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.

Showing 1-10 of 10 results.

A298329 Number of ordered ways of writing n^2 as a sum of n squares of nonnegative integers.

Original entry on oeis.org

1, 1, 2, 6, 5, 90, 582, 4081, 45678, 378049, 3844532, 39039539, 395170118, 4589810849, 53154371025, 660113986997, 8584476248237, 113555197832758, 1572878837435750, 22259911738401660, 324143769099772448, 4869443438412466557, 74837370448784241452, 1182177603062005007658
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 17 2018

Keywords

Examples

			a(3) = 6 because we have [9, 0, 0], [4, 4, 1], [4, 1, 4], [1, 4, 4], [0, 9, 0] and [0, 0, 9].

Links

Crossrefs

[x^(n^b)] (Sum_{k>=0} x^(k^b))^n: A088218 (b=1), this sequence (b=2), A298671 (b=3).
Cf. A037444, A066535, A232173, A287617, A298330, A300446.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0, 1/t!, (s->
     `if`(s*t n!*b(n^2, n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 28 2018
  • Mathematica
    Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^2}], {n, 0, 23}]
  • PARI
    {a(n) = polcoeff((sum(k=0, n, x^(k^2)+x*O(x^(n^2))))^n, n^2)} \\ Seiichi Manyama, Oct 28 2018

Formula

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

A320932 a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 20, 51, 141, 381, 1001, 2796, 7861, 22306, 64129, 185692, 540468, 1585246, 4674464, 13846636, 41216933, 123176849, 369410571, 1111661833, 3355466306, 10156304314, 30821794651, 93761053797, 285859742756, 873355481467, 2673455511946, 8198687383812
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2018

Keywords

Comments

Also the number of nonnegative integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n*(n+1)/2.

Examples

			1*1^2 + 2*1^2 + 3*1^2 + 4*1^2 + 5*1^2 = 15.
1*2^2 + 2*1^2 + 3*0^2 + 4*1^2 + 5*1^2 = 15.
1*0^2 + 2*2^2 + 3*1^2 + 4*1^2 + 5*0^2 = 15.
1*3^2 + 2*1^2 + 3*0^2 + 4*1^2 + 5*0^2 = 15.
1*1^2 + 2*1^2 + 3*2^2 + 4*0^2 + 5*0^2 = 15.
1*2^2 + 2*2^2 + 3*1^2 + 4*0^2 + 5*0^2 = 15.
So a(5) = 6.

Links

Crossrefs

Cf. A000122, A000217, A010052, A173519, A300446, A320931.

Programs

  • Maple
    b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else b(n, i-1); for j while
        i*j^2<=n do %+b(n-i*j^2, i-1) od; % fi
    end:
a:= n-> b(n*(n+1)/2, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 28 2018
  • Mathematica
    nmax = 30; Table[SeriesCoefficient[Product[(EllipticTheta[3, 0, x^k] + 1)/2, {k, 1, n}], {x, 0, n*(n+1)/2}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)
  • PARI
    {a(n) = polcoeff(prod(i=1, n, sum(j=0, sqrtint(n*(n+1)\(2*i)), x^(i*j^2)+x*O(x^(n*(n+1)/2)))), n*(n+1)/2)}

Formula

a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function.

A321139 a(n) = [x^(n^2)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).

Original entry on oeis.org

1, 1, 1, 3, 7, 17, 52, 144, 480, 1732, 5902, 21078, 78434, 289107, 1079949, 4094643, 15574377, 59667023, 230318968, 892694240, 3477119540, 13606993083, 53438614380, 210622413188, 832922044686, 3303392730698, 13137474884294, 52381331536536, 209340904575968
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2018

Keywords

Comments

Also the number of nonnegative integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n^2.

Examples

			1*0^2 + 2*1^2 + 3*1^2 + 4*0^2 + 5*2^2 = 25.
1*0^2 + 2*2^2 + 3*2^2 + 4*0^2 + 5*1^2 = 25.
1*0^2 + 2*3^2 + 3*1^2 + 4*1^2 + 5*0^2 = 25.
1*1^2 + 2*0^2 + 3*0^2 + 4*1^2 + 5*2^2 = 25.
1*1^2 + 2*0^2 + 3*1^2 + 4*2^2 + 5*1^2 = 25.
1*1^2 + 2*2^2 + 3*0^2 + 4*2^2 + 5*0^2 = 25.
1*1^2 + 2*2^2 + 3*2^2 + 4*1^2 + 5*0^2 = 25.
1*2^2 + 2*0^2 + 3*0^2 + 4*2^2 + 5*1^2 = 25.
1*2^2 + 2*0^2 + 3*2^2 + 4*1^2 + 5*1^2 = 25.
1*2^2 + 2*1^2 + 3*1^2 + 4*2^2 + 5*0^2 = 25.
1*2^2 + 2*3^2 + 3*1^2 + 4*0^2 + 5*0^2 = 25.
1*3^2 + 2*0^2 + 3*0^2 + 4*2^2 + 5*0^2 = 25.
1*3^2 + 2*0^2 + 3*2^2 + 4*1^2 + 5*0^2 = 25.
1*3^2 + 2*2^2 + 3*1^2 + 4*0^2 + 5*1^2 = 25.
1*4^2 + 2*0^2 + 3*0^2 + 4*1^2 + 5*1^2 = 25.
1*4^2 + 2*1^2 + 3*1^2 + 4*1^2 + 5*0^2 = 25.
1*5^2 + 2*0^2 + 3*0^2 + 4*0^2 + 5*0^2 = 25.
So a(5) = 17.

