A319574 -id:A319574 - cp's OEIS frontend

A008453 Number of ways of writing n as a sum of 11 squares.

1, 22, 220, 1320, 5302, 15224, 33528, 63360, 116380, 209550, 339064, 491768, 719400, 1095160, 1538416, 1964160, 2624182, 3696880, 4763220, 5686648, 7217144, 9528816, 11676280, 13495680, 16317048, 20787470, 25022184, 27785120, 32503680

Offset: 0

N. J. A. Sloane

nonn

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.

T. D. Noe, Table of n, a(n) for n = 0..10000
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
Shaun Cooper, On the number of representations of certain integers as sums of 11 or 13 squares, J. Number Theory 103 (2) (2003) 135-162
Index entries for sequences related to sums of squares

Row d=11 of A122141 and of A319574, 11th column of A286815.

Cf. A022042.

(sum(x^(m^2),m=-10..10))^11;
# Alternative:
A008453list := proc(len) series(JacobiTheta3(0, x)^11, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A008453list(29); # Peter Luschny, Oct 02 2018

Table[SquaresR[11, n], {n, 0, 28}] (* Ray Chandler, Nov 28 2006 *)

G.f.: theta_3(0,q)^11, where theta_3 is the 3rd Jacobi theta function. - Ilya Gutkovskiy, Jan 13 2017

a(n) = (22/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Extended by Ray Chandler, Nov 28 2006

A319575 a(n) = (2/3)n(n^3 - 6n^2 + 11n - 3).

Original entry on oeis.org

0, 2, 4, 6, 24, 90, 252, 574, 1136, 2034, 3380, 5302, 7944, 11466, 16044, 21870, 29152, 38114, 48996, 62054, 77560, 95802, 117084, 141726, 170064, 202450, 239252, 280854, 327656, 380074, 438540, 503502, 575424, 654786, 742084, 837830, 942552, 1056794, 1181116

Offset: 0

Peter Luschny, Oct 01 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000012 (m=0), A005843 (m=1), A046092 (m=2), A130809 (m=3), this sequence (m=4), A319576 (m=5), A319577 (m=6).
Column n=4 of A122141.
Cf. A319574.

a := n -> (2/3)*n*(n^3 - 6*n^2 + 11*n - 3):
seq(a(n), n=0..38);

A319575[n_] := 2/3*n*(n^3-6*n^2+11*n-3); Array[A319575, 50, 0] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 2, 4, 6, 24}, 50] (* Paolo Xausa, Feb 20 2024 *)

concat(0, Vec(2*x*(1 + x)*(1 - 4*x + 7*x^2) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Oct 02 2018

a(n) = [x^4] JacobiTheta3(x)^n.
a(n) = A319574(n,4).
From Colin Barker, Oct 02 2018: (Start)
G.f.: 2*x*(1 + x)*(1 - 4*x + 7*x^2) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4. (End)

A319576 a(n) = (4/15)n(n - 1)(n^3 - 9n^2 + 26*n - 9).

Original entry on oeis.org

0, 0, 8, 24, 48, 112, 312, 840, 2016, 4320, 8424, 15224, 25872, 41808, 64792, 96936, 140736, 199104, 275400, 373464, 497648, 652848, 844536, 1078792, 1362336, 1702560, 2107560, 2586168, 3147984, 3803408, 4563672, 5440872, 6448000, 7598976, 8908680, 10392984

Offset: 0

Peter Luschny, Oct 01 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000012 (m=0), A005843 (m=1), A046092 (m=2), A130809 (m=3), A319575 (m=4), this sequence (m=5), A319577 (m=6).
Column n=5 of A122141.
Cf. A319574.

a := n -> (4/15)*n*(n - 1)*(n^3 - 9*n^2 + 26*n - 9):
seq(a(n), n=0..41);

A319576[n_] := 4/15*n*(n-1)*(n^3-9*n^2+26*n-9); Array[A319576, 50, 0] (* or *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 8, 24, 48, 112}, 50] (* Paolo Xausa, Feb 20 2024 *)

concat([0,0], Vec(8*x^2*(1 - 3*x + 3*x^2 + 3*x^3) / (1 - x)^6 + O(x^40))) \\ Colin Barker, Oct 02 2018

a(n) = [x^5] JacobiTheta3(x)^n.
a(n) = A319574(n,5).
From Colin Barker, Oct 02 2018: (Start)
G.f.: 8*x^2*(1 - 3*x + 3*x^2 + 3*x^3) / (1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5.
(End)

A319577 a(n) = (4/45)n(n - 2)(n - 1)(n^3 - 12n^2 + 47n - 15).

