A053282 -id:A053282 - cp's OEIS frontend

A053283 Coefficients of the '10th-order' mock theta function X(q).

1, -1, 1, 0, 1, -2, 1, -1, 1, -2, 3, -1, 2, -4, 3, -2, 3, -5, 4, -4, 5, -6, 7, -5, 6, -9, 9, -7, 9, -12, 11, -11, 12, -15, 16, -14, 16, -21, 20, -18, 22, -25, 26, -25, 28, -33, 34, -33, 35, -42, 43, -41, 47, -53, 53, -54, 57, -65, 69, -67, 73, -83, 85, -83, 92, -102, 104, -106, 114, -125, 130, -130, 139, -154

Offset: 0

Dean Hickerson, Dec 19 1999

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 9

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (corrected and extended previous b-file from G. C. Greubel)
Youn-Seo Choi, Tenth order mock theta functions in Ramanujan's lost notebook, Inventiones Mathematicae, 136 (1999) p. 497-569.

Other '10th-order' mock theta functions are at A053281, A053282, A053284.

Series[Sum[(-1)^n q^n^2/Product[1+q^k, {k, 1, 2n}], {n, 0, 10}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[(-1)^k * x^(k^2) / Product[1+x^j, {j, 1, 2*k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 11 2019 *)

G.f.: X(q) = Sum_{n >= 0} (-1)^n q^n^2/((1+q)(1+q^2)...(1+q^(2n))).

a(n) ~ (-1)^n * exp(Pi*sqrt(n/10)) / (2*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A053281 Coefficients of the '10th-order' mock theta function phi(q).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 6, 7, 8, 10, 12, 14, 16, 20, 22, 26, 31, 34, 40, 46, 52, 60, 68, 76, 87, 98, 110, 124, 140, 156, 174, 196, 216, 242, 270, 298, 332, 368, 406, 449, 496, 546, 602, 664, 728, 800, 880, 962, 1056, 1156, 1262, 1381, 1508, 1644, 1794, 1956, 2128

Offset: 0

Dean Hickerson, Dec 19 1999

The alternating sum of the same series, namely phi(q) = Sum_{n>=0} (-1)^n q^(n(n+1)/2)/((1-q)(1-q^3)...(1-q^(2n+1))) = 1 + x^3 - x^7 - x^16 + x^24 + x^39 - x^51 - ..., where the exponents are given by 5n^2 +- 2n. See the Amer. Math. Monthly reference.

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 9.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Youn-Seo Choi, Tenth order mock theta functions in Ramanujan's lost notebook, Inventiones Mathematicae, 136 (1999) p. 497-569.
David Newman, A Recurrence inside a Generating Function: Solution to problem 10681, American Mathematical Monthly, vol. 107 (2000), p. 569.

Other '10th-order' mock theta functions are at A053282, A053283, A053284.

Series[Sum[q^(n(n+1)/2)/Product[1-q^(2k+1), {k, 0, n}], {n, 0, 13}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2) / Product[1-x^(2*j+1), {j, 0, k}], {k, 0, Floor[Sqrt[2*nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 11 2019 *)

G.f.: phi(q) = Sum_{n >= 0} q^(n(n+1)/2)/((1-q)(1-q^3)...(1-q^(2n+1))).

a(n) ~ sqrt(phi) * exp(Pi*sqrt(n/5)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A053284 Coefficients of the '10th-order' mock theta function chi(q).

Original entry on oeis.org

0, 1, -1, 1, -2, 2, -1, 2, -3, 3, -3, 3, -4, 4, -4, 5, -6, 7, -6, 7, -9, 8, -8, 10, -12, 13, -13, 13, -16, 17, -16, 19, -21, 22, -23, 25, -28, 29, -30, 33, -37, 39, -39, 42, -48, 49, -50, 55, -60, 64, -66, 70, -77, 81, -82, 89, -97, 101, -105, 112, -121, 126, -131, 140, -151, 159, -163, 173, -187, 194, -202

Offset: 0

Dean Hickerson, Dec 19 1999

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 9.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (corrected and extended previous b-file from G. C. Greubel)
Youn-Seo Choi, Tenth order mock theta functions in Ramanujan's lost notebook, Inventiones Mathematicae, 136 (1999) pp. 497-569.

