A077285 -id:A077285 - cp's OEIS frontend

A000712 Generating function = Product_{m>=1} 1/(1 - x^m)^2; a(n) = number of partitions of n into parts of 2 kinds.

1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 481, 752, 1165, 1770, 2665, 3956, 5822, 8470, 12230, 17490, 24842, 35002, 49010, 68150, 94235, 129512, 177087, 240840, 326015, 439190, 589128, 786814, 1046705, 1386930, 1831065, 2408658, 3157789, 4126070, 5374390

Offset: 0

N. J. A. Sloane

For n >= 1, a(n) is also the number of conjugacy classes in the automorphism group of the n-dimensional hypercube. This automorphism group is the wreath product of the cyclic group C_2 and the symmetric group S_n, its order is in sequence A000165. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 04 2001
Also, number of noncongruent matrices in GL_n(Z): each Jordan block can only have +1 or -1 on the diagonal. - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 15 2004
a(n) = Sum (k(1)+1)*(k(2)+1)*...*(k(n)+1), where the sum is taken over all (k(1),k(2),...,k(n)) such that k(1)+2*k(2)+...+n*k(n) = n, k(i)>=0, i=1..n, cf. A104510, A077285. - Vladeta Jovovic, Apr 21 2005
Convolution of partition numbers (A000041) with itself. - Graeme McRae, Jun 07 2006
Number of one-to-one partial endofunctions on n unlabeled points. Connected components are either cycles or "lines", hence two for each size. - Franklin T. Adams-Watters, Dec 28 2006
Equals A000716: (1, 3, 9, 22, 561, 108, ...) convolved with A010815. A000716 = the number of partitions of n into parts of 3 kinds = the Euler transform of [3,3,3,...]. - Gary W. Adamson, Oct 26 2008
Paraphrasing the g.f.: 1 + 2x + 5x^2 + ... = s(x) * s(x^2) * s(x^3) * s(x^4) * ...; where s(x) = 1 + 2x + 3x^2 + 4x^3 + ... is (up to a factor x) the g.f. of A000027. - Gary W. Adamson, Apr 01 2010
Also equals number of partitions of 2n in which the odd parts appear as many times in even as in odd positions. - Wouter Meeussen, Apr 17 2013
Also number of ordered pairs (R,S) with R a partition of r, S a partition of s, and r+s=n; see example.  This corresponds to the formula a(n) = sum(r+s==n, p(r)*p(s) ) = Sum_{k=0..n} p(k)*p(n-k). - Joerg Arndt, Apr 29 2013
Also the number of all multi-graphs with exactly n-edges and with vertex degrees 1 or 2. - Ebrahim Ghorbani, Dec 02 2013
If one decomposes k-permutations into cycles and so-called paths, the number of different type of decompositions equals to a(k); see the paper by Chen, Ghorbani, and Wong. - Ebrahim Ghorbani, Dec 02 2013
Let T(n,k) be the number of partitions of n having parts 1 through k of two kinds, with T(n,0) = A000041(n), the number of partitions of n. Then a(n) = T(n,0) + T(n-1,1) + T(n-2,2) + T(n-3,3) + ... - Gregory L. Simay, May 18 2019
Also the number of orbits of projections in the partition monoid P_n under conjugation by permutations. - James East, Jul 21 2020

