cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A298938 Number of ordered ways of writing n^3 as a sum of n squares of nonnegative integers.

Original entry on oeis.org

1, 1, 1, 4, 5, 686, 13942, 455988, 13617853, 454222894, 18323165948, 802161109047, 42149084452070, 2481730049781672, 157265294178424356, 10977302934685469078, 812821237985857557677, 64539935903231450294134, 5504599828399250884049308
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2018

Keywords

Examples

			a(4) = 5 because we have [64, 0, 0, 0], [16, 16, 16, 16], [0, 64, 0, 0], [0, 0, 64, 0] and [0, 0, 0, 64].

Links

Crossrefs

Cf. A000290, A000578, A006456, A030273, A037444, A066535, A218494, A232173, A287617, A298329, A298330, A298935, A298936, A298939.

Programs

  • Mathematica
    Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^3}], {n, 0, 18}]

Formula

a(n) = [x^(n^3)] (Sum_{k>=0} x^(k^2))^n.

A298939 Number of ordered ways of writing n^3 as a sum of n squares of positive integers.

Original entry on oeis.org

1, 1, 1, 4, 1, 286, 7582, 202028, 6473625, 226029577, 8338249868, 391526193477, 19990594900630, 1159906506684446, 74890158861242740, 5119732406649036418, 380146984328280974281, 30198665638519565614034, 2555354508318427693497565
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2018

Keywords

Examples

			a(3) = 4 because we have [25, 1, 1], [9, 9, 9], [1, 25, 1] and [1, 1, 25].

Links

Crossrefs

Cf. A000290, A000578, A006456, A030273, A037444, A066535, A218494, A232173, A287617, A298329, A298330, A298935, A298937, A298938.

Programs

  • Mathematica
    Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^3}], {n, 0, 18}]

Formula

a(n) = [x^(n^3)] (Sum_{k>=1} x^(k^2))^n.

A299032 Number of ordered ways of writing n-th triangular number as a sum of n squares of positive integers.

Original entry on oeis.org

1, 1, 0, 3, 6, 0, 12, 106, 420, 2718, 18240, 120879, 694320, 5430438, 40668264, 300401818, 2369504386, 19928714475, 174151735920, 1543284732218, 14224347438876, 135649243229688, 1331658133954940, 13369350846412794, 138122850643702056, 1462610254141337590
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 01 2018

Keywords

Examples

			a(4) = 6 because fourth triangular number is 10 and we have [4, 4, 1, 1], [4, 1, 4, 1], [4, 1, 1, 4], [1, 4, 4, 1], [1, 4, 1, 4] and [1, 1, 4, 4].

Links

Crossrefs

Cf. A000217, A000290, A066535, A072964, A104383, A126683, A196010, A224677, A224679, A278340, A288126, A298330, A298858, A298939, A299031.

Programs

  • Maple
    b:= proc(n, t) option remember; local i; if n=0 then
      `if`(t=0, 1, 0) elif t<1 then 0 else 0;
      for i while i^2<=n do %+b(n-i^2, t-1) od; % fi
    end:
a:= n-> b(n*(n+1)/2, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 05 2018
  • Mathematica
    Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n (n + 1)/2}], {n, 0, 25}]

Formula

a(n) = [x^(n*(n+1)/2)] (Sum_{k>=1} x^(k^2))^n.

A303333 a(n) = [x^n] (theta_3(x^(1/2))^n + theta_4(x^(1/2))^n)/2, where theta_3() and theta_4() are the Jacobi theta functions.

Original entry on oeis.org

1, 0, 4, 24, 24, 560, 2080, 11088, 74864, 343536, 2050344, 11676280, 61903776, 363737712, 2022013760, 11335886864, 65187410400, 365627715968, 2085523894756, 11894205734280, 67517852274384, 386394626371680, 2205027379874400, 12602057718873040, 72195482578935488, 413235574714857360
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 21 2018

Keywords

Links

Crossrefs

Main diagonal of A297331.
Cf. A066535.

