A003114 -id:A003114 - cp's OEIS frontend

A264591 Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[4](q).

1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 20, 22, 26, 28, 33, 36, 42, 46, 53, 58, 67, 73, 83, 91, 104, 113, 128, 140, 158, 173, 194, 212, 238, 260, 290, 317, 353, 385, 428, 467, 517, 564

Offset: 0

N. J. A. Sloane, Nov 18 2015

nonn

It is conjectured that G[i](q) = 1 + O(q^i) for all i.
For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[4](x). - N. J. A. Sloane, Nov 22 2015
From Wolfdieter Lang, Nov 02 2016: (Start)
The second g.f. given below leads to a combinatorial partition interpretation from (2 + 4 + ... + 2*m) + 2*m = m*(m+3). Take for the sum term m the special M=m+1 part partition [2m,2m,2*(m-1),...,4,2] together with arbitrary partitions of N with part number m' <= M-1 = m added to the first m' parts.
Summing over m>=1 leads to partitions of n = m*(m+3) + N which have no part 1, only one part 2 except for n=4 and for number of parts M >= 3 the difference of parts except of the first two parts has to be at least 2. See the examples below.
A simpler interpretation uses m*(m+3) = 4 + 6 + ... + 2*(m+1), leading to a(n) as the number of partitions of n with parts >= 4 and parts differing by at least 2.
This is in the spirit of MacMahon's and Schur's interpretation of the sum version of the Rogers-Ramanujan identities. See the Hardy and Hardy-Wright references under A003114. (End)

			From _Wolfdieter Lang_, Nov 02 2016: (Start)
a(0) = 1 from the n=0 sum term (undefined product put to 1),
a(n) = 1 for n=4..9 from the partition [n-2,2],
a(10) = 2 from [8,2] and [4,4,2],
a(11) = 2 from [9,2] and [5,4,2],
a(12) = 3 from [10,2], [6,4,2], [5,5,2],
a(18) = 7 from [16,2], all 1+4=5 partitions of 18-10 = 8 with part number <= 2 added to the first two part of [4,4,2] and the new four part partition [6,6,4,2].
The maximal number of parts needed for n is floor((-1+sqrt(9+4*n))/2) = A259361(n+2).
A simpler interpretation:
a(18) = 7 from the partitions of 18 with parts >=4 and parts differing by at least 2: [18], [14,4], [13,5], [12,6], [11,7], [10,8], [8,6,4].
The maximal number of parts needed for n is floor((-3+sqrt(9+4*n))/2).
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
George E. Andrews; R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Handout, Math. Dept., Rutgers University, April 2015.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Dissertation, Math. Dept., Rutgers University, April 2015.
James Lepowsky and Minxian Zhu, A motivated proof of Gordon's identities, The Ramanujan Journal 29.1-3 (2012): 199-211.

For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.

nmax = 100; CoefficientList[Series[Sum[x^(k*(k+3))/Product[1-x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 24 2016 *)

From Wolfdieter Lang, Nov 02 2016: (Start)
G.f.: G[4](q) = (Sum_{n >= 0} (-1)^n*(1 - q^(n+1))*(1 - q^(n+2))*(1 - q^(2*n+3))*q^((5*n+11)*n/2)) / (Product_{j >= 1} (1 - q^j)), from the Andrews-Baxter (AB) reference, eq. (3.7).
G.f.: Sum_{m >= 0} q^(m*(m+3)) / Product_{j=1..m} (1-q^j) from (AB) eq. 51.
This can also be derived from the Hardy (H) or Hardy-Wright reference (see A006141): Put G_4(a,q):= (H_1(a,q) - H_1(a*q,q)) / (a*q) with H_1(a,x) from (H) p. 95, first eq. Then G[4](q) = G_4(q,q). (End)
a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * sqrt(5) * phi^(5/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Dec 24 2016