			Assume there are integers of two kinds: k and k'; then a(3) = 10 since 3 has the following partitions into parts of two kinds: 111, 111', 11'1', 1'1'1', 12, 1'2, 12', 1'2', 3, and 3'. - _W. Edwin Clark_, Jun 24 2011
There are a(4)=20 partitions of 4 into 2 sorts of parts. Here p:s stands for "part p of sort s":
01:  [ 1:0  1:0  1:0  1:0  ]
02:  [ 1:0  1:0  1:0  1:1  ]
03:  [ 1:0  1:0  1:1  1:1  ]
04:  [ 1:0  1:1  1:1  1:1  ]
05:  [ 1:1  1:1  1:1  1:1  ]
06:  [ 2:0  1:0  1:0  ]
07:  [ 2:0  1:0  1:1  ]
08:  [ 2:0  1:1  1:1  ]
09:  [ 2:0  2:0  ]
10:  [ 2:0  2:1  ]
11:  [ 2:1  1:0  1:0  ]
12:  [ 2:1  1:0  1:1  ]
13:  [ 2:1  1:1  1:1  ]
14:  [ 2:1  2:1  ]
15:  [ 3:0  1:0  ]
16:  [ 3:0  1:1  ]
17:  [ 3:1  1:0  ]
18:  [ 3:1  1:1  ]
19:  [ 4:0  ]
20:  [ 4:1  ]
- _Joerg Arndt_, Apr 28 2013
The a(4)=20 ordered pairs (R,S) of partitions for n=4 are
  ([4], [])
  ([3, 1], [])
  ([2, 2], [])
  ([2, 1, 1], [])
  ([1, 1, 1, 1], [])
  ([3], [1])
  ([2, 1], [1])
  ([1, 1, 1], [1])
  ([2], [2])
  ([2], [1, 1])
  ([1, 1], [2])
  ([1, 1], [1, 1])
  ([1], [3])
  ([1], [2, 1])
  ([1], [1, 1, 1])
  ([], [4])
  ([], [3, 1])
  ([], [2, 2])
  ([], [2, 1, 1])
  ([], [1, 1, 1, 1])
This list was created with the Sage command
   for P in PartitionTuples(2,4) : print P;
- _Joerg Arndt_, Apr 29 2013
G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 110*x^7 + 185*x^8 + ...

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Proposition 2.5.2 on page 78.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 501 terms from T. D. Noe)
Arvind Ayyer, Amritanshu Prasad, and Steven Spallone, Macdonald trees and determinants of representations for finite Coxeter groups, arXiv:1812.00608 [math.RT], 2018.
M. Baake, Structure and representations of the hyperoctahedral group, J. Math. Phys. 25 (1984) 3171, table 1.
Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.
Jan Brandts and A Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
E. R. Canfield, C. D. Savage and H. S. Wilf, Regularly spaced subsums of integer partitions, arXiv:math/0308061 [math.CO], 2003.
Alexandre Chaduteau, Nyan Raess, Henry Davenport, and Frank Schindler, Hilbert Space Fragmentation in the Chiral Luttinger Liquid, arXiv:2409.10359 [cond-mat.str-el], 2024. See pp. 8, 11.
Alexandre Chaduteau, Nyan Raess, Henry Davenport, and Frank Schindler, Momentum-space modulated symmetries in the Luttinger liquid, Phys. Rev. B (2025) Vol. 111, Art. No. 165105. See p. 9.
B. F. Chen, E. Ghorbani, and K. B. Wong, Cyclic decomposition of k-permutations and eigenvalues of the arrangement graphs, Electronic J. Combin. 20(4) (2013), #P22.
W. Y. C. Chen, K. Q. Ji and H. S. Wilf, BG-ranks and 2-cores, arXiv:math/0605474 [math.CO], 2006.
W. Edwin Clark, Mohamed Elhamdadi, Xiang-dong Hou, Masahico Saito and Timothy Yeatman, Connected Quandles Associated with Pointed Abelian Groups, arXiv preprint arXiv:1107.5777 [math.RA], 2011.
W. Edwin Clark and Xiang-dong Hou, Galkin Quandles, Pointed Abelian Groups, and Sequence A000712 arXiv:1108.2215 [math.CO], Aug 10, 2011. [added by Jonathan Vos Post]
Shouvik Datta, M. R. Gaberdiel, W. Li, and C. Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016. See Sect. 2.1.
M. De Salvo, D. Fasino, D. Freni, and G. Lo Faro, Fully simple semihypergroups, transitive digraphs, and sequence A000712, Journal of Algebra, Volume 415, 1 October 2014, pp. 65-87.
Mario De Salvo, Dario Fasino, Domenico Freni, and Giovanni Lo Faro, Semihypergroups Obtained by Merging of 0-semigroups with Groups, Filomat (2018) Vol. 32, No. 12, 4177-4194.
Paul Hammond and Richard Lewis, Congruences in ordered pairs of partitions, Int. J. Math. Math. Sci. (2004), no.45--48, 2509--2512.
Ruth Hoffmann, Özgür Akgün, and Christopher Jefferson, Composable Constraint Models for Permutation Enumeration, arXiv:2311.17581 [cs.DM], 2023.
Saud Hussein, An Identity for the Partition Function Involving Parts of k Different Magnitudes, arXiv:1806.05416 [math.NT], 2018.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 129.
Han Mao Kiah, Anshoo Tandon, and Mehul Motani, Generalized Sphere-Packing Bound for Subblock-Constrained Codes, arXiv:1901.00387 [cs.IT], 2019.
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.
Yen-chi R. Lin and Shu-Yen Pan, A recursive relation for bipartition numbers, arXiv:2406.14851 [math.CO], 2024.
P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, and A. M. Läuchli, Chiral spin liquids in triangular lattice SU (N) fermionic Mott insulators with artificial gauge fields, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.
Sylvain Prolhac, Spectrum of the totally asymmetric simple exclusion process on a periodic lattice--first excited states, arXiv preprint arXiv:1404.1315 [cond-mat.stat-mech], 2014.
N. J. A. Sloane, Transforms.
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A000165, A000041, A002107 (reciprocal of g.f.).
Cf. A002720.
Cf. A000716, A010815. - Gary W. Adamson, Oct 26 2008
Row sums of A175012. - Gary W. Adamson, Apr 03 2010
Column k=2 of A144064.
Cf. A008619, A000070, A000990, A285217, A304045.

