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 1-5 of 5 results.

A300446 Expansion of Product_{k>0} (Sum_{m>=0} x^(k*m^2)).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 6, 8, 12, 12, 17, 23, 27, 32, 41, 52, 61, 77, 91, 110, 134, 159, 188, 228, 271, 314, 380, 444, 518, 612, 713, 832, 976, 1128, 1308, 1529, 1756, 2023, 2343, 2698, 3091, 3555, 4072, 4657, 5343, 6074, 6922, 7912, 8986, 10194, 11590, 13135, 14855
Offset: 0

Views

Author

Seiichi Manyama, May 11 2018

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A000041, A000122, A010052, A158441, A232173, A298329, A304329, A320932, A321139.

Programs

  • Magma
    m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(&+[x^(k*j^2):j in [0..2*m]]): k in [1..2*m]]) ));  // G. C. Greubel, Oct 29 2018
  • 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):
seq(a(n), n=0..60);  # Alois P. Heinz, May 11 2018
  • Mathematica
    nmax = 60; CoefficientList[Series[Product[(EllipticTheta[3, 0, x^k] + 1)/2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 25 2018 *)
  • PARI
    N=99; x='x+O('x^N); Vec(prod(i=1, N, sum(j=0, sqrtint(N\i), x^(i*j^2)))) \\ Seiichi Manyama, Oct 28 2018

Formula

G.f.: Product_{k>=1} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Oct 25 2018

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.

A320931 a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 4, 12, 24, 80, 292, 966, 3876, 15554, 61608, 254612, 1065676, 4471672, 19074968, 82043172, 354365492, 1543432514, 6760146292, 29732837780, 131440491584, 583419967664, 2598585783488, 11615321544700, 52079369904384, 234157152231726, 1055628140278948, 4770576024205060
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2018

Keywords

Comments

Also the number of 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

			Solutions (a_1, a_2, ... , a_4) to the equation a_1^2 + 2*a_2^2 + ... + 4*a_4^2 = 10.
-------------------------------------------------------------------------------------
( 1,  1,  1,  1), ( 1,  1,  1, -1),
( 1,  1, -1,  1), ( 1,  1, -1, -1),
( 1, -1,  1,  1), ( 1, -1,  1, -1),
( 1, -1, -1,  1), ( 1, -1, -1, -1),
(-1,  1,  1,  1), (-1,  1,  1, -1),
(-1,  1, -1,  1), (-1,  1, -1, -1),
(-1, -1,  1,  1), (-1, -1,  1, -1),
(-1, -1, -1,  1), (-1, -1, -1, -1),
( 2,  1,  0,  1), ( 2,  1,  0, -1),
( 2, -1,  0,  1), ( 2, -1,  0, -1),
(-2,  1,  0,  1), (-2,  1,  0, -1),
(-2, -1,  0,  1), (-2, -1,  0, -1).

Links

Crossrefs

Cf. A000122, A000217, A173519, A320067, A320932.

Programs

  • Mathematica
    nmax = 25; Table[SeriesCoefficient[Product[EllipticTheta[3, 0, x^k], {k, 1, n}], {x, 0, n*(n+1)/2}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)

Formula

a(n) ~ c * d^n / n^(7/4), where d = 4.818071572655... and c = 0.5869031198... - Vaclav Kotesovec, Oct 29 2018

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

A321208 a(n) = [x^(n*(n+1)*(2*n+1)/6)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).

Original entry on oeis.org

1, 1, 0, 2, 7, 31, 167, 1046, 7949, 60487, 490753, 4232323, 39877499, 401064825, 4191449438, 45993709856, 526379057073, 6284584514360, 77594053714675, 990497759689341, 13053609492660678, 177385290308586391, 2480368806876623852, 35617209442716039028, 524705024124493308382
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 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*n+1)/6.

Examples

			1*0^2 + 2*1^2 + 3*2^2 + 4*3^2 + 5*1^2 = 55.
1*0^2 + 2*1^2 + 3*4^2 + 4*0^2 + 5*1^2 = 55.
1*0^2 + 2*2^2 + 3*3^2 + 4*0^2 + 5*2^2 = 55.
1*0^2 + 2*4^2 + 3*1^2 + 4*0^2 + 5*2^2 = 55.
1*0^2 + 2*5^2 + 3*0^2 + 4*0^2 + 5*1^2 = 55.
1*1^2 + 2*1^2 + 3*1^2 + 4*1^2 + 5*3^2 = 55.
1*1^2 + 2*1^2 + 3*4^2 + 4*1^2 + 5*0^2 = 55.
1*1^2 + 2*3^2 + 3*0^2 + 4*2^2 + 5*2^2 = 55.
1*1^2 + 2*3^2 + 3*0^2 + 4*3^2 + 5*0^2 = 55.
1*1^2 + 2*3^2 + 3*2^2 + 4*1^2 + 5*2^2 = 55.
1*1^2 + 2*3^2 + 3*3^2 + 4*1^2 + 5*1^2 = 55.
1*1^2 + 2*5^2 + 3*0^2 + 4*1^2 + 5*0^2 = 55.
1*2^2 + 2*0^2 + 3*3^2 + 4*1^2 + 5*2^2 = 55.
1*2^2 + 2*1^2 + 3*0^2 + 4*1^2 + 5*3^2 = 55.
1*2^2 + 2*2^2 + 3*3^2 + 4*2^2 + 5*0^2 = 55.
1*2^2 + 2*3^2 + 3*2^2 + 4*2^2 + 5*1^2 = 55.
1*2^2 + 2*4^2 + 3*1^2 + 4*2^2 + 5*0^2 = 55.
1*3^2 + 2*1^2 + 3*1^2 + 4*3^2 + 5*1^2 = 55.
1*3^2 + 2*3^2 + 3*2^2 + 4*2^2 + 5*0^2 = 55.
1*4^2 + 2*0^2 + 3*1^2 + 4*2^2 + 5*2^2 = 55.
1*4^2 + 2*0^2 + 3*1^2 + 4*3^2 + 5*0^2 = 55.
1*4^2 + 2*2^2 + 3*3^2 + 4*1^2 + 5*0^2 = 55.
1*4^2 + 2*3^2 + 3*0^2 + 4*2^2 + 5*1^2 = 55.
1*4^2 + 2*3^2 + 3*2^2 + 4*1^2 + 5*1^2 = 55.
1*4^2 + 2*4^2 + 3*1^2 + 4*1^2 + 5*0^2 = 55.
1*5^2 + 2*1^2 + 3*2^2 + 4*2^2 + 5*0^2 = 55.
1*5^2 + 2*3^2 + 3*1^2 + 4*1^2 + 5*1^2 = 55.
1*5^2 + 2*3^2 + 3*2^2 + 4*0^2 + 5*0^2 = 55.
1*6^2 + 2*0^2 + 3*1^2 + 4*2^2 + 5*0^2 = 55.
1*6^2 + 2*1^2 + 3*2^2 + 4*0^2 + 5*1^2 = 55.
1*7^2 + 2*1^2 + 3*0^2 + 4*1^2 + 5*0^2 = 55.
So a(5) = 31.

Crossrefs

Cf. A000122, A320932, A321181, A321186.

Formula

a(n) = [x^(n*(n+1)*(2*n+1)/6)] Product_{k=1..n} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function.
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