A264592 Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[5](q).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 23, 25, 29, 32, 36, 40, 46, 50, 57, 63, 71, 78, 88, 96, 108, 119, 132, 145, 162, 177, 197, 216, 239, 262, 290, 317, 350, 383, 421, 460, 507, 552, 606, 661, 724, 789, 864, 939, 1027, 1117

Offset: 0

N. J. A. Sloane, Nov 19 2015

nonn

It is conjectured that G[i](q) = 1 + O(q^i) for all i.
From Wolfdieter Lang, Nov 03 2016: (Start)
The generalized Rogers-Ramanujan [R-R] series G[i](q) of Andrews and Baxter [A-B] have a standard combinatorial interpretation of the Schur and MacMahon type (see Hardy [H] and Hardy-Wright [H-W] for the original [R-R] case) inferred from the formula G[i](q) = Sum_{m>=0} q^(m*(m+i-1))/Product_{j=1..m} (1 - q^j) ([A-B], eq. (5.1)). Define GI_k(q) = G[2*k+1](q) and GII_k(q) = G[2*k](q), for k = 0, 1,..., and prove the two formulas I(m,k): m*(m+2*k) = Sum_{j = 1..2*m-1} (2*k + j), and II(m,k): m*(m+2*k+1) = Sum_{j = 1..m} (2*(k + j)) for fixed positive m by induction over k = 0, 1, ... . For GI_k(q) define the special m-part partition SPI(m,k) = [2*k+2*m-1,2*k+2*m-3,...,2*k+1] of m*(m+2*k), and for GII_k(q) the special m-part partition SPII(m,k) [2*(k+1),2*(k+2),...,2*(k+1))] of m*(m+2*k+1).
Then GI_k(q) = 1 + Sum_{n >=1} aI(k,n)*q^n with aI(k,n) the number of partitions of n without parts 1, 2, ..., 2*k, and the parts differ by at least 2. GII_k(q) = 1 + Sum_{n >=1} aII(k,n)*q^n with aII(k,n) the number of partitions of n without parts 1, 2, ..., 2*k+1, and the parts differ by at least 2. The proof can be directly adapted from the one given in [H] or [H-W] for k=1.
For the partitions of n generated by GI_k(q) one needs the maximal part number MmaxI(k,n) = floor(-k + sqrt(k^2 + n)). For the GII_k(q) case MmaxII(k,n) = floor(-(2*k+1) + sqrt((2*k+1)^2 + 4*n)).
The present sequence is aI(2,n), in [A-B] notation generated by G[5](q), giving the number of partitions of n without parts 1, 2, 3 and 4, and the parts differ by at least 2.
(End)

			From _Wolfdieter Lang_, Nov 03 2016: (Start)
a(5) = 1 because the only partition of n = 5 without parts 1, 2, 3 and 4, and parts differing by at least 2 is [5].
a(12) = 2 from the two partitions [12] and [7,5] of n = 12.
a(18) = 5 from the five partitions [18], [13,5], [12,6], [11,7], [10,8] of n = 18.
(End)

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 91-92.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, pp. 290-291.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
George E. Andrews; R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Handout, Math. Dept., Rutgers University, April 2015.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Dissertation, Math. Dept., Rutgers University, April 2015.

For the generalized Rogers-Ramanujan series G[0], G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003113, A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595.

nmax = 100; CoefficientList[Series[Sum[x^(k*(k+4))/Product[1-x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 24 2016 *)

From Wolfdieter Lang, Nov 03 2016: (Start)
G.f.: G[5](q) = GI_2(q) = Sum_{m>=0} q^(m*(m+4))/Product_{j=1..m} (1 - q^j).
  See [A-B], eq. (5.1) for i=5.
a(0) = 1 and a(n) gives the number of partitions of n without part 1 and 2, the parts differing by at least 2.
G.f.: Sum_{m=0} ((-1)^m*(1 - q^(m+1))*(1 - q^(m+2))*(1 - q^(m+3))*(1 - q^(2*(m+2))) * q^(5*(n+3)*n/2)) / Product_{j>=1} (1 - q^j). See [A-B], eq. (3.8) for i=5. (End)
a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * sqrt(5) * phi^(7/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Dec 24 2016

