A000145 -id:A000145 - cp's OEIS frontend

A188671 A000145(n) / 8 - (n^5 + 1).

1, 0, -24, -32, 108, 275, -176, -1056, 45, 3157, 1080, -6541, -836, 16839, 2072, -33824, 1188, 67100, 1672, -95883, 19162, 161083, -8208, -224653, 2707, 371325, 67500, -520025, -1188, 870551, 8512, -1082400, 148334, 1419889, 10428, -1588228

Offset: 1

Michael Somos, Apr 11 2011

sign

Theorem 2 in the Hales reference defines t_p = (n_p - 8(p^5 + 1)) / (32 p^(5/2)) where n_p is the number of ways to express p as a sum of 12 squares.

			x - 24*x^3 - 32*x^4 + 108*x^5 + 275*x^6 - 176*x^7 - 1056*x^8 + 45*x^9 + ...

T. C. Hales, The Mathematical Work of the 2010 Fields Medalists, Notices Amer. Math. Soc, 58 (No. 3, Mar 2011), 453-457. See p. 457, Theorem 2.

Cf. A000145.

{a(n) = if( n<1, 0, polcoeff( sum( k = 1, sqrtint(n), 2 * x^k^2, 1 + x*O(x^n))^12, n) / 8 - (n^5 + 1))}

G.f.: ((Sum_{k} x^k^2)^12 - 1) / 8 - (2*x + 21*x^2 + 76*x^3 + 16*x^4 + 6*x^5 - x^6) / (1 - x)^6.

a(n) = A000145(n) / 8 - (n^5 + 1).

A122141 Array: T(d,n) = number of ways of writing n as a sum of d squares, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 1, 6, 4, 0, 1, 8, 12, 0, 2, 1, 10, 24, 8, 4, 0, 1, 12, 40, 32, 6, 8, 0, 1, 14, 60, 80, 24, 24, 0, 0, 1, 16, 84, 160, 90, 48, 24, 0, 0, 1, 18, 112, 280, 252, 112, 96, 0, 4, 2, 1, 20, 144, 448, 574, 312, 240, 64, 12, 4, 0, 1, 22, 180, 672, 1136, 840, 544, 320, 24, 30, 8, 0

Offset: 1

R. J. Mathar, Oct 29 2006

This is the transpose of the array in A286815.

T(d,n) is divisible by 2d for any n != 0 iff d is a power of 2. - Jianing Song, Sep 05 2018

			Array T(d,n) with rows d = 1,2,3,... and columns n = 0,1,2,3,... reads
  1  2   0   0    2    0     0     0     0     2      0 ...
  1  4   4   0    4    8     0     0     4     4      8 ...
  1  6  12   8    6   24    24     0    12    30     24 ...
  1  8  24  32   24   48    96    64    24   104    144 ...
  1 10  40  80   90  112   240   320   200   250    560 ...
  1 12  60 160  252  312   544   960  1020   876   1560 ...
  1 14  84 280  574  840  1288  2368  3444  3542   4424 ...
  1 16 112 448 1136 2016  3136  5504  9328 12112  14112 ...
  1 18 144 672 2034 4320  7392 12672 22608 34802  44640 ...
  1 20 180 960 3380 8424 16320 28800 52020 88660 129064 ...

Alois P. Heinz, Antidiagonals d = 1..141, flattened
Wikipedia, Sum of squares function (transposed)
Index entries for sequences related to sums of squares

Cf. A066535, A286815.
Cf. A000122 (1st row), A004018 (2nd row), A005875 (3rd row), A000118 (4th row), A000132 (5th row), A000141 (6th row), A008451 (7th row), A000143 (8th row), A008452 (9th row), A000144 (10th row), A008453 (11th row), A000145 (12th row), A276285 (13th row), A276286 (14th row), A276287 (15th row), A000152 (16th row).
Cf. A005843 (2nd column), A046092 (3rd column), A130809 (4th column).
Cf. A010052 (1st row divides 2), A002654 (2nd row divides 4), A046897 (4th row divides 8), A008457 (8th row divides 16), A302855 (16th row divides 32), A302857 (32nd row divides 64).

