, A010054, A053603, A053604, A319816, A340949, A340950, 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, 6):
seq(a(n), n=6..62);  # Alois P. Heinz, Jan 31 2021

nmax = 62; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^6, {x, 0, nmax}], x] // Drop[#, 6] &

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^6, 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.

A341023 Number of partitions of n into 6 distinct nonzero triangular numbers.

Original entry on oeis.org

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

Offset: 56

Ilya Gutkovskiy, Feb 02 2021

nonn

Cf. A000217, A008440, A010054, A307597, A307598, A319816, A340951, A341021, A341022, A341024, A341025, A341026, A341027.

A213791 Expansion of psi(-x)^6 in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 15, -26, 45, -66, 82, -120, 156, -170, 231, -276, 290, -390, 435, -438, 561, -630, 651, -780, 861, -842, 1020, -1170, 1095, -1326, 1431, -1370, 1716, -1740, 1682, -2016, 2145, -2132, 2415, -2550, 2353, -2850, 3120, -2810, 3321, -3486, 3285, -3906, 4005

Offset: 0

Michael Somos, Jun 20 2012

sign

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

			G.f. = 1 - 6*x + 15*x^2 - 26*x^3 + 45*x^4 - 66*x^5 + 82*x^6 - 120*x^7 + ...
G.f. = q^3 - 6*q^7 + 15*q^11 - 26*q^15 + 45*q^19 - 66*q^23 + 82*q^27 + ...

J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. X
J. W. L. Glaisher, Notes on Certain Formulae in Jacobi's Fundamenta Nova, Messenger of Mathematics, 5 (1876), pp. 174-179. see p.176

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. A008440, A134343.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[ x^2, x^4])^6, {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A) * eta(x^4 + A) / eta(x^2 + A) )^6, n))};

Expansion of q^(-3/4) * ( eta(q) * eta(q^4) / eta(q^2) )^6 in powers of q.
Expansion of -1/(8 * r) * ( 1^2 * r^1 / (1 + q) - 3^2 * q^(3/4) / (1 + q^3) - 5^2 * r^5 / (1 + q^5) + 7^2 * q^(7/4) / (1 + q^7) + 9^2 * r^9 / (1 + q^9) - ...) in powers of q where r = q^(3/4) [Glaisher 1876].
Expansion of q^(-1/4) * ( sqrt(k * k') * K / Pi )^3 in powers of q where k, k', K are Jacobi elliptic functions. [Jacobi 1828, p. 108 quoted in Glaisher 1876, p. 176].
Euler transform of period 4 sequence [ -6, 0, -6, -6, ...].
G.f.: (Sum_{k>0} (-x)^((k^2 - k)/2))^6.
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 64^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
a(n) = (-1)^n * A008440(n). Convolution cube of A134343.

A280719 Expansion of (Sum_{k>=0} x^(k(2k-1)))^6.

Original entry on oeis.org

1, 6, 15, 20, 15, 6, 7, 30, 60, 60, 30, 6, 15, 60, 90, 66, 45, 60, 80, 90, 66, 50, 120, 180, 135, 60, 15, 60, 186, 210, 141, 126, 120, 126, 165, 180, 241, 300, 210, 90, 90, 180, 270, 270, 210, 212, 270, 270, 200, 210, 366, 450, 390, 270, 135, 210, 375, 360, 396, 420, 300, 330, 375, 380, 510, 480, 336, 450, 510, 390, 330

Offset: 0

Ilya Gutkovskiy, Feb 10 2017

nonn

Number of ways to write n as an ordered sum of 6 hexagonal numbers (A000384).
a(n) > 0 for all n >= 0.
Every number is the sum of at most 6 hexagonal numbers.
Every number is the sum of at most k k-gonal numbers (Fermat's polygonal number theorem).

			a(6) = 7 because we have:
[6, 0, 0, 0, 0, 0]
[0, 6, 0, 0, 0, 0]
[0, 0, 6, 0, 0, 0]
[0, 0, 0, 6, 0, 0]
[0, 0, 0, 0, 6, 0]
[0, 0, 0, 0, 0, 6]
[1, 1, 1, 1, 1, 1]

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

Cf. A000384, A007536, A008440, A045848, A280718, A282248.

nmax = 70; CoefficientList[Series[Sum[x^(k (2 k - 1)), {k, 0, nmax}]^6, {x, 0, nmax}], x]

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

A286314 Number of representations of 10^n as sum of 6 triangular numbers.

Original entry on oeis.org

6, 231, 20400, 2003001, 200045352, 20000567352, 1959085094400, 200000030000001, 20118337236261000, 1999999999505541852, 200000000030000000001, 19994255180823548693100, 1959183673472326530612252, 200000000000105810631542400, 20118343160415860069040000000

Offset: 0

Seiichi Manyama, May 06 2017

nonn

a(n) is nearly 2*10^(2*n) because a(n) is almost (4*10^n+3)^2 / 8.

			a(0) = 1/8 * (Sum_{d|7, d == 3 mod 4} d^2 - Sum_{d|7, d == 1 mod 4} d^2) = 1/8 * (7^2 - 1^2) = 6.
a(1) = 1/8 * (Sum_{d|43, d == 3 mod 4} d^2 - Sum_{d|43, d == 1 mod 4} d^2) = 1/8 * (43^2 - 1^2) = 231.
a(2) = 1/8 * (Sum_{d|403, d == 3 mod 4} d^2 - Sum_{d|403, d == 1 mod 4} d^2) = 1/8 * (403^2 + 31^2 - 13^2 - 1^2) = 20400.

Seiichi Manyama, Table of n, a(n) for n = 0..17

Cf. A008440, A286315.

a(n) = A008440(10^n).

a(n) = 1/8 * (Sum_{d|4*10^n+3, d == 3 mod 4} d^2 - Sum_{d|4*10^n+3, d == 1 mod 4} d^2).

More terms from Seiichi Manyama, May 07 2017

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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A340951 Number of ways to write n as an ordered sum of 6 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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A341023 Number of partitions of n into 6 distinct nonzero triangular numbers.

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A213791 Expansion of psi(-x)^6 in powers of x where psi() is a Ramanujan theta function.

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PARI

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A280719 Expansion of (Sum_{k>=0} x^(k*(2*k-1)))^6.

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A286314 Number of representations of 10^n as sum of 6 triangular numbers.

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Extensions

A280719 Expansion of (Sum_{k>=0} x^(k(2k-1)))^6.