A001935 -id:A001935 - cp's OEIS frontend

A328546 Number of 12-regular partitions of n (no part is a multiple of 12).

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 100, 133, 173, 226, 290, 374, 475, 605, 762, 960, 1199, 1497, 1856, 2299, 2831, 3482, 4261, 5208, 6337, 7700, 9321, 11266, 13572, 16325, 19578, 23444, 27999, 33389, 39721, 47185, 55929, 66199, 78199, 92246

Offset: 0

N. J. A. Sloane, Oct 19 2019

nonn

Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

f:=(k,M) -> mul(1-q^(k*j),j=1..M);
LRP := (L,M) -> f(L,M)/f(1,M);
s := L -> seriestolist(series(LRP(L,80),q,60));
s(12);

Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 12], 0, 2] ], {n, 0, 46}] (* Robert Price, Jul 28 2020 *)

a(n) ~ exp(Pi*sqrt(2*n*(s-1)/(3*s))) * (s-1)^(1/4) / (2 * 6^(1/4) * s^(3/4) * n^(3/4)) * (1 + ((s-1)^(3/2)*Pi/(24*sqrt(6*s)) - 3*sqrt(6*s) / (16*Pi * sqrt(s-1))) / sqrt(n) + ((s-1)^3*Pi^2/(6912*s) - 45*s/(256*(s-1)*Pi^2) - 5*(s-1)/128) / n), set s=12. - Vaclav Kotesovec, Aug 01 2022

A092877 Expansion of (eta(q^4) / eta(q))^8 in powers of q.

Original entry on oeis.org

1, 8, 44, 192, 718, 2400, 7352, 20992, 56549, 145008, 356388, 844032, 1934534, 4306368, 9337704, 19771392, 40965362, 83207976, 165944732, 325393024, 628092832, 1194744096, 2241688744, 4152367104, 7599231223, 13749863984

Offset: 1

Michael Somos, Mar 19 2004

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q + 8*q^2 + 44*q^3 + 192*q^4 + 718*q^5 + 2400*q^6 + 7352*q^7 + 20992*q^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001935, A001936, A005798, A014103, A093160, A124972.

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ -InverseEllipticNomeQ[ -x] / 16, {x, 0, n}]]; (* Michael Somos, Jun 13 2011 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ With[ {lambda = ModularLambda[ Log[x] / ( Pi I)]}, lambda / (16 * (1 - lambda))], {x, 0, n}]]; (* Michael Somos, Jun 13 2011 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^4] / QPochhammer[ q])^8, {q, 0, n}]; (* Michael Somos, Aug 09 2015 *)
a[1] = 1; a[n_] := a[n] = (8/(n-1))*Sum[DivisorSum[k, Identity, Mod[#, 4] != 0&]*a[n-k], {k, 1, n-1}]; Array[a, 26] (* Jean-François Alcover, Mar 01 2018, after Seiichi Manyama *)
eta[q_]:= q^(1/6) QPochhammer[q]; a[n_]:=SeriesCoefficient[(eta[q^4] / eta[q])^8, {q, 0, n}]; Table[a[n], {n, 4, 35}] (* Vincenzo Librandi, Oct 18 2018 *)

{a(n) = if( n<0, 0, polcoeff( x * prod(k=1, (n+1)\2, (1 + x^(2*k)) / (1 - x^(2*k-1)), 1 + x * O(x^n))^8, n))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^8, n))};