Links

Crossrefs

Cf. A000122, A010052, A206226, A300446, A320932.

Programs

  • Maple
    b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else b(n, i-1); for j while
        i*j^2<=n do %+b(n-i*j^2, i-1) od; % fi
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 28 2018
  • Mathematica
    nmax = 25; Table[SeriesCoefficient[Product[(EllipticTheta[3, 0, x^k] + 1)/2, {k, 1, n}], {x, 0, n^2}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)
  • PARI
    {a(n) = polcoeff(prod(i=1, n, sum(j=0, sqrtint(n^2\i), x^(i*j^2)+x*O(x^(n^2)))), n^2)}

Formula

a(n) = [x^(n^2)] Product_{k=1..n} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function.

A304329 Expansion of Product_{k>0} (Sum_{m>=0} x^(k*m^3)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 7, 8, 11, 13, 16, 20, 24, 30, 36, 44, 51, 62, 74, 88, 103, 122, 145, 169, 197, 231, 268, 312, 362, 419, 485, 557, 642, 737, 846, 967, 1108, 1262, 1442, 1640, 1865, 2118, 2398, 2719, 3074, 3474, 3922, 4421, 4980, 5604, 6294, 7070, 7929
Offset: 0

Views

Author

Seiichi Manyama, May 11 2018

Keywords

Comments

Also the number of partitions of n in which each part occurs a cube number (>=0) of times.

Examples

			n | Partitions of n in which each part occurs a cube number (>=0) of times
--+-----------------------------------------------------------------------
1 | 1;
2 | 2;
3 | 3 = 2+1;
4 | 4 = 3+1;
5 | 5 = 4+1 = 3+2;
6 | 6 = 5+1 = 4+2 = 3+2+1;
7 | 7 = 6+1 = 5+2 = 4+3 = 4+2+1;
8 | 8 = 7+1 = 6+2 = 5+3 = 5+2+1 = 4+3+1 = 1+1+1+1+1+1+1+1;

Links

Crossrefs

Cf. A000041, A300446.

Programs

  • Maple
    b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else b(n, i-1); for j while
        i*j^3<=n do %+b(n-i*j^3, i-1) od; % fi
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 11 2018
  • Mathematica
    terms = 100;
Product[Sum[x^(k*m^3), {m, 0, Ceiling[terms^(1/3)]}], {k, 1, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Mar 08 2021 *)

A304332 Expansion of Product_{k>0} (1 + Sum_{m>0} x^(k*m!)).

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 9, 14, 17, 24, 29, 40, 49, 64, 77, 101, 122, 156, 187, 235, 281, 349, 416, 514, 608, 742, 877, 1062, 1252, 1502, 1766, 2108, 2467, 2928, 3419, 4039, 4701, 5524, 6411, 7505, 8688, 10130, 11695, 13587, 15648, 18118, 20819, 24034, 27555, 31712
Offset: 0

Views

Author

Seiichi Manyama, May 11 2018

Keywords

Comments

Also the number of partitions of n in which each part occurs a factorial number of times.

Examples

			n | Partitions of n in which each part occurs a factorial number of times
--+----------------------------------------------------------------------
1 | 1;
2 | 2 = 1+1;
3 | 3 = 2+1;
4 | 4 = 3+1 = 2+2 = 2+1+1;
5 | 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1;
6 | 6 = 5+1 = 4+2 = 4+1+1 = 3+2+1 = 3+3 = 2+2+1+1 = 1+1+1+1+1+1;
7 | 7 = 6+1 = 5+2 = 5+1+1 = 4+3 = 4+2+1 = 3+3+1 = 3+2+2 = 3+2+1+1;

Links

Crossrefs

Cf. A000041, A300446.

