A008438 -id:A008438 - cp's OEIS frontend

A239050 a(n) = 4*sigma(n).

4, 12, 16, 28, 24, 48, 32, 60, 52, 72, 48, 112, 56, 96, 96, 124, 72, 156, 80, 168, 128, 144, 96, 240, 124, 168, 160, 224, 120, 288, 128, 252, 192, 216, 192, 364, 152, 240, 224, 360, 168, 384, 176, 336, 312, 288, 192, 496, 228, 372, 288, 392, 216, 480, 288, 480, 320, 360, 240, 672, 248, 384, 416, 508

Offset: 1

Omar E. Pol, Mar 09 2014

4 times the sum of divisors of n.

a(n) is also the total number of horizontal cells in the terraces of the n-th level of an irregular stepped pyramid (starting from the top) where the structure of every three-dimensional quadrant arises after the 90-degree zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a square formed by four cells (see links and examples). - Omar E. Pol, Jul 04 2016

			For n = 4 the sum of divisors of 4 is 1 + 2 + 4 = 7, so a(4) = 4*7 = 28.
For n = 5 the sum of divisors of 5 is 1 + 5 = 6, so a(5) = 4*6 = 24.
.
Illustration of initial terms:                                    _ _ _ _ _ _
.                                           _ _ _ _ _ _          |_|_|_|_|_|_|
.                           _ _ _ _       _|_|_|_|_|_|_|_     _ _|           |_ _
.             _ _ _ _     _|_|_|_|_|_    |_|_|       |_|_|   |_|               |_|
.     _ _    |_|_|_|_|   |_|       |_|   |_|           |_|   |_|               |_|
.    |_|_|   |_|   |_|   |_|       |_|   |_|           |_|   |_|               |_|
.    |_|_|   |_|_ _|_|   |_|       |_|   |_|           |_|   |_|               |_|
.            |_|_|_|_|   |_|_ _ _ _|_|   |_|_         _|_|   |_|               |_|
.                          |_|_|_|_|     |_|_|_ _ _ _|_|_|   |_|_             _|_|
.                                          |_|_|_|_|_|_|         |_ _ _ _ _ _|
.                                                                |_|_|_|_|_|_|
.
n:     1          2             3                4                     5
S(n):  1          3             4                7                     6
a(n):  4         12            16               28                    24
.
For n = 1..5, the figure n represents the reflection in the four quadrants of the symmetric representation of S(n) = sigma(n) = A000203(n). For more information see A237270 and A237593.
The diagram also represents the top view of the first four terraces of the stepped pyramid described in Comments section. - _Omar E. Pol_, Jul 04 2016

Antti Karttunen, Table of n, a(n) for n = 1..10000
Omar E. Pol, Diagram of the triangle before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Folding the first eight rows of triangle
Index entries for sequences related to sigma(n)

Alternating row sums of A239662.
Partial sums give A243980.
k times sigma(n), k=1..6: A000203, A074400, A272027, this sequence, A274535, A274536.
k times sigma(n), k = 1..10: A000203, A074400, A272027, this sequence, A274535, A274536, A319527, A319528, A325299, A326122.
Cf. A008438, A017113, A062731, A112610, A144613, A193553, A196020, A235791, A236104, A237270, A237593, A239052, A239053, A239660, A239662, A244050, A262626.

[4*SumOfDivisors(n): n in [1..70]]; // Vincenzo Librandi, Jul 30 2019

with(numtheory): seq(4*sigma(n), n=1..64); # Omar E. Pol, Jul 04 2016

Array[4 DivisorSigma[1, #] &, 64] (* Michael De Vlieger, Nov 16 2017 *)

a(n) = 4 * sigma(n); \\ Omar E. Pol, Jul 04 2016

a(n) = 4*A000203(n) = 2*A074400(n).
a(n) = A000203(n) + A272027(n). - Omar E. Pol, Jul 04 2016
Dirichlet g.f.: 4*zeta(s-1)*zeta(s). - Ilya Gutkovskiy, Jul 04 2016
Conjecture: a(n) = sigma(3*n) = A144613(n) iff n is not a multiple of 3. - Omar E. Pol, Oct 02 2018
The conjecture above is correct. Write n = 3^e*m, gcd(3, m) = 1, then sigma(3*n) = sigma(3^(e+1))*sigma(m) = ((3^(e+2) - 1)/2)*sigma(m) = ((3^(e+2) - 1)/(3^(e+1) - 1))*sigma(3^e*m), and (3^(e+2) - 1)/(3^(e+1) - 1) = 4 if and only if e = 0. - Jianing Song, Feb 03 2019

A193553 Sum of divisors of 4*n.