Expansion of q * (psi(-q) / phi(-q))^8 = q * (psi(q^2) / psi(-q))^8 = q * (psi(q) / phi(-q^2))^8 = q * (psi(q^2) / phi(-q))^4 = q * (chi(q) / chi(-q^2)^2)^8 = q / (chi(-q) * chi(-q^2))^8 = q / (chi(q) * chi(-q)^2)^8 = q * (f(-q^4) / f(-q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jun 13 2011
Euler transform of period 4 sequence [ 8, 8, 8, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 16*u*v - 16*v^2 - 256*u*v^2.
G.f.: x * (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^8.
G.f.: theta_2^4 / (16*theta_4^4) = lambda / (16 * (1 - lambda)).
G.f.: exp( Integral theta_3(x)^4/x dx ). - Paul D. Hanna, May 03 2010
a(n) = (-1)^n * A005798(n).
a(2*n) = 8 * A014103(n). - Michael Somos, Aug 09 2015
Convolution inverse of A124972, 8th power of A001935, 4th power of A001936, square of A093160. - Michael Somos, Aug 09 2015
a(n) ~ exp(2*Pi*sqrt(n))/(512*n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
a(1) = 1, a(n) = (8/(n-1))*Sum_{k=1..n-1} A046897(k)*a(n-k) for n > 1. - Seiichi Manyama, Apr 01 2017

A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^n.

Original entry on oeis.org

1, 1, 5, 22, 101, 481, 2330, 11425, 56549, 281911, 1413465, 7120136, 36006362, 182681916, 929461993, 4740491107, 24229115109, 124069449335, 636376573943, 3268955179686, 16814509004601, 86593280920756, 446437797872016, 2303948443259841, 11900990745759578, 61526182236027756

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Main diagonal of A296068.

Cf. A001935, A001936, A001937, A001939, A001940, A001941, A092877, A093160, A295831, A296043, A296045.

Table[SeriesCoefficient[Product[((1 + x^(2 k))/(1 - x^(2 k - 1)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[((1 - x^(4 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[(EllipticTheta[2, 0, x]/EllipticTheta[2, Pi/4, x^(1/2)]/(16 x)^(1/8))^n, {x, 0, n}], {n, 0, 25}]
(* Calculation of constant d: *) With[{k = 4}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=1} ((1 - x^(4*k))/(1 - x^k))^n.

a(n) ~ c * d^n / sqrt(n), where d = 5.2749356339591798618290252741994029798069148326559... and c = 0.2726256757090475625917361066565981461455343437... - Vaclav Kotesovec, Dec 05 2017

A286653 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^(k*j))/(1 - x^j).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 3, 4, 3, 0, 1, 1, 2, 3, 4, 5, 4, 0, 1, 1, 2, 3, 5, 6, 7, 5, 0, 1, 1, 2, 3, 5, 6, 9, 9, 6, 0, 1, 1, 2, 3, 5, 7, 10, 12, 13, 8, 0, 1, 1, 2, 3, 5, 7, 10, 13, 16, 16, 10, 0, 1, 1, 2, 3, 5, 7, 11, 14, 19, 22, 22, 12, 0

Offset: 0

Ilya Gutkovskiy, May 11 2017

A(n,k) is the number of partitions of n in which no parts are multiples of k.

A(n,k) is also the number of partitions of n into at most k-1 copies of each part.

			Square array begins:
  1,  1,  1,  1,  1,  1,  ...
  0,  1,  1,  1,  1,  1,  ...
  0,  1,  2,  2,  2,  2,  ...
  0,  2,  2,  3,  3,  3,  ...
  0,  2,  4,  4,  5,  5,  ...
  0,  3,  5,  6,  6,  7,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Partition Function b_k
Index entries for sequences related to partitions

Columns k=1-13 give: A000007, A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546, A341714.
Main diagonal gives A000041.
Mirror of A061198.
Cf. A061199, A210485.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(k*i*(i+1)/2[0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
A:= (n, k)-> b(n$2, k-1)[1]:
seq(seq(A(n, 1+d-n), n=0..d), d=0..16);  # Alois P. Heinz, Oct 17 2018

Table[Function[k, SeriesCoefficient[Product[(1 - x^(i k))/(1 - x^i), {i, Infinity}], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[x^k, x^k]/QPochhammer[x, x], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 - x^(k*j))/(1 - x^j).

