A050456 -id:A050456 - cp's OEIS frontend

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

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

A000144 Number of ways of writing n as a sum of 10 squares.

Original entry on oeis.org

1, 20, 180, 960, 3380, 8424, 16320, 28800, 52020, 88660, 129064, 175680, 262080, 386920, 489600, 600960, 840500, 1137960, 1330420, 1563840, 2050344, 2611200, 2986560, 3358080, 4194240, 5318268, 5878440, 6299520, 7862400, 9619560

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 20*x + 180*x^2 + 960*x^3 + 3380*x^4 + 8424*x^5 + 16320*x^6 + ...

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.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York 1959, p. 135 section 9.3. MR0106147 (21 #4881)

T. D. Noe, Table of n, a(n) for n = 0..10000
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.
J. Liouville, Nombre des représentations d’un entier quelconque sous la forme d’une somme de dix carrés, Journal de mathématiques pures et appliquées 2e série, tome 11 (1866), p. 1-8.
Index entries for sequences related to sums of squares

Row d=10 of A122141 and of A319574, 10th column of A286815.

(sum(x^(m^2),m=-10..10))^10;
# Alternative:
A000144list := proc(len) series(JacobiTheta3(0, x)^10, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000144list(30); # Peter Luschny, Oct 02 2018

Table[SquaresR[10, n], {n, 0, 30}] (* Ray Chandler, Jun 29 2008; updated by T. D. Noe, Jan 23 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^10, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^k)^10 * (1 + x^k)^30 / (1 + x^(2*k))^20, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 24 2017 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^10, n))}; /* Michael Somos, Sep 12 2005 */

Q = DiagonalQuadraticForm(ZZ, [1]*10)
Q.representation_number_list(37) # Peter Luschny, Jun 20 2014

Euler transform of period 4 sequence [ 20, -30, 20, -10, ...]. - Michael Somos, Sep 12 2005
Expansion of eta(q^2)^50 / (eta(q) * eta(q^4))^20 in powers of q. - Michael Somos, Sep 12 2005
a(n) = 4/5 * (A050456(n) + 16*A050468(n) + 8*A030212(n)) if n>0. - Michael Somos, Sep 12 2005
a(n) = (20/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

A322143 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, d==1 (mod 4)} d^k - Sum_{d|n, d==3 (mod 4)} d^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, -2, 1, 1, 1, -8, 1, 2, 1, 1, -26, 1, 6, 0, 1, 1, -80, 1, 26, -2, 0, 1, 1, -242, 1, 126, -8, -6, 1, 1, 1, -728, 1, 626, -26, -48, 1, 1, 1, 1, -2186, 1, 3126, -80, -342, 1, 7, 2, 1, 1, -6560, 1, 15626, -242, -2400, 1, 73, 6, 0, 1, 1, -19682, 1, 78126, -728, -16806, 1, 703, 26, -10, 0

Offset: 1

Ilya Gutkovskiy, Nov 28 2018

			Square array begins:
  1,  1,   1,    1,    1,     1,  ...
  1,  1,   1,    1,    1,     1,  ...
  0, -2,  -8,  -26,  -80,  -242,  ...
  1,  1,   1,    1,    1,     1,  ...
  2,  6,  26,  126,  626,  3126,  ...
  0, -2,  -8,  -26,  -80,  -242,  ...

Index entries for sequences mentioned by Glaisher

Columns k=0..12 give A002654, A050457, A002173, A050459, A050456, A321821, A321822, A321823, A321824, A321825, A321826, A321827, A321828.

Cf. A109974, A322081, A322082, A322083, A322084.

Table[Function[k, SeriesCoefficient[Sum[(-1)^(j - 1) (2 j - 1)^k x^(2 j - 1)/(1 - x^(2 j - 1)), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

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

A050459 a(n) = Sum_{d|n, d==1 mod 4} d^3 - Sum_{d|n, d==3 mod 4} d^3.