Original entry on oeis.org

0, 0, 0, 24, 96, 240, 544, 1288, 3136, 7392, 16320, 33528, 64416, 116688, 200928, 331240, 525952, 808384, 1207680, 1759704, 2508000, 3504816, 4812192, 6503112, 8662720, 11389600, 14797120, 19014840, 24189984, 30488976, 38099040, 47229864, 58115328, 71015296

Offset: 0

Peter Luschny, Oct 01 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000012 (m=0), A005843 (m=1), A046092 (m=2), A130809 (m=3), A319575 (m=4), A319576 (m=5), this sequence (m=6).
Column n=6 of A122141.
Cf. A319574.

a := n -> (4/45)*n*(n - 2)*(n - 1)*(n^3 - 12*n^2 + 47*n - 15):
seq(a(n), n=0..41);

A319577[n_]:=4/45*n*(n-2)*(n-1)*(n^3-12*n^2+47*n-15); Array[A319577, 50, 0] (*or*)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 0, 24, 96, 240, 544}, 50] (* Paolo Xausa, Feb 20 2024 *)

concat([0,0,0], Vec(8*x^3*(3 - 9*x + 9*x^2 + 5*x^3) / (1 - x)^7 + O(x^40))) \\ Colin Barker, Oct 02 2018

a(n) = [x^6] JacobiTheta3(x)^n.
a(n) = A319574(n,6).
From Colin Barker, Oct 02 2018: (Start)
G.f.: 8*x^3*(3 - 9*x + 9*x^2 + 5*x^3) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6.
(End)

A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, -1, -2, 1, 0, 0, -1, -3, 1, 0, 0, 2, 0, -4, 1, 0, 1, 1, 5, 2, -5, 1, 0, 0, 2, 0, 8, 5, -6, 1, 0, 1, -2, 0, -5, 10, 9, -7, 1, 0, 0, 0, -7, -4, -15, 10, 14, -8, 1, 0, 0, -2, 0, -10, -6, -30, 7, 20, -9, 1, 0, 0, -2, 0, 8, -5, 0, -49, 0, 27, -10, 1

Offset: 0

Peter Luschny, Oct 02 2018

The columns are generated by polynomials whose coefficients constitute the triangle of signed D'Arcais numbers A078521 when multiplied with n!.

			[ 0] 1,   0,   0,    0,     0,    0,     0,     0,     0,     0, ... A000007
[ 1] 1,  -1,  -1,    0,     0,    1,     0,     1,     0,     0, ... A010815
[ 2] 1,  -2,  -1,    2,     1,    2,    -2,     0,    -2,    -2, ... A002107
[ 3] 1,  -3,   0,    5,     0,    0,    -7,     0,     0,     0, ... A010816
[ 4] 1,  -4,   2,    8,    -5,   -4,   -10,     8,     9,     0, ... A000727
[ 5] 1,  -5,   5,   10,   -15,   -6,    -5,    25,    15,   -20, ... A000728
[ 6] 1,  -6,   9,   10,   -30,    0,    11,    42,     0,   -70, ... A000729
[ 7] 1,  -7,  14,    7,   -49,   21,    35,    41,   -49,  -133, ... A000730
[ 8] 1,  -8,  20,    0,   -70,   64,    56,     0,  -125,  -160, ... A000731
[ 9] 1,  -9,  27,  -12,   -90,  135,    54,   -99,  -189,   -85, ... A010817
[10] 1, -10,  35,  -30,  -105,  238,     0,  -260,  -165,   140, ... A010818
    A001489,  v , A167541, v , A319931,  v ,         diagonal: A008705
           A080956       A319930      A319932

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003.

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Vaclav Kotesovec, The integration of q-series
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
M. Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Transpose of A286354.

Cf. A078521, A319574 (JacobiTheta3).

# DedekindEta is defined in A000594
for n in 0:10
    DedekindEta(10, n) |> println
end

DedekindEta := (x, n) -> mul(1-x^j, j=1..n):
A319933row := proc(n, len) series(DedekindEta(x, len)^n, x, len+1):
seq(coeff(%, x, j), j=0..len-1) end:
seq(print([n], A319933row(n, 10)), n=0..10);

eta[x_, n_] := Product[1 - x^j, {j, 1, n}];
A[n_, k_] := SeriesCoefficient[eta[x, k]^n, {x, 0, k}];
Table[A[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)

from sage.modular.etaproducts import qexp_eta
def A319933row(n, len):
    return (qexp_eta(ZZ['q'], len+4)^n).list()[:len]
for n in (0..10):
    print(A319933row(n, 10))

A000156 Number of ways of writing n as a sum of 24 squares.

Original entry on oeis.org

1, 48, 1104, 16192, 170064, 1362336, 8662720, 44981376, 195082320, 721175536, 2319457632, 6631997376, 17231109824, 41469483552, 93703589760, 200343312768, 407488018512, 793229226336, 1487286966928, 2697825744960, 4744779429216

Offset: 0

N. J. A. Sloane

The Carlitz paper has the wrong formula on p. 505, eq. (3). The factor in front of tau(n/2) should be -2^16 (not -2^12). The mistake appeared in the previous equation derived from eq. (2) where theta_3^(24) * 256*k^4*k'^4 was replaced by 2^8*g(q^2) which produces the factor 2^8*256 = 2^16. (One should subtract on p. 504 the second equation in the middle from the negative of the first equation. There is also a sign mistake in the sum term of the third equation from the bottom.) - Wolfdieter Lang, Sep 24 2016

Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 195, eq. (15.1).
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 107.
G. H. Hardy, Ramanujan, 1940, Cambridge, reprinted with additional corrections and comments by AMS Chelsea Publishing, 1999, 2002, Providence, Rhode Island, ch. IX., pp. 153-155.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.