Original entry on oeis.org

7, 15, 28, 31, 42, 60, 56, 63, 91, 90, 84, 124, 98, 120, 168, 127, 126, 195, 140, 186, 224, 180, 168, 252, 217, 210, 280, 248, 210, 360, 224, 255, 336, 270, 336, 403, 266, 300, 392, 378, 294, 480, 308, 372, 546, 360, 336, 508, 399, 465, 504, 434, 378, 600, 504, 504, 560, 450, 420, 744, 434, 480, 728, 511, 588, 720

Offset: 1

Joerg Arndt, Jul 30 2011

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), this sequence (k=4), A283118 (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).

Cf. A008438, A008586, A182820.

DivisorSigma[1,4*Range[70]] (* Harvey P. Dale, Jan 27 2015 *)

PARI
```
vector(66, n, sigma(4*n, 1))
```

a(n) = sigma(4*n) = A000203(4*n).
a(n) = 3*sigma(2*n) - 2*sigma(n); the relation is the special case e=1, p=2 of the relation sigma(t^2*n) = (t+1)*sigma(t*n) - t*sigma(n) where t=p^e (p a prime).
G.f. is x times the logarithmic derivative of the g.f. of A182820.
a(2*n-1) = 7 * A008438(n) = 7 * sigma(2*n-1); special case of sigma(2^k*(2*n-1)) = (2^(k+1)-1) * sigma(2*n-1).
Sum_{k=1..n} a(k) = (11*Pi^2/24) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022
G.f.: Sum_{k>=1} k*x^(k/gcd(k, 4))/(1 - x^(k/gcd(k, 4))). - Miles Wilson, Sep 29 2024

A008439 Expansion of Jacobi theta constant theta_2^5 /32.

Original entry on oeis.org

1, 5, 10, 15, 25, 31, 35, 55, 60, 60, 90, 90, 95, 135, 125, 126, 170, 180, 175, 215, 220, 195, 285, 280, 245, 340, 300, 320, 405, 350, 351, 450, 465, 415, 515, 480, 425, 620, 590, 505, 655, 625, 590, 755, 660, 650, 805, 770, 755, 855, 841, 730, 1045, 960, 770, 1100

Offset: 0

N. J. A. Sloane

nonn

Also number of ways of writing n as a sum of five triangular numbers. - N. J. A. Sloane, Jun 01 2013

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.

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.

a002129[n_]:=-Sum[(-1)^d*d, {d, Divisors[n]}]; a[n_]:=a[n]=If[n==0, 1, 5 Sum[a002129[k] a[n - k], {k, n}]/n]; Table[a[n], {n, 0, 100}] (* Indranil Ghosh, Aug 02 2017 *)

a(0) = 1, a(n) = (5/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017

G.f.: exp(Sum_{k>=1} 5*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017

More terms from Seiichi Manyama, May 05 2017

A097057 Number of integer solutions to a^2 + b^2 + 2c^2 + 2d^2 = n.

Original entry on oeis.org

1, 4, 8, 16, 24, 24, 32, 32, 24, 52, 48, 48, 96, 56, 64, 96, 24, 72, 104, 80, 144, 128, 96, 96, 96, 124, 112, 160, 192, 120, 192, 128, 24, 192, 144, 192, 312, 152, 160, 224, 144, 168, 256, 176, 288, 312, 192, 192, 96, 228, 248, 288, 336, 216, 320, 288, 192, 320, 240, 240

Offset: 0

N. J. A. Sloane, Sep 15 2004

a^2 + b^2 + 2*c^2 + 2*d^2 is another (cf. A000118) of Ramanujan's 54 universal quaternary quadratic forms. - Michael Somos, Apr 01 2008

			1 + 4*q + 8*q^2 + 16*q^3 + 24*q^4 + 24*q^5 + 32*q^6 + 32*q^7 + 24*q^8 + ...

