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 1-6 of 6 results.

A053274 Coefficients of the '6th-order' mock theta function gamma(q).

Original entry on oeis.org

1, 1, -1, 0, 2, -2, -1, 3, -2, 0, 3, -4, -1, 5, -3, -1, 6, -6, -2, 7, -6, 0, 9, -8, -3, 11, -9, -2, 13, -13, -3, 17, -12, -3, 19, -18, -5, 22, -19, -3, 27, -24, -7, 33, -26, -5, 36, -34, -9, 44, -35, -9, 51, -45, -11, 58, -49, -9, 68, -59, -16, 78, -65, -15, 88, -79, -19, 104, -84, -19, 117, -102, -26, 133, -112, -24, 152, -131
Offset: 0

Views

Author

Dean Hickerson, Dec 19 1999

Keywords

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 17

Links

Crossrefs

Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053271, A053272, A053273.
Cf. A328988, A328989.

Programs

  • Mathematica
    Series[Sum[q^n^2/Product[1+q^k+q^(2k), {k, 1, n}], {n, 0, 10}], {q, 0, 100}]
  • PARI
    a(n) = polcoeff(sum(k=0, 50, q^(k^2)/prod(j=1, k, 1+q^j+q^(2*j)), q*O(q^n)), n);
for(n=0,50, print1(a(n), ", ")) \\ G. C. Greubel, May 18 2018
  • PARI
    my(N=80, x='x+O('x^N)); Vec(1+1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1+x^k)*(1-x^k)^2/(1+x^k+x^(2*k)))) \\ Seiichi Manyama, May 23 2023

Formula

G.f.: gamma(q) = Sum_{n >= 0} q^n^2/((1+q+q^2)(1+q^2+q^4)...(1+q^n+q^(2n))).
From Seiichi Manyama, May 23 2023: (Start)
a(n) = A328988(n) - A328989(n) for n > 0.
G.f.: 1 + (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1+x^k) * (1-x^k)^2 / (1+x^k+x^(2*k)). (End)

A328989 Number of partitions of n with rank congruent to 1 mod 3.

Original entry on oeis.org

0, 1, 1, 1, 3, 4, 4, 8, 10, 13, 20, 26, 32, 46, 59, 75, 101, 129, 161, 211, 264, 331, 421, 526, 649, 815, 1004, 1235, 1526, 1869, 2275, 2787, 3382, 4097, 4967, 5994, 7205, 8678, 10396, 12437, 14869, 17727, 21076, 25067, 29713, 35174, 41596, 49094, 57827, 68087
Offset: 1

Views

Author

N. J. A. Sloane, Nov 09 2019

Keywords

Comments

Also number of partitions of n with rank congruent to 2 mod 3. - Seiichi Manyama, May 23 2023

Links

Crossrefs

Cf. A000041, A053274, A328988.

Programs

  • Maple
    b:= proc(n, i, r) option remember; `if`(n=0 or i=1,
      `if`(irem(r+n, 3)=0, 1, 0), b(n, i-1, r)+
        b(n-i, min(n-i, i), irem(r+1, 3)))
    end:
a:= proc(n) option remember; add(
      b(n-i, min(n-i, i), modp(2-i, 3)), i=1..n)
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Nov 11 2019
  • Mathematica
    b[n_, i_, r_] := b[n, i, r] = If[n == 0 || i == 1, If[Mod[r + n, 3] == 0, 1, 0], b[n, i - 1, r] + b[n - i, Min[n - i, i], Mod[r + 1, 3]]];
a[n_] := a[n] = Sum[b[n - i, Min[n - i, i], Mod[2 - i, 3]], {i, 1, n}];
Array[a, 60] (* Jean-François Alcover, Feb 29 2020, after Alois P. Heinz *)
  • PARI
    my(N=60, x='x+O('x^N)); concat(0, Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k+1)/2)*(1+x^k)/(1+x^k+x^(2*k))))) \\ Seiichi Manyama, May 23 2023

Formula

a(n) = (A000041(n) - A328988(n))/2. - Alois P. Heinz, Nov 11 2019
From Seiichi Manyama, May 23 2023: (Start)
a(n) = (A000041(n) - A053274(n))/3.
G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k+1)/2) * (1+x^k) / (1+x^k+x^(2*k)). (End)

Extensions

a(22)-a(50) from Lars Blomberg, Nov 11 2019

A363233 Number of partitions of n with rank a multiple of 4.