A001937 Expansion of (psi(x^2) / psi(-x))^3 in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 3, 9, 22, 48, 99, 194, 363, 657, 1155, 1977, 3312, 5443, 8787, 13968, 21894, 33873, 51795, 78345, 117312, 174033, 255945, 373353, 540486, 776848, 1109040, 1573209, 2218198, 3109713, 4335840, 6014123, 8300811, 11402928, 15593702, 21232521, 28790667, 38884082

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

The Cayley reference is actually to A187053. - Michael Somos, Jul 26 2012

			1 + 3*x + 9*x^2 + 22*x^3 + 48*x^4 + 99*x^5 + 194*x^6 + 363*x^7 + 657*x^8 + ...
q^3 + 3*q^11 + 9*q^19 + 22*q^27 + 48*q^35 + 99*q^43 + 194*q^51 + 363*q^59 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001936, A079006, A001935, A083365, A001938, A093160, A001939, A001940, A001941.

g100:= mul((1+x^(2*k))/(1-x^(2*k-1)),k=1..50)^3:
S:= series(g100,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Nov 30 2015

CoefficientList[ Series[Product[(1 - x^k)^(-3*Boole[Mod[k, 4] != 0]), {k, 1, 101}], {x, 0, 100}], x] (* Olivier GERARD, May 06 2009 *)
QP = QPochhammer; s = (QP[q^4]/QP[q])^3 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^3, n))} /* Michael Somos, Mar 06 2011 */

Expansion of q^(-3/8) * (eta(q^4) / eta(q))^3 in powers of q. - Michael Somos, Jul 26 2012
Euler transform of period 4 sequence [ 3, 3, 3, 0, ...]. - Michael Somos, Mar 06 2011
Convolution cube of A001935. A187053(n) = (-1)^n * a(n). - Michael Somos, Mar 06 2011
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)))^3.
a(n) ~ 3^(1/4) * exp(sqrt(3*n/2)*Pi) / (16*2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Corrected and extended by Simon Plouffe

Checked and more terms from Olivier GERARD, May 06 2009

A066447 Number of basis partitions (or basic partitions) of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 8, 10, 13, 16, 20, 26, 32, 40, 50, 61, 74, 90, 108, 130, 156, 186, 222, 264, 313, 370, 436, 512, 600, 702, 818, 952, 1106, 1282, 1484, 1715, 1978, 2278, 2620, 3008, 3448, 3948, 4512, 5150, 5872, 6684, 7600, 8632, 9791, 11094, 12558, 14198, 16036, 18096, 20398

Offset: 0

Herbert S. Wilf, Dec 29 2001

The k-th successive rank of a partition pi = (pi_1, pi_2, ..., pi_s) of the integer n is r_k = pi_k - pi'_k, where pi' denotes the conjugate partition. A partition pi is basic if the number of dots in its Ferrers diagram is the least among all the Ferrers diagrams of partitions with the same rank vector.

The g.f. Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 - x^k) = Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 + x^k - 2*x^k) == Sum_{n >= 0} x^(n^2) (mod 2). It follows that a(n) is odd iff n is a square (Nolan et al., equation 6, p. 282). - Peter Bala, Jan 08 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
George E. Andrews, Partition Identities for Two-Color Partitions, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See p. 79.
J. M. Nolan, C. D. Savage and H. S. Wilf, Basis partitions, Discrete Math. 179 (1998), 277-283.

Row sums of A066448.
Cf. A001130, A001935, A003114, A306734, A333374.
Cf. A192918, A376841.

b := proc(n,d); option remember; if n=0 and d=0 then RETURN(1) elif n<=0 or d<=0 then RETURN(0) else RETURN(b(n-d,d)+b(n-2*d+1,d-1)+b(n-3*d+1,d-1)) fi: end: A066447 := n->add(b(n,d),d=0..n);

nmax = 60; CoefficientList[Series[Sum[x^(n^2)*Product[(1 + x^k)/(1 - x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 17 2020 *)
nmax = 60; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k - 1), {x, 0, nmax}]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Mar 17 2020 *)

N=66; x='x+O('x^N); s=sum(n=0,N,x^(n^2)*prod(k=1,n,(1+x^k)/(1-x^k))); Vec(s) /* Joerg Arndt, Apr 07 2011 */

G.f.: Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 - x^k) [Given in Nolan et al. reference]. - Joerg Arndt, Apr 07 2011
Limit_{n->infinity} a(n) / A333374(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... - Vaclav Kotesovec, Mar 17 2020
a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.193340468476900308848561788251945... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 + cosh(arccosh(53*sqrt(11/2)/64)/3) / (3*sqrt(22))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - Vaclav Kotesovec, Mar 19 2020, updated Oct 10 2024

A264905 Expansion of Product_{k>=1} (1 + x^k + x^(3*k)).