Other '10th-order' mock theta functions are at A053281, A053282, A053283.

Series[Sum[(-1)^n q^(n+1)^2/Product[1+q^k, {k, 1, 2n+1}], {n, 0, 9}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[(-1)^k * x^((k+1)^2)/Product[1+x^j, {j, 1, 2*k+1}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 11 2019 *)

G.f.: chi(q) = Sum_{n >= 0} (-1)^n q^(n+1)^2/((1+q)(1+q^2)...(1+q^(2n+1))).

a(n) ~ -(-1)^n * sqrt(phi) * exp(Pi*sqrt(n/10)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A304705 Number of partitions (d1,d2,...,dm) of n such that d1/1 >= d2/2 >= ... >= dm/m and 0 < d1 <= d2 <= ... <= dm.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 6, 5, 6, 8, 9, 9, 12, 11, 14, 17, 16, 17, 23, 22, 27, 31, 30, 33, 40, 41, 46, 50, 54, 57, 70, 70, 77, 88, 92, 99, 111, 115, 129, 142, 152, 160, 175, 183, 199, 223, 234, 255, 283, 299, 328, 347, 370, 390, 430, 455, 489, 523, 557, 592, 642, 674, 724, 784

Offset: 0

Seiichi Manyama, May 17 2018

nonn

			n | Partition (d1,d2,...,dm)    | (d1/1, d2/2, ... , dm/m)
--+-----------------------------+---------------------------------------------
1 | (1)                         | (1)
2 | (2)                         | (2)
  | (1, 1)                      | (1, 1/2)
3 | (3)                         | (3)
  | (1, 2)                      | (1, 1)
  | (1, 1, 1)                   | (1, 1/2, 1/3)
4 | (4)                         | (4)
  | (2, 2)                      | (2, 1)
  | (1, 1, 1, 1)                | (1, 1/2, 1/3, 1/4)
5 | (5)                         | (5)
  | (2, 3)                      | (2, 3/2)
  | (1, 2, 2)                   | (1, 1, 2/3)
  | (1, 1, 1, 1, 1)             | (1, 1/2, 1/3, 1/4, 1/5)
6 | (6)                         | (6)
  | (2, 4)                      | (2, 2)
  | (3, 3)                      | (3, 3/2)
  | (1, 2, 3)                   | (1, 1, 1)
  | (2, 2, 2)                   | (2, 1, 2/3)
  | (1, 1, 1, 1, 1, 1)          | (1, 1/2, 1/3, 1/4, 1/5, 1/6)
7 | (7)                         | (7)
  | (3, 4)                      | (3, 2)
  | (2, 2, 3)                   | (2, 1, 1)
  | (1, 2, 2, 2)                | (1, 1, 2/3, 1/2)
  | (1, 1, 1, 1, 1, 1, 1)       | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7)
8 | (8)                         | (8)
  | (3, 5)                      | (3, 5/2)
  | (4, 4)                      | (4, 2/1)
  | (2, 3, 3)                   | (2, 3/2, 1)
  | (2, 2, 2, 2)                | (2, 1, 2/3, 1/2)
  | (1, 1, 1, 1, 1, 1, 1, 1)    | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8)
9 | (9)                         | (9)
  | (3, 6)                      | (3, 3)
  | (4, 5)                      | (4, 5/2)
  | (2, 3, 4)                   | (2, 3/2, 4/3)
  | (3, 3, 3)                   | (3, 3/2, 1)
  | (1, 2, 3, 3)                | (1, 1, 1, 3/4)
  | (1, 2, 2, 2, 2)             | (1, 1, 2/3, 1/2, 2/5)
  | (1, 1, 1, 1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9)

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A053251, A053282, A304706, A304707, A304708.