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 373 Entry 31.
Jesse Ira Deutsch, Bumby's technique and a result of Liouville on a quadratic form, Integers 8 (2008), no. 2, A2, 20 pp. MR2438287 (2009g:11047).
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.29).
S. Ramanujan, Collected Papers, Chap. 20, Cambridge Univ. Press 1927 (Proceedings of the Camb. Phil. Soc., 19 (1917), 11-21).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Jesse Ira Deutsch, A quaternionic proof of the representation formula of a quaternary quadratic form, J. Number Theory 113 (2005), no. 1, 149-174. MR2141762 (2006b:11033).
Y. Mimura, Almost Universal Quadratic Forms.
Olivia X. M. Yao and Ernest X. W. Xia, Combinatorial proofs of five formulas of Liouville, Discrete Math. 318 (2014), 1-9. MR3141622.

a^2 + b^2 + 2*c^2 + m*d^2 = n: this sequence (m=2), A320124 (m=3), A320125 (m=4), A320126 (m=5), A320127 (m=6), A320128 (m=7), A320130 (m=8), A320131 (m=9), A320132 (m=10), A320133 (m=11), A320134 (m=12), A320135 (m=13), A320136 (m=14).

Cf. A000118, A004011, A008438, A097723, A112610.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}] (* Michael Somos, Jul 05 2011 *)
f[p_, e_] := (p^(e+1)-1)/(p-1); f[2, 1] = 2; f[2, e_] := 6; a[0] = 1; a[1] = 4; a[n_] := 4 * Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* Amiram Eldar, Aug 22 2023 *)

{a(n) = local(t); if( n<1, n>=0, t = 2^valuation( n, 2); 4 * sigma(n/t) * if( t>2, 6, t))} \\ Michael Somos, Sep 17 2004

{a(n) = local(A = x * O(x^n)); polcoeff( (eta(x^2 + A) * eta(x^4 + A))^6 / (eta(x + A) * eta(x^8 + A))^4, n)} \\ Michael Somos, Sep 17 2004

{a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0, 0, 0; 0, 1, 0, 0; 0, 0, 2, 0; 0, 0, 0, 2], n)[n])} \\ Michael Somos, Oct 29 2005

A097057(n)=if(n,sigma(n>>n=valuation(n,2))*if(n>1,24,4<M. F. Hasler, May 07 2018

Euler transform of period 8 sequence [4, -2, 4, -8, 4, -2, 4, -4, ...]. - Michael Somos, Sep 17 2004
Multiplicative with a(n) = 4*b(n), b(2) = 2, b(2^e) = 6 if e > 1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p > 2. - Michael Somos, Sep 17 2004
Expansion of (eta(q^2) * eta(q^4))^6 / (eta(q) * eta(q^8))^4 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 05 2011
G.f.: (theta_3(q) * theta_3(q^2))^2.
G.f.: Product_{k>0} ((1-x^(2k))(1-x^(4k)))^6/((1-x^k)(1-x^(8k)))^4.
G.f.: 1 + Sum_{k>0} 8 * x^(4*k) / (1 + x^(4*k))^2 + 4 * x^(2*k-1) / (1 - x^(2*k-1))^2 = 1 + Sum_{k>0} (2 + (-1)^k) * 4*k * x^(2*k) / (1 + x^(2*k)) + 4*(2*k - 1) * x^(2*k-1) / (1 - x^(2*k - 1)). - Michael Somos, Oct 22 2005
a(2*n) = A000118(n). a(2*n + 1) = 4 * A008438(n). a(4*n) = A004011(n). a(4*n + 1) = 4 * A112610(n). a(4*n + 2) = 8 * A008438(n). a(4*n + 3) = 16 * A097723(n). - Michael Somos, Jul 05 2011

Added keyword mult and minor edits by M. F. Hasler, May 07 2018

A008440 Expansion of Jacobi theta constant theta_2^6 /(64q^(3/2)).

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, 3722, 4350

Offset: 0

N. J. A. Sloane

Number of representations of n as sum of 6 triangular numbers. - Michel Marcus, Oct 24 2012. See the Ono et al. link.