Programs

  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x^(1/2)]^n + EllipticTheta[4, 0, x^(1/2)]^n)/2, {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n, {x, 0, 2 n}], {n, 0, 25}]
Table[SeriesCoefficient[EllipticTheta[3, 0, Sqrt[x]]^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jun 26 2019 *)

Formula

a(n) = A297331(n,n).
a(n) ~ c * d^n / sqrt(n), where d = 5.84456473064455581274428417... and c = 0.14104739588693592503498... - Vaclav Kotesovec, Jun 26 2019

A338464 Number of ways to write 2*n as an ordered sum of n squares of positive integers.

Original entry on oeis.org

1, 0, 0, 3, 0, 0, 15, 0, 8, 84, 0, 110, 495, 0, 1092, 3018, 120, 9520, 18870, 2907, 77520, 120270, 43890, 606188, 780023, 531300, 4620200, 5161377, 5651100, 34622172, 35045340, 55234560, 256503672, 245772464, 508930224, 1886151225, 1788167610, 4491607230
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 31 2021

Keywords

Comments

Also number of ways to write n as an ordered sum of n nonnegative numbers one less than a square.

Links

Crossrefs

Cf. A000290, A005563, A066535, A111178, A287617, A303333, A337165.

Programs

  • Maple
    b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..39);  # Alois P. Heinz, Feb 04 2021
  • Mathematica
    Table[SeriesCoefficient[((EllipticTheta[3, 0, x] - 1)/2)^n, {x, 0, 2 n}], {n, 0, 37}]

Formula

a(n) = [x^(2*n)] ((theta_3(x) - 1) / 2)^n, where theta_3() is the Jacobi theta function.
a(n) = [x^n] (Sum_{k>=0} x^(k*(k + 2)))^n.
a(n) = A337165(2n,n). - Alois P. Heinz, Feb 04 2021

A299031 Number of ordered ways of writing n-th triangular number as a sum of n squares of nonnegative integers.

Original entry on oeis.org

1, 1, 0, 3, 18, 60, 252, 1576, 10494, 64152, 458400, 3407019, 27713928, 225193982, 1980444648, 17626414158, 165796077562, 1593587604441, 15985672426992, 163422639872978, 1729188245991060, 18743981599820280, 208963405365941380, 2378065667103672024, 27742569814633730608
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 01 2018

Keywords

Examples

			a(3) = 3 because third triangular number is 6 and we have [4, 1, 1], [1, 4, 1] and [1, 1, 4].

Links

Crossrefs

Cf. A000217, A000290, A066535, A072964, A104383, A126683, A196010, A224677, A224679, A278340, A288126, A298329, A298858, A298938, A299032.

Programs

  • Mathematica
    Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n (n + 1)/2}], {n, 0, 24}]

Formula

a(n) = [x^(n*(n+1)/2)] (Sum_{k>=0} x^(k^2))^n.

A303172 Number of ordered ways of writing n as a sum of n square pyramidal numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 31, 106, 281, 631, 1306, 2806, 6931, 19306, 55070, 150816, 391161, 977501, 2426071, 6141865, 16000186, 42465571, 112950916, 297793651, 776866355, 2015237231, 5233754306, 13668689206, 35908153534, 94633042267, 249398115466, 656105299636, 1723150461561
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 19 2018

Keywords

Links

Crossrefs

Main diagonal of A290430.
Cf. A000330, A066535, A279220, A287617, A303170.

Programs

  • Mathematica
    Table[SeriesCoefficient[Sum[x^(k (k + 1) (2 k + 1)/6), {k, 0, n}]^n, {x, 0, n}], {n, 0, 32}]

Formula

a(n) = [x^n] (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^n.
a(n) = A290430(n,n).

A361431 Number of ways to write n^2 as an ordered sum of n^2 squares of integers.