b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0,
      b(n, r, i+1, t)+`if`(i/t>r, 0, b(n-i, i/t, i, t+1))))
    end:
a:= n-> b(n$2, 1$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018

b[n_, r_, i_, t_] := b[n, r, i, t] = If[n == 0, 1, If[i > n, 0, b[n, r, i + 1, t] + If[i/t > r, 0, b[n - i, i/t, i, t + 1]]]];
a[n_] := b[n, n, 1, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

A304706 Number of partitions (d1,d2,...,dm) of n such that d1/1 > d2/2 > ... > dm/m and 0 < d1 <= d2 <= ... <= dm.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 6, 5, 6, 6, 8, 7, 11, 10, 11, 12, 15, 14, 18, 17, 20, 23, 27, 25, 31, 32, 35, 38, 43, 43, 51, 54, 59, 63, 71, 73, 85, 89, 96, 102, 113, 120, 134, 141, 149, 161, 175, 183, 203, 213, 233, 252, 280, 293, 319, 338, 360, 383, 409, 430, 468, 493, 531, 565

Offset: 0

Seiichi Manyama, May 17 2018

nonn

			n | Partition (d1,d2,...,dm)    | (d1/1, d2/2, ... , dm/m)
--+-----------------------------+---------------------------------------------
1 | (1)                         | (1)
2 | (2)                         | (2)
  | (1, 1)                      | (1, 1/2)
3 | (3)                         | (3)
  | (1, 1, 1)                   | (1, 1/2, 1/3)
4 | (4)                         | (4)
  | (2, 2)                      | (2, 1)
  | (1, 1, 1, 1)                | (1, 1/2, 1/3, 1/4)
5 | (5)                         | (5)
  | (2, 3)                      | (2, 3/2)
  | (1, 1, 1, 1, 1)             | (1, 1/2, 1/3, 1/4, 1/5)
6 | (6)                         | (6)
  | (3, 3)                      | (3, 3/2)
  | (2, 2, 2)                   | (2, 1, 2/3)
  | (1, 1, 1, 1, 1, 1)          | (1, 1/2, 1/3, 1/4, 1/5, 1/6)
7 | (7)                         | (7)
  | (3, 4)                      | (3, 2)
  | (1, 1, 1, 1, 1, 1, 1)       | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7)
8 | (8)                         | (8)
  | (3, 5)                      | (3, 5/2)
  | (4, 4)                      | (4, 2/1)
  | (2, 3, 3)                   | (2, 3/2, 1)
  | (2, 2, 2, 2)                | (2, 1, 2/3, 1/2)
  | (1, 1, 1, 1, 1, 1, 1, 1)    | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8)
9 | (9)                         | (9)
  | (4, 5)                      | (4, 5/2)
  | (2, 3, 4)                   | (2, 3/2, 4/3)
  | (3, 3, 3)                   | (3, 3/2, 1)
  | (1, 1, 1, 1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9)

Cf. A053251, A053282, A304705, A304707, A304708.

b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0,
      b(n, r, i+1, t)+`if`(i/t>=r, 0, b(n-i, i/t, i, t+1))))
    end:
a:= n-> b(n, n+1, 1$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018

b[n_, r_, i_, t_] := b[n, r, i, t] = If[n == 0, 1, If[i > n, 0, b[n, r, i + 1, t] + If[i/t >= r, 0, b[n - i, i/t, i, t + 1]]]];
a[n_] := b[n, n + 1, 1, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

a(n) <= A304705(n).

A304707 Number of partitions (d1,d2,...,dm) of n such that d1/1 >= d2/2 >= ... >= dm/m and d1 < d2 < ... < dm.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 4, 3, 4, 5, 4, 5, 7, 5, 7, 8, 8, 10, 12, 10, 11, 14, 14, 14, 18, 17, 20, 23, 22, 26, 30, 29, 32, 35, 34, 37, 43, 44, 48, 54, 54, 59, 67, 70, 76, 81, 84, 89, 97, 101, 110, 119, 123, 129, 139, 145, 155, 171, 176, 189, 201, 211, 228, 245, 257, 274, 295