A264593 Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[6](q).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 11, 13, 14, 17, 18, 21, 23, 26, 28, 33, 35, 40, 44, 50, 54, 62, 67, 76, 83, 93, 101, 114, 123, 138, 150, 167, 181, 202, 219, 243, 264, 292, 317, 351, 380, 419, 455, 500, 542, 596, 645, 707, 766, 838, 907, 992, 1072

Offset: 0

N. J. A. Sloane, Nov 19 2015

nonn

It is conjectured that G[i](q) = 1 + O(q^i) for all i.

For n >=1 a(n) gives the number of partitions of n without parts 1, 2, 3, 4, and 5, and the parts differ by at least 2. For the proof see a comment given in A264592. - Wolfdieter Lang, Nov 10 2016

			a(18) = 4 because the four partitions of 18 without parts 1, 2, 3, 4 and 5, and the parts differ by at least 2 are [18], [12, 6], [11, 7], [10, 8]. - _Wolfdieter Lang_, Nov 10 2016

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
George E. Andrews; R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Handout, Math. Dept., Rutgers University, April 2015.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Dissertation, Math. Dept., Rutgers University, April 2015.

For the generalized Rogers-Ramanujan series G[0], G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003113, A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595.

nmax = 100; CoefficientList[Series[Sum[x^(k*(k+5))/Product[1-x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 24 2016 *)

From Wolfdieter Lang, Nov 10 2016: (Start)
G.f.: G[6](q) = GII_2(q) = Sum_{m>=0} q^(m*(m+5)) / Product_{j =1..m} (1 - q^j).
See Andrews and Baxter [A-B], eq. (5.1) for i=6.
G.f.: Sum_{m=0} ((-1)^m*(1 - q^(m+1))*(1 - q^(m+2))*(1 - q^(m+3))*(1 - q^(m+4))*(1 - q^(2*m+5))*q^((5*m+19)*m/2)) / Product_{j>=1} (1 - q^j). See [A-B] eq. (3.8) for i=6. (End)
a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * sqrt(5) * phi^(9/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Dec 24 2016

A264594 Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[7](q).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 10, 12, 13, 15, 17, 19, 21, 24, 26, 29, 32, 36, 39, 44, 48, 54, 59, 66, 72, 81, 88, 98, 107, 119, 129, 143, 156, 172, 187, 206, 224, 247, 268, 294, 320, 351, 381, 417, 453, 495, 537, 586, 635, 693, 750, 816

Offset: 0

N. J. A. Sloane, Nov 19 2015

nonn

It is conjectured that G[i](q) = 1 + O(q^i) for all i.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Handout, Math. Dept., Rutgers University, April 2015.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Dissertation, Math. Dept., Rutgers University, April 2015.

For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595.

nmax = 100; CoefficientList[Series[Sum[x^(k*(k+6))/Product[1-x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 24 2016 *)

a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * sqrt(5) * phi^(11/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Dec 24 2016

A264595 Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[8](q).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 9, 9, 11, 12, 14, 15, 18, 19, 22, 24, 27, 29, 33, 35, 40, 43, 48, 52, 59, 63, 71, 77, 86, 93, 104, 112, 125, 135, 149, 161, 179, 192, 212, 229, 252, 272, 299, 322, 354, 382, 418, 451, 494, 532, 581, 627, 683

Offset: 0

N. J. A. Sloane, Nov 19 2015

nonn

It is conjectured that G[i](q) = 1 + O(q^i) for all i.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Handout, Math. Dept., Rutgers University, April 2015.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Dissertation, Math. Dept., Rutgers University, April 2015.