Original entry on oeis.org

1, 1, -26, 1, 126, -26, -342, 1, 703, 126, -1330, -26, 2198, -342, -3276, 1, 4914, 703, -6858, 126, 8892, -1330, -12166, -26, 15751, 2198, -18980, -342, 24390, -3276, -29790, 1, 34580, 4914, -43092, 703, 50654, -6858, -57148, 126, 68922, 8892, -79506, -1330

Offset: 1

N. J. A. Sloane, Dec 23 1999

Multiplicative because it is the Inverse Möbius transform of [1 0 -3^3 0 5^3 0 -7^3 ...], which is multiplicative. - Christian G. Bower, May 18 2005

Seiichi Manyama, Table of n, a(n) for n = 1..10000
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. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Column k=3 of A322143.
Glaisher's E_i (i=0..12): A002654, A050457, A002173, A050459, A050456, A321821, A321822, A321823, A321824, A321825, A321826, A321827, A321828.
Cf. A050451, A050454.

A050459 := proc(n) local a; a := 0 ; for d in numtheory[divisors](n) do if d mod 4 = 1 then a := a+d^3 ; elif d mod 4 = 3 then a := a-d^3 ; end if; end do;  a ; end proc:
seq(A050459(n),n=1..100) ; # R. J. Mathar, Jan 07 2011

s[n_, r_] := DivisorSum[n, #^3 &, Mod[#, 4]==r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Dec 06 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^3)^(e+1)-1)/(p^3-1), ((-p^3)^(e+1)-1)/(-p^3-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)

a(n) = A050451(n) - A050454(n).
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^3*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^3)^(e+1)-1)/(p^3-1) if p == 1 (mod 4) and ((-p^3)^(e+1)-1)/(-p^3-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^3*sin(d*Pi/2). - Ridouane Oudra, Jun 02 2024

A204372 Expansion of phi(x)^2 * (5 * phi(-x)^8 + 64 * x * psi(-x)^8) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

5, 4, 4, -320, 4, 2504, -320, -9600, 4, 25924, 2504, -58560, -320, 114248, -9600, -200320, 4, 334088, 25924, -521280, 2504, 768000, -58560, -1119360, -320, 1565004, 114248, -2099840, -9600, 2829128, -200320, -3694080, 4, 4684800

Offset: 0

Michael Somos, Jan 14 2012

sign

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

			G.f. = 5 + 4*x + 4*x^2 - 320*x^3 + 4*x^4 + 2504*x^5 - 320*x^6 - 9600*x^7 + 4*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A050456, A050468.

A := Basis( ModularForms( Gamma1(4), 5), 34); 5*A[1] + 4*A[2] + 4*A[3]; /* Michael Somos, May 04 2015 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2 (5 EllipticTheta[ 4, 0, q]^8 + 4 EllipticTheta[ 2, Pi/4, q^(1/2)]^8), {q, 0, n}]; (* Michael Somos, May 03 2015 *)
a[ n_] := If[ n < 1, 5 Boole[n == 0], 4 DivisorSum[ n, #^4 KroneckerSymbol[ -4, #] &]]; (* Michael Somos, May 04 2015 *)

{a(n) = if( n<1, 5 * (n==0), 4 * sumdiv( n, d, d^4 * kronecker( -4, d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^2 + A)^2 * (5 * eta(x + A)^8 / eta(x^4 + A)^4 + 64 * x * eta(x^4 + A)^4 ), n))};

Expansion of eta(q)^4 * eta(q^2)^2 * (5 * eta(q)^8 / eta(q^4)^4 + 64 * q * eta(q^4)^4 ) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4*t)) = 2048 (t/i)^5 g(t) where q = exp(2*Pi*i*t) and g(t) is the g.f. for A050468.
G.f.: 5 + 4 * Sum_{k>0} (-1)^(k-1) * (2*k - 1)^4 * x^(2*k - 1) / (1 - x^(2*k - 1)).
a(n) = 4 * A050456(n) if n>0.

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

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A000144 Number of ways of writing n as a sum of 10 squares.

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A322143 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, d==1 (mod 4)} d^k - Sum_{d|n, d==3 (mod 4)} d^k.

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A050459 a(n) = Sum_{d|n, d==1 mod 4} d^3 - Sum_{d|n, d==3 mod 4} d^3.

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A204372 Expansion of phi(x)^2 * (5 * phi(-x)^8 + 64 * x * psi(-x)^8) in powers of x where phi(), psi() are Ramanujan theta functions.

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