Offset: 0

Seiichi Manyama, May 17 2018

nonn

			n | Partition (d1,d2,...,dm)    | (d1/1, d2/2, ... , dm/m)
--+-----------------------------+-------------------------
1 | (1)                         | (1)
2 | (2)                         | (2)
3 | (3)                         | (3)
  | (1, 2)                      | (1, 1)
4 | (4)                         | (4)
5 | (5)                         | (5)
  | (2, 3)                      | (2, 3/2)
6 | (6)                         | (6)
  | (2, 4)                      | (2, 2)
  | (1, 2, 3)                   | (1, 1, 1)
7 | (7)                         | (7)
  | (3, 4)                      | (3, 2)
8 | (8)                         | (8)
  | (3, 5)                      | (3, 5/2)
9 | (9)                         | (9)
  | (3, 6)                      | (3, 3)
  | (4, 5)                      | (4, 5/2)
  | (2, 3, 4)                   | (2, 3/2, 4/3)

Cf. A053251, A053282, A304705, A304706, A304708.

b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0,
      b(n, r, i+1, t) +`if`(i/t>r, 0, b(n-i, i/t, i+1, t+1))))
    end:
a:= n-> b(n$2, 1$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018

b[n_, r_, i_, t_] := b[n, r, i, t] = If[n == 0, 1, If[i > n, 0, b[n, r, i + 1, t] + If[i/t > r, 0, b[n - i, i/t, i + 1, t + 1]]]];
a[n_] := b[n, n, 1, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

A304708 Number of partitions (d1,d2,...,dm) of n such that d1/1 > d2/2 > ... > dm/m and d1 < d2 < ... < dm.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 3, 5, 5, 4, 5, 6, 6, 7, 8, 8, 9, 10, 12, 11, 13, 13, 16, 16, 15, 18, 21, 22, 26, 25, 28, 31, 33, 33, 35, 39, 41, 46, 47, 50, 53, 59, 63, 68, 74, 77, 84, 90, 93, 98, 105, 111, 119, 129, 132, 138, 149, 157, 169, 178, 189, 201, 211, 227

Offset: 0

Seiichi Manyama, May 17 2018

nonn

			n | Partition (d1,d2,...,dm)    | (d1/1, d2/2, ... , dm/m)
--+-----------------------------+-------------------------
1 | (1)                         | (1)
2 | (2)                         | (2)
3 | (3)                         | (3)
4 | (4)                         | (4)
5 | (5)                         | (5)
  | (2, 3)                      | (2, 3/2)
6 | (6)                         | (6)
7 | (7)                         | (7)
  | (3, 4)                      | (3, 2)
8 | (8)                         | (8)
  | (3, 5)                      | (3, 5/2)
9 | (9)                         | (9)
  | (4, 5)                      | (4, 5/2)
  | (2, 3, 4)                   | (2, 3/2, 4/3)

Cf. A053251, A053282, A304705, A304706, A304707.

b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0,
      b(n, r, i+1, t)+`if`(i/t>=r, 0, b(n-i, i/t, i+1, t+1))))
    end:
a:= n-> b(n, n+1, 1$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018

b[n_, r_, i_, t_] := b[n, r, i, t] = If[n == 0, 1, If[i > n, 0, b[n, r, i + 1, t] + If[i/t >= r, 0, b[n - i, i/t, i + 1, t + 1]]]];
a[n_] := b[n, n + 1, 1, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

a(n) <= A304707(n).

A376624 G.f.: Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 - x^(2j-1))^2.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 13, 18, 23, 33, 44, 57, 77, 99, 125, 163, 207, 259, 328, 407, 503, 626, 769, 938, 1149, 1397, 1687, 2044, 2458, 2943, 3531, 4213, 5011, 5957, 7055, 8334, 9838, 11580, 13594, 15948, 18661, 21790, 25425, 29593, 34386, 39918, 46250, 53501, 61824, 71325

Offset: 0

Vaclav Kotesovec, Sep 30 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A053282, A072223, A376542, A376622, A376581, A376658.

nmax=60; CoefficientList[Series[Sum[x^(k*(k+1)/2)/Product[1-x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} Product_{j=1..k} x^j/(1 - x^(2*j-1))^2.

a(n) ~ (r^(3/4)/sqrt(8*(1 + 3*r^2))) * A376658^sqrt(n) / sqrt(n), where r = A072223 = 0.52488859865640479389948613854128391569... is the smallest real root of the equation (1 - r^2)^2 = r.