For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595.

nmax = 100; CoefficientList[Series[Sum[x^(k*(k+7))/Product[1-x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 24 2016 *)

a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * sqrt(5) * phi^(13/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Dec 24 2016

A327690 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A003114.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 14, 19, 31, 43, 64, 88, 131, 176, 250, 337, 471, 626, 859, 1133, 1532, 2008, 2674, 3479, 4595, 5933, 7745, 9952, 12888, 16451, 21142, 26842, 34260, 43283, 54878, 68993, 87017, 108884, 136564, 170191, 212441, 263646, 327616, 405034, 501203, 617423, 760964

Offset: 0

Seiichi Manyama, Sep 22 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)

Cf. A003114, A327688, A327691.

nmax = 50; CoefficientList[Series[Product[1/(QPochhammer[x^(5*j - 4)] * QPochhammer[x^(5*j - 1)]), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)

G.f.: Product_{i>=1} Product_{j>=1} 1 / ((1-x^(i*(5*j-1))) * (1-x^(i*(5*j-4)))).

A145467 Convolution square of A003114.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 15, 20, 28, 38, 52, 68, 91, 118, 153, 196, 252, 318, 403, 504, 632, 784, 973, 1196, 1473, 1800, 2198, 2668, 3238, 3908, 4714, 5660, 6789, 8112, 9683, 11516, 13685, 16210, 19178, 22628, 26671, 31354, 36821, 43140, 50489, 58968, 68796

Offset: 0

Michael Somos, Oct 11 2008

nonn

			1/q + 2*q^29 + 3*q^59 + 4*q^89 + 7*q^119 + 10*q^149 + 15*q^179 + ...

Georg Fischer, Table of n, a(n) for n = 0..1000

Cf. A003114, A145466, A001622.

# Using the function EULER from Transforms (see link at the bottom of the page).
[1,op(EULER([seq(op([2,0,0,2,0]),n=1..9)]))]; # Peter Luschny, Aug 19 2020
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*add(`if`(irem(d, 5) in {1, 4}, 2*d, 0),
       d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 19 2020

nmax = 50; CoefficientList[Series[Product[1/((1 - x^(5*k - 1))*(1 - x^(5*k - 4)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 14 2018 *)

{a(n) = local(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k - 1) / (1 - x^k) * (1 + x * O(x^(n - k^2))), 1)^2, n))}

a(n) = A145466(5*n).
Expansion of G(x)^2 in powers of x where G() is a Rogers-Ramanujan function.
a(n) ~ exp(2*Pi*sqrt(2*n/15)) * phi / (3^(1/4) * 10^(3/4) * n^(3/4)) * (1 - (3*sqrt(15/2)/(16*Pi) + Pi/(15*sqrt(30)))/sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 14 2018
Euler transform of the period 5 sequence [2, 0, 0, 2, 0, ...]. - Georg Fischer, Aug 19 2020

A217569 Expansion of H(q)*G(q^11) where H and G are respectively the g.f. of A003106 and A003114 (Rogers-Ramanujan functions).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 31, 35, 42, 47, 55, 62, 72, 81, 94, 105, 121, 136, 155, 175, 199, 222, 252, 282, 318, 355, 400, 445, 501, 556, 624, 693, 774, 857, 957, 1059, 1178, 1302, 1446, 1596, 1769, 1951, 2158, 2376, 2624, 2885, 3182, 3495, 3847, 4221, 4642

Offset: 0

Joerg Arndt, Oct 07 2012

nonn

Also the expansion of 1+q^2*H(q^11)*G(q); that is, H(q)*G(q^11) - q^2*G(q)*H(q^11) = 1, we also have H(q)*G(q)^11 - q^2*G(q)*H(q)^11 = 1 + 11*q*(G(q)*H(q))^6, see the Ramanujan reference.
Number of partitions of n into parts t such that t mod 55 is in {2, 3, 7, 8, 11, 12, 13, 17, 18, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 44, 47, 48, 52, 53}.
With E(q) = Product_{n>=1} (1-q^n) we have G(q)*H(q) - E(q^5)/E(q), G(q) = ( E(q^8)/E(q^2) * (G(q^16) + q*H(-q^4)) ), and H(q) = ( E(q^8)/E(q^2) * (q^3*H(q^16) + G(-q^4)) ), see the Berkovich/Yesilyurt reference.