A122141 := proc(d,n) local i,cnts ; cnts := 0 ; for i from -trunc(sqrt(n)) to trunc(sqrt(n)) do if n-i^2 >= 0 then if d > 1 then cnts := cnts+procname(d-1,n-i^2) ; elif n-i^2 = 0 then cnts := cnts+1 ; fi ; fi ; od ; cnts ;
end:
for diag from 1 to 14 do for n from 0 to diag-1 do d := diag-n ; printf("%d,",A122141(d,n)) ; od ; od;
# second Maple program:
A:= proc(d, n) option remember; `if`(n=0, 1, `if`(n<0 or d<1, 0,
      A(d-1, n) +2*add(A(d-1, n-j^2), j=1..isqrt(n))))
    end:
seq(seq(A(h-n, n), n=0..h-1), h=1..14); # Alois P. Heinz, Jul 16 2014

Table[ SquaresR[d - n, n], {d, 1, 12}, {n, 0, d - 1}] // Flatten (* Jean-François Alcover, Jun 13 2013 *)
A[d_, n_] := A[d, n] = If[n==0, 1, If[n<0 || d<1, 0, A[d-1, n] + 2*Sum[A[d-1, n-j^2], {j, 1, Sqrt[n]}]]]; Table[A[h-n, n], {h, 1, 14}, {n, 0, h-1}] // Flatten (* Jean-François Alcover, Feb 28 2018, after Alois P. Heinz *)

from sympy.core.power import isqrt
from functools import cache
@cache
def T(d, n):
  if n == 0: return 1
  if n < 0 or d < 1: return 0
  return T(d-1, n) + sum(T(d-1, n-(j**2)) for j in range(1, isqrt(n)+1)) * 2  # Darío Clavijo, Feb 06 2024

T(n,n) = A066535(n). - Alois P. Heinz, Jul 16 2014

A286815 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>=1} (1 - x^(2j))^5/((1 - x^j)(1 - x^(4*j)))^2)^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 0, 0, 1, 6, 4, 0, 0, 1, 8, 12, 0, 2, 0, 1, 10, 24, 8, 4, 0, 0, 1, 12, 40, 32, 6, 8, 0, 0, 1, 14, 60, 80, 24, 24, 0, 0, 0, 1, 16, 84, 160, 90, 48, 24, 0, 0, 0, 1, 18, 112, 280, 252, 112, 96, 0, 4, 2, 0, 1, 20, 144, 448, 574, 312, 240, 64, 12

Offset: 0

Seiichi Manyama, May 27 2017

A(n,k) is the number of ways of writing n as a sum of k squares.

This is the transpose of the array in A122141.

			Square array begins:
   1, 1, 1,  1,  1, ...
   0, 2, 4,  6,  8, ...
   0, 0, 4, 12, 24, ...
   0, 0, 0,  8, 32, ...
   0, 2, 4,  6, 24, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Sum of squares function

Columns k=0-16 give: A000007, A000122, A004018, A005875, A000118, A000132, A000141, A008451, A000143, A008452, A000144, A008453, A000145, A276285, A276286, A276287, A000152.
Diagonal gives A066535.
Cf. A122141.

A:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      A(n, k-1) +2*add(A(n-j^2, k-1), j=1..isqrt(n))))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 27 2017

A[n_, k_] := A[n, k] = If[n == 0, 1, If[n < 0 || k < 1, 0, A[n, k-1] + 2*Sum[A[n-j^2, k-1], {j, 1, Sqrt[n]}]]];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 28 2018, after Alois P. Heinz *)

G.f. of column k: (Product_{j>=1} (1 - x^(2*j))^5/((1 - x^j)*(1 - x^(4*j)))^2)^k.

A000735 Expansion of Product_{k>=1} (1 - x^k)^12.

Original entry on oeis.org

1, -12, 54, -88, -99, 540, -418, -648, 594, 836, 1056, -4104, -209, 4104, -594, 4256, -6480, -4752, -298, 5016, 17226, -12100, -5346, -1296, -9063, -7128, 19494, 29160, -10032, -7668, -34738, 8712, -22572, 21812, 49248, -46872, 67562, 2508, -47520, -76912, -25191, 67716

Offset: 0

N. J. A. Sloane

Glaisher (1905, 1907) calls this sequence {Omega(m): m=1,3,5,7,9,11,...}. - N. J. A. Sloane, Nov 24 2018
Number 9 of the 74 eta-quotients listed in Table I of Martin (1996). See g.f. B(q) below: cusp form of weight 6 and level 4.
Grosswald uses b_n where b_{2n+1} = a(n).
Cynk and Hulek on page 14 in "The Example of Ahlgren" refer to a_p of the unique normalized weight 6 level 4 cusp form. - Michael Somos, Aug 24 2012
Expansion of q^(-1/2) * k(q) * k'(q)^4 * (K(q) / (Pi/2))^6 / 4 in powers of q where k(), k'(), K() are Jacobi elliptic functions. In Glaisher 1907 denoted by Omega(m) defined in section 62 on page 37. - Michael Somos, May 19 2013

			G.f. A(x) = 1 - 12*x + 54*x^2 - 88*x^3 - 99*x^4 + 540*x^5 - 418*x^6 - 648*x^7 + ...
G.f. B(q) = q - 12*q^3 + 54*q^5 - 88*q^7 - 99*q^9 + 540*q^11 - 418*q^13 - 648*q^15 + ...