A376628 G.f.: Sum_{k>=0} x^(k(k+1)) / Product_{j=1..k} (1 - x^(2j-1)).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 8, 10, 10, 12, 14, 15, 17, 19, 21, 23, 27, 29, 31, 37, 39, 43, 49, 52, 58, 64, 70, 76, 84, 92, 99, 111, 119, 129, 143, 153, 167, 183, 197, 213, 233, 251, 271, 295, 317, 343, 372, 400, 430, 466, 500, 538, 582, 622, 670

Offset: 0

Vaclav Kotesovec, Sep 30 2024

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003106, A053251, A053254, A053258, A053282, A333179.

nmax=100; CoefficientList[Series[Sum[x^(k*(k+1))/Product[1-x^(2*j-1), {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} Product_{j=1..k} x^(2*j)/(1 - x^(2*j-1)).
a(n) ~ exp(Pi*sqrt(n/6)) / (2^(5/2) * sqrt(n)).
Conjectural g.f.: (1 + q * nu(-q))/(1 + q) = 1 + Sum_{k >= 0} q^(k+2)*Product_{j = 1..k} 1 + q^(2*j+1), where nu(q) is the g.f. of A053254. - Peter Bala, Jan 03 2025

A304869 Triangle read by rows: T(n, k) gives the number of partitions (d1,d2,...,dk) of n such that 0 < d1/1 <= d2/2 <= ... <= dk/k for 1 <= k <= A003056(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 3, 2, 1, 1, 3, 3, 1, 1, 4, 4, 1, 1, 4, 4, 2, 1, 4, 5, 2, 1, 5, 6, 3, 1, 1, 5, 7, 4, 1, 1, 5, 8, 5, 1, 1, 6, 9, 6, 2, 1, 6, 10, 7, 2, 1, 6, 11, 9, 3, 1, 7, 13, 10, 4, 1, 1, 7, 14, 12, 5, 1, 1, 7, 15, 14, 6, 1

Offset: 1

Seiichi Manyama, May 20 2018

			The partitions (d1,d2) of 9 such that 0 < d1/1 <= d2/2 are (1, 8), (2, 7) and (3, 6). So T(9, 2) = 3.
First few rows are:
  1;
  1;
  1, 1;
  1, 1;
  1, 1;
  1, 2, 1;
  1, 2, 1;
  1, 2, 1;
  1, 3, 2;
  1, 3, 2, 1;
  1, 3, 3, 1;
  1, 4, 4, 1;
  1, 4, 4, 2;
  1, 4, 5, 2;
  1, 5, 6, 3, 1;

Seiichi Manyama, Rows n = 1..100, flattened

Row sums give A053282.

Cf. A304871.

cp's OEIS Frontend

A053283 Coefficients of the '10th-order' mock theta function X(q).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

A053281 Coefficients of the '10th-order' mock theta function phi(q).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A053284 Coefficients of the '10th-order' mock theta function chi(q).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

A304705 Number of partitions (d1,d2,...,dm) of n such that d1/1 >= d2/2 >= ... >= dm/m and 0 < d1 <= d2 <= ... <= dm.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A304706 Number of partitions (d1,d2,...,dm) of n such that d1/1 > d2/2 > ... > dm/m and 0 < d1 <= d2 <= ... <= dm.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A304707 Number of partitions (d1,d2,...,dm) of n such that d1/1 >= d2/2 >= ... >= dm/m and d1 < d2 < ... < dm.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

A304708 Number of partitions (d1,d2,...,dm) of n such that d1/1 > d2/2 > ... > dm/m and d1 < d2 < ... < dm.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A376624 G.f.: Sum_{k>=0} x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^(2*j-1))^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A376624 G.f.: Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 - x^(2j-1))^2.

A376628 G.f.: Sum_{k>=0} x^(k(k+1)) / Product_{j=1..k} (1 - x^(2j-1)).