Alexander Berkovich, Hamza Yesilyurt, On Rogers-Ramanujan functions, binary quadratic forms and eta-quotients, arXiv:1204.1092v2 [math.NT], 2012.
Srinivasa Ramanujan, Algebraic relations between certain infinite products, Proceedings of the London Mathematical Society, vol.2, no.18, 1920.

Cf. A003106 and A003114 (Rogers-Ramanujan functions H and G).

Cf. A121591 (expansion of q*(G(q)*H(q))^6).

N=66; q='q+O('q^N );
S=2+2*ceil(sqrt(N));
G(q)=sum(n=0,S,q^(n^2)/prod(k=1,n,1-q^k)); /* g.f. of  A003114 */
H(q)=sum(n=0,S,q^(n^2+n)/prod(k=1,n,1-q^k)); /* g.f. of A003106 */
Vec(H(q)*G(q^11)) /* show terms */
/* checking the modular equations, all expressions are zero:
( H(q)*G(q)^11 - q^2*G(q)*H(q)^11 ) - ( 1 + 11*q*(G(q)*H(q))^6 )
( H(q)*G(q^11) - q^2*G(q)*H(q^11) ) - ( 1 )
E(q)=prod(n=1,N, 1-q^n);
G(q)*H(q) - E(q^5)/E(q)
G(q) - ( E(q^8)/E(q^2) * (G(q^16) + q*H(-q^4)) )
H(q) - ( E(q^8)/E(q^2) * (q^3*H(q^16) + G(-q^4)) )
*/

N=66; q='q+O('q^N );
E=[2,3,7,8,11,12,13,17,18,22,23,27,28,32,33,37,38,42,43,44,47,48,52,53];
Vec( 1 / prod(K=0, N\55+1,  prod(k=1,24, 1 - q^(K*55+E[k]) ) ) )

G.f.: H(q)*G(q^11) where G(q) = Sum_{n>=0} q^(n^2)/Product_{k=1..n} (1-q^k) and H(q) = Sum_{n>=0} q^(n^2+n)/Product_{k=1..n} (1-q^k).

G.f.: 1 / Product_{k>=0} (1 - q^k) where k (mod 55) is restricted to the set {2, 3, 7, 8, 11, 12, 13, 17, 18, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 44, 47, 48, 52, 53} (the set has 24 elements).

A047993 Number of balanced partitions of n: the largest part equals the number of parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 2, 4, 4, 6, 7, 11, 11, 16, 19, 25, 29, 40, 45, 60, 70, 89, 105, 134, 156, 196, 232, 285, 336, 414, 485, 591, 696, 839, 987, 1187, 1389, 1661, 1946, 2311, 2702, 3201, 3731, 4400, 5126, 6018, 6997, 8195, 9502, 11093, 12849, 14949, 17281, 20062

Offset: 1

Wouter Meeussen

Useful in the creation of plane partitions with C3 or C3v symmetry.
The function T[m,a,b] used here gives the partitions of m whose Ferrers plot fits within an a X b box.
Central terms of triangle in A063995: a(n) = A063995(n,0). - Reinhard Zumkeller, Jul 24 2013
Sequence enumerates the collection of partitions of size n that are in the monoid of Dyson rank=0, or balanced partitions, under the binary operation A*B = (a1,a2,...,a[k-1],k)*(b1,...,b[n-1,n) = (a1*b1,...,a1*n,a2*b1,...,a2*n,...,k*b1,...,k*n), where A is a partition with k parts and B is a partition with n parts, and A*B is a partition with k*n parts. Note that the rank of A*B is 0, as required. For example, the product of the rank 0 partitions (1,2,3) of 6 and (1,1,3) of 5 is the rank 0 partition (1,1,2,2,3,3,3,6,9) of 30. There is no rank zero partition of 2, as shown in the sequence. It can be seen that any element of the monoid that partitions an odd prime p or a composite number of form 2p cannot be a product of smaller nontrivial partitions, whether in this monoid or not. - Richard Locke Peterson, Jul 15 2018
The "multiplication" given above was noted earlier by Franklin T. Adams-Watters in A122697. - Richard Peterson, Jul 19 2023
The Heinz numbers of these integer partitions are given by A106529. - Gus Wiseman, Mar 09 2019