T. D. Noe, Table of n, a(n) for n = 0..10000
L. Carlitz, On the representation of an integer as the sum of twenty-four squares, Indagationes Mathematicae (Proceedings), 58 (1955) 504-506.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
Index entries for sequences related to sums of squares

Row d=24 of A122141 and of A319574, 24th column of A286815.

(sum(x^(m^2),m=-10..10))^24; seq(coeff(%,x,n), n=0..30);
# Alternative:
A000156list := proc(len) series(JacobiTheta3(0, x)^24, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000156list(21); # Peter Luschny, Oct 02 2018

Table[SquaresR[24, n], {n, 0, 20}] (* Ray Chandler, Nov 28 2006 *)

first(n)=my(x='x); x+=O(x^(n+1)); Vec((2*sum(k=1,sqrtint(n),x^k^2) + 1)^24) \\ Charles R Greathouse IV, Jul 29 2016

From Wolfdieter Lang, Sep 24 2016: (Start)
For n >= 1: a(n) = (16*sigma^*{11} - 128*(512*tau(n/2) + (-1)^n*259*tau(n)))/691, with sigma^*{11} = sigma_{11}^{e}(n) - sigma_{11}^{o}(n) if n even and sigma_{11}(n) otherwise. Here sigma_{11}(n) = A013959(n) and 0 if n is not an integer, sigma_{11}^{e}(n) and sigma_{11}^{o}(n) are the sums of the 11th power of the odd and even positive divisors of n, respectively. Ramanujan's tau(n) = A000594(n) and 0 if n is not an integer. This is from Hardy, ch. IX., p. 155, eqs. (9.17.1) and (9.17.2), and p.142 for the definition of sigma^*_{nu}(n). See also the Ash and Gross reference.
Another version, see the corrected Carlitz reference:
a(n) = (2^4*(sigma_{11}(n)- 2*sigma_{11}(n/2) + 2^{12}*sigma_{11}(n/4)) - 2^7*259*(-1)^n*tau(n) - 2^16*tau(n/2))/691, n >= 1.
(End)
a(n) = (48/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Extended by Ray Chandler, Nov 28 2006

A319935 T(n,k) = [x^n] JacobiTheta3(0,x)^k, for 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, 0, 2, 0, 0, 4, 0, 0, 0, 8, 0, 2, 4, 6, 24, 0, 0, 8, 24, 48, 112, 0, 0, 0, 24, 96, 240, 544, 0, 0, 0, 0, 64, 320, 960, 2368, 0, 0, 4, 12, 24, 200, 1020, 3444, 9328, 0, 2, 4, 30, 104, 250, 876, 3542, 12112, 34802, 0, 8, 24, 144, 560, 1560, 4424, 14112, 44640, 129064, 339064

Offset: 0

Peter Luschny, Oct 06 2018

			Triangle starts:
[0] 1
[1] 0, 2
[2] 0, 0, 4
[3] 0, 0, 0,  8
[4] 0, 2, 4,  6,  24
[5] 0, 0, 8, 24,  48, 112
[6] 0, 0, 0, 24,  96, 240,  544
[7] 0, 0, 0,  0,  64, 320,  960, 2368
[8] 0, 0, 4, 12,  24, 200, 1020, 3444,  9328
[9] 0, 2, 4, 30, 104, 250,  876, 3542, 12112, 34802

T(n,n) = A066535(n), row sums A320025.

Cf. A319574, A319934.

A319935row := proc(n) local ser;
ser := j -> series(JacobiTheta3(0, x)^j, x, n+1);
seq(coeff(ser(j), x, n), j=0..n) end:
seq(A319935row(n), n=0..10);

cp's OEIS Frontend

A008453 Number of ways of writing n as a sum of 11 squares.

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A319575 a(n) = (2/3)*n*(n^3 - 6*n^2 + 11*n - 3).

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A319576 a(n) = (4/15)*n*(n - 1)*(n^3 - 9*n^2 + 26*n - 9).

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A319577 a(n) = (4/45)*n*(n - 2)*(n - 1)*(n^3 - 12*n^2 + 47*n - 15).

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A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

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A000156 Number of ways of writing n as a sum of 24 squares.

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Maple

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A319935 T(n,k) = [x^n] JacobiTheta3(0,x)^k, for 0 <= k <= n, triangle read by rows.

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A319575 a(n) = (2/3)n(n^3 - 6n^2 + 11n - 3).

A319576 a(n) = (4/15)n(n - 1)(n^3 - 9n^2 + 26*n - 9).

A319577 a(n) = (4/45)n(n - 2)(n - 1)(n^3 - 12n^2 + 47n - 15).