Original entry on oeis.org

1, 0, 1, 1, 3, 1, 5, 4, 10, 8, 16, 17, 29, 29, 48, 53, 81, 89, 130, 149, 208, 238, 325, 381, 506, 592, 770, 910, 1165, 1374, 1738, 2057, 2571, 3038, 3761, 4451, 5461, 6447, 7855, 9270, 11219, 13214, 15899, 18703, 22386, 26276, 31306, 36691, 43525, 50902, 60149, 70221, 82679, 96325
Offset: 1

Views

Author

Seiichi Manyama, May 23 2023

Keywords

Crossrefs

Cf. A000041, A328988, A340601, A363237, A363238, A363239.

Programs

  • Maple
    b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
     `if`(irem(i-c, 4)=0, 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
    end:
a:= n-> b(n, 1$2):
seq(a(n), n=1..54);  # Alois P. Heinz, May 23 2023
  • PARI
    my(N=60, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)*(1+x^(4*k))/(1-x^(4*k))))

Formula

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k) * (1+x^(4*k)) / (1-x^(4*k)).

A363237 Number of partitions of n with rank a multiple of 5.

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 3, 4, 6, 8, 12, 15, 21, 27, 34, 47, 59, 77, 98, 125, 160, 200, 251, 315, 390, 488, 602, 744, 913, 1120, 1370, 1669, 2029, 2462, 2975, 3597, 4327, 5203, 6237, 7466, 8919, 10634, 12653, 15035, 17824, 21114, 24950, 29455, 34705, 40844, 47991, 56317, 65987, 77231, 90252
Offset: 1

Views

Author

Seiichi Manyama, May 23 2023

Keywords

Crossrefs

Cf. A000041, A328988, A340601, A363233, A363238, A363239.

Programs

  • Maple
    b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
     `if`(irem(i-c, 5)=0, 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
    end:
a:= n-> b(n, 1$2):
seq(a(n), n=1..55);  # Alois P. Heinz, May 23 2023
  • PARI
    my(N=60, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)*(1+x^(5*k))/(1-x^(5*k))))

Formula

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k) * (1+x^(5*k)) / (1-x^(5*k)).

A363238 Number of partitions of n with rank a multiple of 6.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 5, 2, 6, 6, 10, 11, 21, 19, 32, 37, 51, 59, 90, 97, 138, 162, 215, 253, 340, 392, 514, 610, 771, 916, 1166, 1367, 1711, 2032, 2503, 2965, 3647, 4293, 5237, 6188, 7469, 8808, 10613, 12459, 14920, 17530, 20862, 24457, 29029, 33924, 40099, 46829, 55101, 64215, 75386
Offset: 1

Views

Author

Seiichi Manyama, May 23 2023

Keywords

Crossrefs

Cf. A000041, A328988, A340601, A363233, A363237, A363239.

Programs

  • Maple
    b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
     `if`(irem(i-c, 6)=0, 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
    end:
a:= n-> b(n, 1$2):
seq(a(n), n=1..55);  # Alois P. Heinz, May 23 2023
  • PARI
    my(N=60, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)*(1+x^(6*k))/(1-x^(6*k))))

Formula

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k) * (1+x^(6*k)) / (1-x^(6*k)).

A363239 Number of partitions of n with rank a multiple of 7.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 4, 4, 6, 8, 11, 15, 19, 26, 33, 43, 55, 70, 89, 114, 144, 179, 225, 280, 348, 430, 532, 653, 800, 978, 1193, 1449, 1758, 2127, 2569, 3091, 3717, 4455, 5334, 6369, 7596, 9039, 10739, 12734, 15080, 17822, 21039, 24791, 29176, 34277, 40227, 47133, 55165, 64468
Offset: 1

Views

Author

Seiichi Manyama, May 23 2023

Keywords

Crossrefs

Cf. A000041, A328988, A340601, A363233, A363237, A363238.

Programs

  • Maple
    b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
     `if`(irem(i-c, 7)=0, 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
    end:
a:= n-> b(n, 1$2):
seq(a(n), n=1..55);  # Alois P. Heinz, May 23 2023
  • PARI
    my(N=60, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)*(1+x^(7*k))/(1-x^(7*k))))

Formula

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k) * (1+x^(7*k)) / (1-x^(7*k)).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.