J. W. L. Glaisher, On the representations of a number as a sum of four squares, and on some allied arithmetical functions, Quarterly Journal of Pure and Applied Mathematics, 36 (1905), 305-358. See p. 340.
Glaisher, J. W. L. (1906). The arithmetical functions P(m), Q(m), Omega(m). Quart. J. Math, 37, 36-48.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
Newman, Morris; A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 5).
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
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.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Index entries for expansions of Product_{k >= 1} (1-x^k)^m
Index entries for sequences mentioned by Glaisher

Cf. A000145, A000729, A005758, A029751.
A209676 is the same except for signs.
This is a bisection of A227239.

# DedekindEta is defined in A000594.
A000735List(len) = DedekindEta(len, 12)
A000735List(42) |> println # Peter Luschny, Mar 10 2018

Basis( CuspForms( Gamma0(4), 6), 85) [1]; /* Michael Somos, Dec 09 2013 */

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> -12): seq(a(n), n=0..45); # Alois P. Heinz, Sep 08 2008

CoefficientList[ Take[ Expand[ Product[(1 - x^k)^12, {k, 42}]], 42], x]
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^12, {q, 0, n}]; (* Michael Somos, May 19 2013 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - q^k, {k, n}]^12, {q, 0, n}]; (* Michael Somos, May 19 2013 *)

{a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^12, n))}; /* Michael Somos, Sep 21 2005 */

CuspForms( Gamma0(4), 6, prec=85).0; # Michael Somos, May 28 2013

Expansion of q^(-1/2) * eta(q)^12 in powers of q.
Euler transform of period 1 sequence [-12, ...]. - Michael Somos, Sep 21 2005
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^4*w^2 + 48*(u*v*w)^2 + 4906*u^2*w^4 - u^6. - Michael Somos, Sep 21 2005
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^5 * b(p^(e-2)). - Michael Somos, Mar 08 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 64 (t/i)^6 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 24 2012
G.f.: (Product_{k>0} (1 - x^k))^12.
A000145(n) = A029751(n) + 16*a(n). - Michael Somos, Sep 21 2005
a(n) = (-1)^n * A209676(n).
Convolution inverse of A005758. Convolution square of A000729.
a(0) = 1, a(n) = -(12/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017
G.f.: exp(-12*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018

A286346 Expansion of eta(q)^24 / eta(q^2)^12 in powers of q.

Original entry on oeis.org

1, -24, 264, -1760, 7944, -25872, 64416, -133056, 253704, -472760, 825264, -1297056, 1938336, -2963664, 4437312, -6091584, 8118024, -11368368, 15653352, -19822176, 24832944, -32826112, 42517728, -51425088, 61903776, -78146664, 98021616, -115331264, 133522752

Offset: 0

Seiichi Manyama, May 08 2017

sign

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

Cf. A000145, A013973 (E_6).

nmax = 20; CoefficientList[Series[Product[((1 - x^k)/(1 + x^k))^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 10 2018 *)
a[n_] := (-1)^n SquaresR[12, n];
a /@ Range[0, 30] (* Jean-François Alcover, Feb 21 2021 *)

q = 'q + O('q^50); Vec(eta(q)^24 / eta(q^2)^12) \\ Michel Marcus, Jul 07 2018

a(n) = (-1)^n * A000145(n).

Euler Transform of [-24, -12, -24, -12, -24, -12, -24, -12, ...]. - Simon Plouffe, Jun 23 2018

A276285 Number of ways of writing n as a sum of 13 squares.