Original entry on oeis.org

1, 2, 24, 34802, 509145568, 142743029326162, 715761543475698773496, 63014651062141097287201438690, 96683719664587866428237173383906926464, 2573179910450886540215919614478751310457090316706, 1184101051443285881265166362742300236491599013268534224381864
Offset: 0

Views

Author

Alois P. Heinz, Mar 11 2023

Keywords

Examples

			a(2) = 24: 4 = x^2 + y^2 + z^2 + u^2 has 24 solutions (x,y,z,u): 16 permutations of (+/-1,+/-1,+/-1,+/-1) and 8 permutations of (+/-2,0,0,0).

Links

Crossrefs

Cf. A000290, A066535.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
      b(n, t-1) +2*add(b(n-j^2, t-1), j=1..isqrt(n))))
    end:
a:= n-> b(n^2$2):
seq(a(n), n=0..10);

Formula

a(n) = [x^(n^2)] (Sum_{j=-oo..oo} x^(j^2))^(n^2).
a(n) = A066535(n^2).

A294592 a(n) = [x^n] (theta_3(x)/theta_4(x))^n, where theta_() is the Jacobi theta function.

Original entry on oeis.org

1, 4, 32, 304, 3072, 32024, 340352, 3666016, 39878656, 437091892, 4819567552, 53401892240, 594093969408, 6631726263608, 74242911364864, 833237193123104, 9371924860764160, 105614054423502408, 1192210691317862048, 13478559927485340144, 152589996020498655232, 1729590806617202662528
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 03 2017

Keywords

Links

Crossrefs

Cf. A007096, A014969, A014970, A014972, A066535, A080054, A103261, A270919, A289522, A291697.

Programs

  • Maple
    S:= series((JacobiTheta3(0,x)/JacobiTheta4(0,x))^n,x,51):
seq(coeff(S,x,n),n=0..50); # Robert Israel, Nov 03 2017
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x]/EllipticTheta[4, 0, x])^n, {x, 0, n}], {n, 0, 21}]
Table[SeriesCoefficient[Product[((1 + x^(2 k + 1))/(1 - x^(2 k + 1)))^(2 n), {k, 0, n}], {x, 0, n}], {n, 0, 21}]
Table[SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^(2 n), {x, 0, n}], {n, 0, 21}]
(* Calculation of constant d: *) 1/r /. FindRoot[{s == EllipticTheta[3, 0, r*s]/EllipticTheta[4, 0, r*s], EllipticTheta[4, 0, r*s] + r*s*Derivative[0, 0, 1][EllipticTheta][4, 0, r*s] == r*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s]}, {r, 1/10}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jan 17 2024 *)

Formula

a(n) = [x^n] Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^(2*n).
From Vaclav Kotesovec, Nov 05 2017: (Start)
a(n) ~ c * d^n / sqrt(n), where
d = 11.61255065799699699891360038489317237925475956178123836149123386457... and
c = 0.34456510029264878768512693687607064416428117641473856418257649837... (End)

A319456 a(n) = [x^n] Product_{k>=1} ((1 - x^k)*(1 - x^(2*k)))^n.

Original entry on oeis.org

1, -1, -3, 14, -11, -81, 282, -57, -2043, 5405, 2417, -46476, 94522, 110512, -943407, 1505289, 2807589, -16888311, 23645199, 46006542, -265972791, 472882620, 187884672, -3981273597, 14234579226, -19187383356, -78662039004, 502118911904, -847583768679, -2627514175002
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 19 2018

Keywords

Crossrefs

Cf. A002171, A002288, A008705, A030204, A030211, A066535, A296043, A296044, A319457.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 - x^k) (1 - x^(2 k)))^n , {k, 1, n}], {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[(QPochhammer[x] QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[Exp[n Sum[(DivisorSigma[1, 2 k] - 4 DivisorSigma[1, k]) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 29}]

Formula

a(n) = [x^n] Product_{k>=1} (1 - x^(2*k))^(2*n)/(1 + x^k)^n.
a(n) = [x^n] exp(n*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k).
Previous Showing 11-20 of 24 results. Next

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