Programs

  • Maple
    b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else b(n, i-1); for j while
        i*j!<=n do %+b(n-i*j!, i-1) od; % fi
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 11 2018

A308286 Expansion of Product_{i>=1, j>=1} theta_3(x^(i*j)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 4, 12, 20, 40, 84, 140, 252, 456, 752, 1260, 2128, 3392, 5436, 8760, 13582, 21092, 32744, 49620, 75104, 113448, 168508, 249620, 368840, 538412, 783480, 1136652, 1634000, 2341280, 3344680, 4743684, 6706120, 9452392, 13245800, 18504888, 25777520, 35735376
Offset: 0

Views

Author

Ilya Gutkovskiy, May 18 2019

Keywords

Links

Crossrefs

Cf. A000005, A000122, A006171, A007425, A300446, A301554, A305050, A308288, A320067, A320908, A321241.

Programs

  • Mathematica
    nmax = 37; CoefficientList[Series[Product[Product[EllipticTheta[3, 0, x^(i j)], {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Product[EllipticTheta[3, 0, x^k]^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} theta_3(x^k)^tau(k), where tau = number of divisors (A000005).
G.f.: Product_{i>=1, j>=1} (Sum_{k=-oo..+oo} x^(i*j*k^2)).
G.f.: Product_{i>=1, j>=1, k>=1} (1 - x^(i*j*k))*(1 + x^(i*j*k))^3/(1 + x^(2*i*j*k))^2.
G.f.: Product_{k>=1} (1 - x^k)^tau_3(k)*(1 + x^k)^(3*tau_3(k))/(1 + x^(2*k))^(2*tau_3(k)), where tau_3 = A007425.

A321181 a(n) = [x^((n*(n+1)/2)^2)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).

Original entry on oeis.org

1, 1, 2, 7, 28, 262, 3428, 52289, 1147221, 30161625, 893291633, 30894822277, 1214415301634, 52617692115135, 2528123847871538, 133088227043557512, 7574733515354756765, 466116310963215784930, 30810712157925101729430, 2173319693639115252360852, 163247410881483617710298406
Offset: 0

Views

Author

Seiichi Manyama, Oct 29 2018

Keywords

Comments

Also the number of nonnegative integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = (n*(n+1)/2)^2.

Examples

			1* 0^2 + 2*0^2 + 3*0^2 + 4*5^2 = 100.
1* 0^2 + 2*6^2 + 3*2^2 + 4*2^2 = 100.
1* 1^2 + 2*2^2 + 3*3^2 + 4*4^2 = 100.
1* 1^2 + 2*2^2 + 3*5^2 + 4*2^2 = 100.
1* 1^2 + 2*4^2 + 3*1^2 + 4*4^2 = 100.
1* 1^2 + 2*6^2 + 3*3^2 + 4*0^2 = 100.
1* 2^2 + 2*4^2 + 3*0^2 + 4*4^2 = 100.
1* 2^2 + 2*4^2 + 3*4^2 + 4*2^2 = 100.
1* 3^2 + 2*0^2 + 3*3^2 + 4*4^2 = 100.
1* 3^2 + 2*0^2 + 3*5^2 + 4*2^2 = 100.
1* 3^2 + 2*6^2 + 3*1^2 + 4*2^2 = 100.
1* 4^2 + 2*0^2 + 3*4^2 + 4*3^2 = 100.
1* 4^2 + 2*2^2 + 3*2^2 + 4*4^2 = 100.
1* 4^2 + 2*4^2 + 3*4^2 + 4*1^2 = 100.
1* 4^2 + 2*6^2 + 3*2^2 + 4*0^2 = 100.
1* 5^2 + 2*0^2 + 3*5^2 + 4*0^2 = 100.
1* 5^2 + 2*2^2 + 3*1^2 + 4*4^2 = 100.
1* 5^2 + 2*4^2 + 3*3^2 + 4*2^2 = 100.
1* 5^2 + 2*6^2 + 3*1^2 + 4*0^2 = 100.
1* 6^2 + 2*0^2 + 3*0^2 + 4*4^2 = 100.
1* 6^2 + 2*0^2 + 3*4^2 + 4*2^2 = 100.
1* 7^2 + 2*2^2 + 3*3^2 + 4*2^2 = 100.
1* 7^2 + 2*4^2 + 3*1^2 + 4*2^2 = 100.
1* 8^2 + 2*0^2 + 3*0^2 + 4*3^2 = 100.
1* 8^2 + 2*2^2 + 3*2^2 + 4*2^2 = 100.
1* 8^2 + 2*4^2 + 3*0^2 + 4*1^2 = 100.
1* 9^2 + 2*0^2 + 3*1^2 + 4*2^2 = 100.
1*10^2 + 2*0^2 + 3*0^2 + 4*0^2 = 100.
So a(4) = 28.

Crossrefs

Cf. A000122, A000537, A300446, A320932.