			From _Joerg Arndt_, Oct 08 2012: (Start)
a(12) = 7 because the partitions of 12 where the largest part equals the number of parts are
   2 + 3 + 3 + 4,
   2 + 2 + 4 + 4,
   1 + 3 + 4 + 4,
   1 + 2 + 2 + 2 + 5,
   1 + 1 + 2 + 3 + 5,
   1 + 1 + 1 + 4 + 5, and
   1 + 1 + 1 + 1 + 2 + 6.
(End)
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(1) = 1 through a(13) = 11 integer partitions:
  1  21  22  311  321  322   332   333    4222   4322    4332    4333
                       331   4211  4221   4321   4331    4422    4432
                       4111        4311   4411   4421    4431    4441
                                   51111  52111  52211   52221   52222
                                                 53111   53211   53221
                                                 611111  54111   53311
                                                         621111  54211
                                                                 55111
                                                                 622111
                                                                 631111
                                                                 7111111
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Erich Friedman, Illustration of initial terms

Cf. A000700, A063995, A064173, A064174.
Cf. A003114, A006141, A039900, A090858, A106529, A324516, A324518, A324520.
Column 1 of A350879.

a047993 = flip a063995 0  -- Reinhard Zumkeller, Jul 24 2013

A047993 := proc(n)
     a := 0 ;
     for p in combinat[partition](n) do
        r := max(op(p))-nops(p) ;
        if r = 0 then
             a := a+1 ;
        end if;
     end do:
     a ;
 end proc:
seq(A047993(n),n=1..20) ; # Emeric Deutsch, Dec 11 2004

Table[ Count[Partitions[n], par_List/; First[par]===Length[par]], {n, 12}] or recur: Sum[T[n-(2m-1), m-1, m-1], {m, Ceiling[Sqrt[n]], Floor[(n+1)/2]}] with T[m_, a_, b_]/; b < a := T[m, b, a]; T[m_, a_, b_]/; m > a*b := 0; T[m_, a_, b_]/; (2m > a*b) := T[a*b-m, a, b]; T[m_, 1, b_] := If[b < m, 0, 1]; T[0, , ] := 1; T[m_, a_, b_] := T[m, a, b]=Sum[T[m-a*i, a-1, b-i], {i, 0, Floor[m/a]}];
Table[Sum[ -(-1)^k*(p[n-(3*k^2-k)/2] - p[n-(3*k^2+k)/2]), {k, 1, Floor[(1+Sqrt[1+24*n])/6]}] /. p -> PartitionsP, {n, 1, 64}] (* Wouter Meeussen *)
(* also *)
Table[Count[IntegerPartitions[n], q_ /; Max[q] == Length[q]], {n, 24}]
(* Clark Kimberling, Feb 13 2014 *)
nmax = 100; p = 1; s = 1; Do[p = Normal[Series[p*x^2*(1 - x^(2*k - 1))*(1 + x^k)/(1 - x^k), {x, 0, nmax}]]; s += p;, {k, 1, nmax + 1}]; Take[CoefficientList[s, x], nmax] (* Vaclav Kotesovec, Oct 16 2024 *)

N=66;  q='q + O('q^N );
S=2+2*ceil(sqrt(N));
gf= sum(k=1, S,  (-1)^k * ( q^((3*k^2+k)/2) - q^((3*k^2-k)/2) ) ) / prod(k=1,N, 1-q^k );
/* Joerg Arndt, Oct 08 2012 */