Original entry on oeis.org

1, 1, 1, 3, 2, 4, 6, 7, 8, 13, 16, 18, 26, 29, 38, 49, 58, 68, 90, 101, 125, 156, 181, 214, 263, 304, 358, 435, 505, 589, 701, 812, 939, 1115, 1275, 1485, 1736, 1991, 2286, 2667, 3038, 3489, 4028, 4588, 5240, 6036, 6833, 7787, 8904, 10078, 11429, 13020, 14698

Offset: 0

Vaclav Kotesovec, Nov 28 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000009, A000726, A001935, A100405, A266647, A266648, A266649, A266650, A266686, A275820.

nmax = 100; CoefficientList[Series[Product[1+x^k+x^(3*k), {k,1,nmax}], {x,0,nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[2]] = 1; p[[4]] = 1; Do[Do[p[[j+1]] = p[[j+1]] + p[[j - k + 1]] + If[j < 3*k, 0, p[[j - 3*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + exp(-x) + exp(-3*x)) dx = 0.9953865985263189816963357718655148864441174218433250148867... . - Vaclav Kotesovec, Jan 05 2016

A022568 Expansion of Product_{m>=1} (1+x^m)^3.

Original entry on oeis.org

1, 3, 6, 13, 24, 42, 73, 120, 192, 302, 465, 702, 1046, 1536, 2226, 3195, 4536, 6378, 8896, 12306, 16896, 23045, 31224, 42048, 56310, 75000, 99384, 131072, 172071, 224910, 292774, 379608, 490338, 631104, 809472, 1034814, 1318707, 1675344, 2122176, 2680602, 3376728, 4242432, 5316562, 6646272

Offset: 0

N. J. A. Sloane

The g.f. Product_{m >= 1} (1 + x^m)^3 = Product_{m >= 1} (1 - x^m + 2*x^m)^3 == Product_{m >= 1} (1 - x^m)^3 == Sum_{m >= 0} (-1)^m*(2*m + 1)*q^(m*(m+1)/2) (mod 2) by an identity of Jacobi. It follows that a(n) is odd iff n = m*(m + 1)/2 for some nonnegative integer m. - Peter Bala, Jan 07 2025

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
Mohammed L. Nadji and Moussa Ahmia, Congruences for L-regular tripartitions for L in {2, 3}, Integers (2024) Vol. 24, Art. No. A86. See p. 2.

Cf. A000009, A001935.

Column k=3 of A286335.

Coefficients(&*[(1+x^m)^3:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018

nmax=50; CoefficientList[Series[Product[(1+q^m)^3,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
nmax = 50; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 3; poly[[3]] = 3; poly[[4]] = 1; Do[Do[Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];, {p, 1, 3}], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Mar 31 2018 *)

x='x+O('x^51); Vec(prod(m=1, 50, (1 + x^m)^3)) \\ Indranil Ghosh, Apr 03 2017

a(n) ~ exp(Pi * sqrt(n)) / (8 * n^(3/4)) * (1 + (Pi/16 - 3/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Mar 05 2015, extended Jan 16 2017
a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(3*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
G.f.: Sum_{n >= 0} q^(n*(n+1)/2) / Sum_{n in Z} (-1)^n * q^(n^2). - Peter Bala, Jan 07 2025

A061199 Upper right triangle read by columns where T(n,k), with k >= n, is the number of partitions of k where no part appears more than n times; also partitions of k where no parts are multiples of (n+1).