			G.f. = 1 + 6*x + 15*x^2 + 26*x^3 + 45*x^4 + 66*x^5 + 82*x^6 + ... - _Michael Somos_, Jun 25 2019
G.f. = q^3 + 6*q^7 + 15*q^11 + 26*q^15 + 45*q^19 + 66*q^23 + 82*q^27 + ...

B. C. Berndt, Fragments by Ramanujan on Lambert series, in Number Theory and Its Applications, K. Gyory and S. Kanemitsu, eds., Kluwer, Dordrecht, 1999, pp. 35-49, see Entry 6.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
B. C. Berndt, Fragments by Ramanujan on Lambert series.
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. Theorem 4.

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, A002173.

CoefficientList[(QPochhammer[q^2]^2 / QPochhammer[q])^6 + O[q]^50, q] (* Jean-François Alcover, Nov 05 2015 *)
a[ n_] := If[ n < 0, 0, -DivisorSum[ 4 n + 3, Re[I^(# - 1)] #^2 &] / 8]; (* Michael Somos, Jun 25 2019 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(8*n+1)-1)\2, x^((k^2+k)/2), x * O(x^n))^6, n))}; /* Michael Somos, May 23 2006 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^6, n))}; /* Michael Somos, May 23 2006 */

{a(n)= -sumdiv(4*n + 3, d, real(I^(d-1))*d^2)/8}; /* Michael Somos, Oct 24 2012 */

Expansion of Ramanujan phi^6(q) in powers of q.
Expansion of q^(-3/4)(eta(q^2)^2/eta(q))^6 in powers of q.
Euler transform of period 2 sequence [6, -6, ...]. - Michael Somos, May 23 2006
G.f.: (Sum_{n>=0} x^((n^2+n)/2))^6.
a(n) = (-1/8)*Sum_{d divides (4n+3)} Chi_2(4;d)*d^2. - Michel Marcus, Oct 24 2012. See the Ono et al. link. Theorem 4.
a(n) =(-1/8)*A002173(4*n+3). This is the preceding formula. - Wolfdieter Lang, Jan 12 2017
a(0) = 1, a(n) = (6/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 6*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017

A226252 Number of ways of writing n as the sum of 7 triangular numbers.

Original entry on oeis.org

1, 7, 21, 42, 77, 126, 175, 253, 357, 434, 567, 735, 833, 1057, 1302, 1400, 1708, 2037, 2191, 2597, 3003, 3151, 3619, 4242, 4389, 4935, 5691, 5740, 6594, 7434, 7371, 8400, 9303, 9506, 10626, 11592, 11585, 12761, 14427, 14203, 15519, 17241, 16808, 18788, 20559, 19950, 21882, 23898, 23786

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 7th power of g.f. for A010054.
a(0) = 1, a(n) = (7/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 7*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017

A226253 Number of ways of writing n as the sum of 9 triangular numbers.

Original entry on oeis.org

1, 9, 36, 93, 198, 378, 633, 990, 1521, 2173, 2979, 4113, 5370, 6858, 8955, 11055, 13446, 16830, 20031, 23724, 28836, 33381, 38520, 45729, 52203, 59121, 68922, 77461, 86283, 99747, 110547, 121500, 138870, 152034, 166725, 188568, 204156, 221760, 248310, 268713, 289422, 321786, 345570, 369036

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 9th power of g.f. for A010054.
a(0) = 1, a(n) = (9/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 9*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017

A226254 Number of ways of writing n as the sum of 10 triangular numbers from A000217.

Original entry on oeis.org

1, 10, 45, 130, 300, 612, 1105, 1830, 2925, 4420, 6341, 9000, 12325, 16290, 21645, 27932, 34980, 44370, 54900, 66430, 81702, 98050, 115440, 138330, 162565, 187800, 220545, 254800, 289265, 334890, 382058, 427350, 488700, 550420, 609960, 691812, 770185, 845750, 949365, 1049400, 1145580, 1274580

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. Case k=10, Theorem 6.

Cf. A000217, A030212, A050456.

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 10th power of g.f. for A010054.
a(n) = (A050456(4*n+5) - A030212(4*n+5))/640. See the Ono et al. link, case k=10, Theorem 6. - Wolfdieter Lang, Jan 13 2017
a(0) = 1, a(n) = (10/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 10*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017

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

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

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