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^(2*k-1)*prod(j=1, k-1, (1-x^(k+j-1))/(1-x^j)))) \\ Seiichi Manyama, Jan 24 2022

a(n) = p(n-1) - p(n-2) - p(n-5) + p(n-7) + ... + (-1)^k*(p(n-(3*k^2-k)/2) - p(n-(3*k^2+k)/2)) + ..., where p() is A000041(). E.g., a(20) = p(19) - p(18) - p(15) + p(13) + p(8) - p(5) = 490 - 385 - 176 + 101 + 22 - 7 = 45. - Vladeta Jovovic, Aug 04 2004
G.f.: ( Sum_{k>=1} (-1)^k * ( x^((3*k^2+k)/2) - x^((3*k^2-k)/2) ) ) / Product_{k>=1} (1-x^k). - Vladeta Jovovic, Aug 05 2004
a(n) ~ exp(Pi*sqrt(2*n/3))*Pi / (48*sqrt(2)*n^(3/2)) ~ p(n) * Pi / (4*sqrt(6*n)), where p(n) is the partition function A000041. - Vaclav Kotesovec, Oct 06 2016
G.f.: Sum_{n>=0} [2n,n]q q^(2*n), where [m,n]_q are the q-binomial coefficients. - _Mamuka Jibladze, Aug 12 2021
G.f.: Sum_{k>=1} x^(2*k-1) * Product_{j=1..k-1} (1-x^(k+j-1))/(1-x^j). - Seiichi Manyama, Jan 24 2022

A001156 Number of partitions of n into squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 8, 9, 10, 10, 12, 13, 14, 14, 16, 19, 20, 21, 23, 26, 27, 28, 31, 34, 37, 38, 43, 46, 49, 50, 55, 60, 63, 66, 71, 78, 81, 84, 90, 98, 104, 107, 116, 124, 132, 135, 144, 154, 163, 169, 178, 192, 201, 209, 220, 235, 247, 256

Offset: 0

N. J. A. Sloane

Number of partitions of n such that number of parts equal to k is multiple of k for all k. - Vladeta Jovovic, Aug 01 2004
Of course p_{4*square}(n)>0. In fact p_{4*square}(32n+28)=3 times p_{4*square}(8n+7) and p_{4*square}(72n+69) is even. These seem to be the only arithmetic properties the function p_{4*square(n)} possesses. Similar results hold for partitions into positive squares, distinct squares and distinct positive squares. - Michael David Hirschhorn, May 05 2005
The Heinz numbers of these partitions are given by A324588. - Gus Wiseman, Mar 09 2019

			p_{4*square}(23)=1 because 23 = 3^2 + 3^2 + 2^2 + 1^2 and there is no other partition of 23 into squares.
G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 +...
such that the g.f. A(x) satisfies the identity [_Paul D. Hanna_]:
A(x) = 1/((1-x)*(1-x^4)*(1-x^9)*(1-x^16)*(1-x^25)*...)
A(x) = 1 + x/(1-x) + x^4/((1-x)*(1-x^4)) + x^9/((1-x)*(1-x^4)*(1-x^9)) + x^16/((1-x)*(1-x^4)*(1-x^9)*(1-x^16)) + ...
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(14) = 6 integer partitions into squares are:
  (941)
  (911111)
  (44411)
  (44111111)
  (41111111111)
  (11111111111111)
while the a(14) = 6 integer partitions in which the multiplicity of k is a multiple of k for all k are:
  (333221)
  (33311111)
  (22222211)
  (2222111111)
  (221111111111)
  (11111111111111)
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301.
J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301. (Annotated scanned copy)
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.
M. D. Hirschhorn and J. A. Sellers, On a problem of Lehmer on partitions into squares, The Ramanujan Journal 8 (2004), 279-287.
F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sci. 16E, 237-240, 1997.
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.
Florentin Smarandache, Sequences of Numbers Involved in Unsolved Problems, arXiv:math/0604019 [math.GM], 2006.
Eric Weisstein's World of Mathematics, Partition
Eric Weisstein's World of Mathematics, Smarandache Sequences
Eric Weisstein's World of Mathematics, Square Number