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 2, 3, 0, 2, 4, 4, 5, 0, 3, 5, 6, 6, 7, 0, 4, 7, 9, 10, 10, 11, 0, 5, 9, 12, 13, 14, 14, 15, 0, 6, 13, 16, 19, 20, 21, 21, 22, 0, 8, 16, 22, 25, 27, 28, 29, 29, 30, 0, 10, 22, 29, 34, 37, 39, 40, 41, 41, 42, 0, 12, 27, 38, 44, 49, 51, 53, 54, 55, 55, 56, 0, 15, 36

Offset: 0

Henry Bottomley, Apr 20 2001

			T(2,4) = 4 since the possible partitions of 4 are on the first definition (no term more than twice) 1+1+2, 2+2, 1+3, or 4 and on the second definition (no term a multiple of 3) 1+1+1+1, 1+1+2, 2+2, or 4.
Triangle T(n,k) begins:
1, 0, 0, 0, 0, 0,  0,  0,  0,  0, ...
   1, 1, 2, 2, 3,  4,  5,  6,  8, ...
      2, 2, 4, 5,  7,  9, 13, 16, ...
         3, 4, 6,  9, 12, 16, 22, ...
            5, 6, 10, 13, 19, 25, ...
               7, 10, 14, 20, 27, ...
                  11, 14, 21, 28, ...
                      15, 21, 29, ...
                          22, 29, ...
                              30, ...

Alois P. Heinz, Columns k = 0..140, flattened

Rows effectively include A000007, A000009, A000726, A001935, A035959.
Main diagonal is A000041.
A061198 is the same table but includes cases where n>k.
T(n,2*n) gives: A232623.

b:= proc(n, i, k) option remember;
      `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k), j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(k$2, n):
seq(seq(T(n, k), n=0..k), k=0..12);  # Alois P. Heinz, Nov 27 2013

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1, k], {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[k, k, n]; Table[Table[T[n, k], {n, 0, k}], {k, 0, 12}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

A246584 Number of overcubic partitions of n.

Original entry on oeis.org

1, 2, 6, 12, 26, 48, 92, 160, 282, 470, 784, 1260, 2020, 3152, 4896, 7456, 11290, 16836, 24962, 36556, 53232, 76736, 110012, 156384, 221156, 310482, 433776, 602200, 832224, 1143696, 1565088, 2131072, 2890266, 3902344, 5249356, 7032576, 9389022, 12488368

Offset: 0

N. J. A. Sloane, Sep 03 2014

nonn

Convolution of A001935 and A002513. - Vaclav Kotesovec, Aug 16 2019

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Michael D. Hirschhorn, A note on overcubic partitions, New Zealand J. Math., 42:229-234, 2012.
Bernard L. S. Lin, Arithmetic properties of overcubic partition pairs, Electronic Journal of Combinatorics 21(3) (2014), #P3.35.
James A. Sellers, Elementary proofs of congruences for the cubic and overcubic partition functions, Australasian Journal of Combinatorics, 60(2) (2014), 191-197.

Trisections: A246585, A246586, A246587.

# to get 140 terms:
ph:=add(q^(n^2),n=-12..12);
ph:=series(ph,q,140);
g1:=1/(subs(q=-q,ph)*subs(q=-q^2,ph));
g1:=series(g1,q,140);
seriestolist(%);
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
      `if`(irem(d, 4)=2, 3, 2), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 17 2019

nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) / ((1-x^k) * (1-x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 16 2019 *)
nmax = 50; CoefficientList[Series[Product[(1+x^(2*k)) / (1-x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 16 2019 *)

G.f.: Product_{k>=1} (1+x^k) * (1+x^(2*k)) / ((1-x^k) * (1-x^(2*k))). - Vaclav Kotesovec, Aug 16 2019

a(n) ~ 3^(3/4) * exp(sqrt(3*n/2)*Pi) / (2^(19/4)*n^(5/4)). - Vaclav Kotesovec, Aug 16 2019

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