A000152 -id:A000152

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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.

+10
29

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

Alois P. Heinz, Antidiagonals d = 1..141, flattened

Wikipedia, Sum of squares function (transposed)

Index entries for sequences related to sums of squares

FORMULA

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

EXAMPLE

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 ...

MAPLE

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

MATHEMATICA

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 *)

PROG

(Python)

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

CROSSREFS

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).

KEYWORD

nonn,tabl

AUTHOR

R. J. Mathar, Oct 29 2006

STATUS

approved

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.

+10
26

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

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.

LINKS

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Wikipedia, Sum of squares function

FORMULA

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

EXAMPLE

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, ...

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

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.

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, May 27 2017

STATUS

approved

A276285

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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.

LINKS

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

FORMULA

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

MATHEMATICA

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

CROSSREFS

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).

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Aug 27 2016

STATUS

approved

A195391

(7*n)!

+10
5

1, 5040, 87178291200, 51090942171709440000, 304888344611713860501504000000, 10333147966386144929666651337523200000000, 1405006117752879898543142606244511569936384000000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..60

PROG

(Magma) [Factorial(7*n): n in [0..10]]

(PARI) a(n)=(7*n)! \\ Charles R Greathouse IV, Dec 27 2011

CROSSREFS

Cf. A000152, A010050, A100732, A100733, A100734, A195390, A271947.

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Sep 24 2011

STATUS

approved

A319307

Expansion of theta_4(q)^16 in powers of q = exp(Pi i t).

+10
5

1, -32, 480, -4480, 29152, -140736, 525952, -1580800, 3994080, -8945824, 18626112, -36714624, 67978880, -118156480, 197120256, -321692928, 509145568, -772845120, 1143441760, -1681379200, 2428524096, -3392205824, 4658843520, -6411152640, 8705492608, -11488092896

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

Expansion of eta(q)^32 / eta(q^2)^16 in powers of q.

CROSSREFS

theta_4(q)^b: A002448 (b=1), A104794 (b=2), A213384 (b=3), A096727 (b=4), A035016 (b=8), A286346 (b=12), this sequence (b=16), A319308 (b=20), A319309 (b=24), A319310 (b=28).

Cf. A000152.

KEYWORD

sign

AUTHOR

Seiichi Manyama, Sep 16 2018

STATUS

approved

A022047

Theta series of D_16 lattice.

+10
4

1, 480, 29152, 525952, 3994080, 18626112, 67978880, 197120256, 509145568, 1143441760, 2428524096, 4658843520, 8705492608, 15000919872, 25593401600, 40864033536, 65187410400, 98098623936, 148724365664, 213678303360, 310833534528, 430766443520, 605650030464, 813970121472

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 118.

LINKS

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

MATHEMATICA

terms = 18; s = EllipticTheta[3, 0, x]^16 + O[x]^(2 terms); Partition[ CoefficientList[s, x], 2][[All, 1]][[1 ;; terms]] (* Jean-François Alcover, Jul 06 2017 *)

CROSSREFS

a(n) = A000152(2*n).

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Seiichi Manyama, Oct 16 2018

STATUS

approved

A195392

(8*n)!

+10
3

1, 40320, 20922789888000, 620448401733239439360000, 263130836933693530167218012160000000, 815915283247897734345611269596115894272000000000, 12413915592536072670862289047373375038521486354677760000000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..55

PROG

(Magma) [Factorial(8*n): n in [0..10]]

CROSSREFS

Cf. A000152, A010050, A100732, A100733, A100734, A195390, A195391

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Sep 24 2011

STATUS

approved

A302855

Expansion of ((1 + 2 * Sum_{k>=1} q^(k^2))^16 - 1) / 32.

+10
3

1, 15, 140, 911, 4398, 16436, 49400, 124815, 279557, 582066, 1147332, 2124340, 3692390, 6160008, 10052904, 15910799, 24151410, 35732555, 52543100, 75891378, 106006432, 145588860, 200348520, 272046644, 359002903, 468778746, 615548600, 799793800, 1014602070, 1277001048

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

a(n) = A000152(n) / 32.

CROSSREFS

Expansion of ((1 + 2 * Sum_{k>=1} q^(k^2))^(2^m) - 1) / 2^(m + 1): A002654 (m=1), A046897 (m=2), A008457 (m=3), this sequence (m=4), A302857 (m=5).

Cf. A000152, A186690.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Apr 14 2018

STATUS

approved

A195393

a(n) = (9*n)!.

+10
2

1, 362880, 6402373705728000, 10888869450418352160768000000, 371993326789901217467999448150835200000000, 119622220865480194561963161495657715064383733760000000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..50

MATHEMATICA

(9Range[0, 10])! (* Harvey P. Dale, Jan 25 2023 *)

PROG

(Magma) [Factorial(9*n): n in [0..10]]

CROSSREFS

Cf. A000152, A010050, A100732, A100733, A100734, A195390, A195391, A195392.

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Sep 24 2011

STATUS

approved

A004417

Expansion of (Sum x^(n^2), n = -inf .. inf )^(-16).

+10
1

1, -32, 544, -6528, 61984, -495040, 3453312, -21581568, 123040288, -648624288, 3194776000, -14823993472, 65231647104, -273714726080, 1100198199040, -4252621927680, 15859616674336, -57229459033664

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

a(n) ~ (-1)^n * exp(Pi*sqrt(m*n)) * m^((m+1)/4) / (2^(3*(m+1)/2) * n^((m+3)/4)), set m = 16 for this sequence. - Vaclav Kotesovec, Aug 18 2015

From Ilya Gutkovskiy, Sep 20 2018: (Start)

G.f.: 1/theta_3(x)^16, where theta_3() is the Jacobi theta function.

G.f.: Product_{k>=1} 1/((1 - x^(2*k))*(1 + x^(2*k-1))^2)^16. (End)

MATHEMATICA

nmax = 30; CoefficientList[Series[Product[((1 + (-x)^k)/(1 - (-x)^k))^16, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)

PROG

(PARI) q='q+O('q^99); Vec(((eta(q)*eta(q^4))^2/eta(q^2)^5)^16) \\ Altug Alkan, Sep 20 2018

CROSSREFS

Cf. A000122, A000152.

KEYWORD

sign

AUTHOR

N. J. A. Sloane

STATUS

approved

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