Extensions

a(17)-a(20) from Alois P. Heinz, Oct 29 2018

A304393 Expansion of Product_{k>0} (1 + Sum_{m>=0} x^(k*2^m)).

Original entry on oeis.org

1, 1, 2, 2, 5, 5, 8, 10, 17, 19, 27, 33, 48, 56, 76, 92, 126, 146, 192, 228, 298, 352, 444, 528, 667, 783, 969, 1145, 1414, 1658, 2017, 2365, 2878, 3352, 4027, 4703, 5634, 6548, 7773, 9033, 10705, 12381, 14573, 16857, 19790, 22800, 26631, 30655, 35723, 41005
Offset: 0

Views

Author

Seiichi Manyama, May 12 2018

Keywords

Comments

Also the number of partitions of n in which each part occurs a power of 2 (cf. A000079) of times.

Examples

			n | Partitions of n in which each part occurs a power of 2 (cf. A000079) of times
--+------------------------------------------------------------------------------
1 | 1;
2 | 2 = 1+1;
3 | 3 = 2+1;
4 | 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1;
5 | 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1;
6 | 6 = 5+1 = 4+2 = 4+1+1 = 3+2+1 = 3+3 = 2+2+1+1 = 2+1+1+1+1;
7 | 7 = 6+1 = 5+2 = 5+1+1 = 4+3 = 4+2+1 = 3+3+1 = 3+2+2 = 3+2+1+1 = 3+1+1+1+1;

Links

Crossrefs

Cf. A000079, A055922, A300446, A304332.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*2^j, i-1), j=0..ilog2(n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 13 2018
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
   b[n, i-1] + Sum[b[n-i*2^j, i-1], {j, 0, Floor@Log2[n/i]}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 14 2023, after Alois P. Heinz *)

A320119 Expansion of Product_{k>0} 1/(Sum_{m>=0} x^(k*m^2)).

Original entry on oeis.org

1, -1, 0, -1, 0, 1, -1, 2, -2, 0, 4, -5, 6, -7, 5, 3, -14, 22, -29, 23, 1, -27, 57, -92, 96, -50, -38, 157, -281, 341, -270, 57, 304, -777, 1156, -1175, 695, 357, -1897, 3447, -4274, 3677, -1045, -3723, 9521, -14161, 15044, -9633, -3494, 22711, -42509, 54373, -48146
Offset: 0

Views

Author

Seiichi Manyama, Oct 06 2018

Keywords

Crossrefs

Cf. A300446, A317665, A320120.

Formula

Convolution inverse of A300446.

A321238 a(n) = [x^(n^3)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).

Original entry on oeis.org

1, 1, 1, 4, 16, 87, 911, 8081, 82494, 1108584, 14559487, 206462480, 3300362073, 54235076625, 939612600043, 17366394088532, 332129019947772, 6615538793829307, 137564490944940832, 2954281836759475893, 65572183746807351880, 1503752010271535590284, 35476544827929325305961
Offset: 0

Views

Author

Seiichi Manyama, Nov 01 2018

Keywords

Comments

Also the number of nonnegative integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n^3.

Examples

			1*0^2 + 2*0^2 + 3*0^2 + 4*4^2 = 64.
1*0^2 + 2*0^2 + 3*4^2 + 4*2^2 = 64.
1*1^2 + 2*0^2 + 3*3^2 + 4*3^2 = 64.
1*1^2 + 2*4^2 + 3*3^2 + 4*1^2 = 64.
1*2^2 + 2*2^2 + 3*4^2 + 4*1^2 = 64.
1*2^2 + 2*4^2 + 3*2^2 + 4*2^2 = 64.
1*4^2 + 2*0^2 + 3*2^2 + 4*3^2 = 64.
1*4^2 + 2*0^2 + 3*4^2 + 4*0^2 = 64.
1*4^2 + 2*4^2 + 3*0^2 + 4*2^2 = 64.
1*4^2 + 2*4^2 + 3*2^2 + 4*1^2 = 64.
1*5^2 + 2*0^2 + 3*1^2 + 4*3^2 = 64.
1*5^2 + 2*2^2 + 3*3^2 + 4*1^2 = 64.
1*5^2 + 2*4^2 + 3*1^2 + 4*1^2 = 64.
1*6^2 + 2*0^2 + 3*2^2 + 4*2^2 = 64.
1*7^2 + 2*2^2 + 3*1^2 + 4*1^2 = 64.
1*8^2 + 2*0^2 + 3*0^2 + 4*0^2 = 64.
So a(4) = 16.

Crossrefs

Cf. A300446, A321139, A321239.

Programs

  • PARI
    {a(n) = polcoeff(prod(i=1, n, sum(j=0, sqrtint(n^3\i), x^(i*j^2)+x*O(x^(n^3)))), n^3)}
Showing 1-10 of 10 results.

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