Cf. A000041, A000161 (partitions into 2 squares), A000290, A033461, A131799, A218494, A285218, A304046.
Cf. A078134 (first differences).
Cf. A003108, A037444, A046042, A259792, A259793, A294529.
Row sums of A243148.
Euler trans. of A010052 (see also A308297).
Cf. A001462, A003114, A006141, A011757, A039900, A047993, A052335, A062457, A064174, A078135, A109298, A117144.
Cf. A324524, A324572, A324587, A324588.

a001156 = p (tail a000290_list) where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Oct 31 2012, Aug 14 2011

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^(k^2)): k in [1..(m+2)]]) )); // G. C. Greubel, Nov 11 2018

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+ `if`(i^2>n, 0, b(n-i^2, i))))
    end:
a:= n-> b(n, isqrt(n)):
seq(a(n), n=0..120);  # Alois P. Heinz, May 30 2014

CoefficientList[ Series[Product[1/(1 - x^(m^2)), {m, 70}], {x, 0, 68}], x] (* Or *)
Join[{1}, Table[Length@PowersRepresentations[n, n, 2], {n, 68}]] (* Robert G. Wilson v, Apr 12 2005, revised Sep 27 2011 *)
f[n_] := Length@ IntegerPartitions[n, All, Range@ Sqrt@ n^2]; Array[f, 67] (* Robert G. Wilson v, Apr 14 2013 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2>n, 0, b[n-i^2, i]]]]; a[n_] := b[n, Sqrt[n]//Floor]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Nov 02 2015, after Alois P. Heinz *)

{a(n)=polcoeff(1/prod(k=1, sqrtint(n+1), 1-x^(k^2)+x*O(x^n)), n)} \\ Paul D. Hanna, Mar 09 2012

{a(n)=polcoeff(1+sum(m=1, sqrtint(n+1), x^(m^2)/prod(k=1, m, 1-x^(k^2)+x*O(x^n))), n)} \\ Paul D. Hanna, Mar 09 2012

G.f.: Product_{m>=1} 1/(1-x^(m^2)).
G.f.: Sum_{n>=0} x^(n^2) / Product_{k=1..n} (1 - x^(k^2)). - Paul D. Hanna, Mar 09 2012
a(n) = (1/n)*Sum_{k=1..n} A035316(k)*a(n-k). - Vladeta Jovovic, Nov 20 2002
a(n) = f(n,1,3) with f(x,y,z) = if xReinhard Zumkeller, Nov 08 2009
Conjecture (Jan Bohman, Carl-Erik Fröberg, Hans Riesel, 1979): a(n) ~ c * n^(-alfa) * exp(beta*n^(1/3)), where c = 1/18.79656, beta = 3.30716, alfa = 1.16022. - Vaclav Kotesovec, Aug 19 2015
From Vaclav Kotesovec, Dec 29 2016: (Start)
Correct values of these constants are:
1/c = sqrt(3) * (4*Pi)^(7/6) / Zeta(3/2)^(2/3) = 17.49638865935104978665...
alfa = 7/6 = 1.16666666666666666...
beta = 3/2 * (Pi/2)^(1/3) * Zeta(3/2)^(2/3) = 3.307411783596651987...
a(n) ~ 3^(-1/2) * (4*Pi*n)^(-7/6) * Zeta(3/2)^(2/3) * exp(2^(-4/3) * 3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3)). [Hardy & Ramanujan, 1917]
(End)

More terms from Eric W. Weisstein

More terms from Gh. Niculescu (ghniculescu(AT)yahoo.com), Oct 08 2006

cp's OEIS Frontend

A264591 Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[4](q).

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A264592 Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[5](q).

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A264593 Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[6](q).

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A264594 Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[7](q).

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A264595 Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[8](q).

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A327690 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A003114.

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A145467 Convolution square of A003114.

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A217569 Expansion of H(q)*G(q^11) where H and G are respectively the g.f. of A003106 and A003114 (Rogers-Ramanujan functions).

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