Original entry on oeis.org

1, 26, 312, 2288, 11466, 41808, 116688, 265408, 535704, 1031914, 1899664, 3214224, 5043376, 7801744, 12066912, 17689152, 24443978, 34039200, 48210760, 64966096, 83323344, 109157152, 145532816, 185245632, 227110416, 284788010, 363737712

Offset: 0

Ilya Gutkovskiy, Aug 27 2016

More generally, the ordinary generating function for the number of ways of writing n as a sum of k squares is theta_3(0, q)^k = 1 + 2*k*q + 2*(k - 1)*k*q^2 + (4/3)*(k - 2)*(k - 1)*k*q^3 + (2/3)*((k - 3)*(k - 2)*(k - 1) + 3)*k*q^4 + (4/15) *(k - 1)*k*(k^3 - 9*k^2 + 26*k - 9)*q^5 + ..., where theta is the Jacobi theta functions.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Sum of Squares Function
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

13th column of A286815. - Seiichi Manyama, May 27 2017
Row d=13 of A122141.
Cf. Number of ways of writing n as a sum of k squares: A004018 (k = 2), A005875 (k = 3), A000118 (k = 4), A000132 (k = 5), A000141 (k = 6), A008451 (k = 7), A000143 (k = 8), A008452 (k = 9), A000144 (k = 10), A008453 (k = 11), A000145 (k = 12), this sequence (k = 13), A000152 (k = 16).

Mathematica
```
Table[SquaresR[13, n], {n, 0, 26}]
```

G.f.: theta_3(0,q)^13, where theta_3(x,q) is the third Jacobi theta function.

a(n) = (26/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

A029751 Average theta series of odd unimodular lattices in dimension 12.

Original entry on oeis.org

1, 8, 248, 1952, 7928, 25008, 60512, 134464, 253688, 474344, 775248, 1288416, 1934432, 2970352, 4168384, 6101952, 8118008, 11358864, 14704664, 19808800, 24782928, 32809216, 39940896, 51490752, 61899872, 78150008, 92080912

Offset: 0

N. J. A. Sloane

nonn

R. A. Rankin, Modular Forms, p. 240 ff.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A000145, A000735.

a[0] = 1; a[n_] := (-1)^(n-1)*8*DivisorSum[n, (-1)^(n + n/#)*#^5&]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 06 2017, translated from PARI *)

a(n)=if(n<1, n==0, (-1)^(n-1)*8*sumdiv(n,d,(-1)^(n+n/d)*d^5)) /* Michael Somos, Sep 21 2005 */

G.f.: 1 + 8*Sum_{k>0} k^5 x^k/(1+(-x)^k). - Michael Somos, Sep 21 2005

A000145(n) = a(n) + 16*A000735(n). - Michael Somos, Sep 21 2005

A004413 Expansion of ( Sum_{n = -infinity..infinity} x^(n^2) )^(-12).

Original entry on oeis.org

1, -24, 312, -2912, 21816, -139152, 783328, -3986112, 18650424, -81251896, 332798544, -1291339296, 4776117216, -16922753616, 57683178432, -189821722688, 604884735288, -1871370360240, 5633654421720

Offset: 0

N. J. A. Sloane

sign

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

Cf. A000122, A000145.

nmax = 30; CoefficientList[Series[Product[((1 + (-x)^k)/(1 - (-x)^k))^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)

q='q+O('q^99); Vec(((eta(q)*eta(q^4))^2/eta(q^2)^5)^12) \\ Altug Alkan, Sep 20 2018

a(n) ~ (-1)^n * exp(Pi*sqrt(m*n)) * m^((m+1)/4) / (2^(3*(m+1)/2) * n^((m+3)/4)), set m = 12 for this sequence. - Vaclav Kotesovec, Aug 18 2015
From Ilya Gutkovskiy, Sep 20 2018: (Start)
G.f.: 1/theta_3(x)^12, where theta_3() is the Jacobi theta function.
G.f.: Product_{k>=1} 1/((1 - x^(2*k))*(1 + x^(2*k-1))^2)^12. (End)

cp's OEIS Frontend

A188671 A000145(n) / 8 - (n^5 + 1).

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A122141 Array: T(d,n) = number of ways of writing n as a sum of d squares, read by ascending antidiagonals.

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A286815 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>=1} (1 - x^(2*j))^5/((1 - x^j)*(1 - x^(4*j)))^2)^k.

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A000735 Expansion of Product_{k>=1} (1 - x^k)^12.

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A286346 Expansion of eta(q)^24 / eta(q^2)^12 in powers of q.

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A276285 Number of ways of writing n as a sum of 13 squares.

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A029751 Average theta series of odd unimodular lattices in dimension 12.

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A286815 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>=1} (1 - x^(2j))^5/((1 - x^j)(1 - x^(4*j)))^2)^k.