a000712 = p a008619_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, Nov 06 2012

# DedekindEta is defined in A000594.
A000712List(len) = DedekindEta(len, -2)
A000712List(39) |> println # Peter Luschny, Mar 09 2018

with(combinat): A000712:= n-> add(numbpart(k)*numbpart(n-k), k=0..n): seq(A000712(n), n=0..40); # Emeric Deutsch

CoefficientList[ Series[ Product[1/(1 - x^n)^2, {n, 40}], {x, 0, 37}], x]; (* Robert G. Wilson v, Feb 03 2005 *)
Table[Count[Partitions[2*n], q_ /; Tr[(-1)^Mod[Flatten[Position[q, ?OddQ]], 2]] === 0], {n, 12}] (* _Wouter Meeussen, Apr 17 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^-2, {x, 0, n}]; (* Michael Somos, Oct 12 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@n], {n, 0, 15}] (* Robert Price, Jun 15 2020 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / eta(x + A)^2, n))}; /* Michael Somos, Nov 14 2002 */

Vec(1/eta('x+O('x^66))^2) /* Joerg Arndt, Jun 25 2011 */

from sympy import npartitions
def A000712(n): return (sum(npartitions(k)*npartitions(n-k) for k in range(n+1>>1))<<1) + (0 if n&1 else npartitions(n>>1)**2) # Chai Wah Wu, Sep 25 2023

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(0, 1, 2, 2)
b = EulerTransform(a)
print([b(n) for n in range(40)]) # Peter Luschny, Nov 11 2020

a(n) = Sum_{k=0..n} p(k)*p(n-k), where p(n) = A000041(n).
Euler transform of period 1 sequence [ 2, 2, 2, ...]. - Michael Somos, Jul 22 2003
a(n) = A006330(n) + A001523(n). - Michael Somos, Jul 22 2003
a(0) = 1, a(n) = (1/n)*Sum_{k=0..n-1} 2*a(k)*sigma_1(n-k). - Joerg Arndt, Feb 05 2011
a(n) ~ (1/12)*3^(1/4)*n^(-5/4)*exp((2/3)*sqrt(3)*Pi*sqrt(n)). - Joe Keane (jgk(AT)jgk.org), Sep 13 2002
G.f.: Product_{i>=1} (1 + x^i)^(2*A001511(i)) (see A000041). - Jon Perry, Jun 06 2004
More precise asymptotics: a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(3/4)*n^(5/4)) * (1 - (Pi/(12*sqrt(3)) + 15*sqrt(3)/(16*Pi)) / sqrt(n) + (Pi^2/864 + 315/(512*Pi^2) + 35/192)/n). - Vaclav Kotesovec, Jan 22 2017
From Peter Bala, Jan 26 2016: (Start)
a(n) is odd iff n = 2*m and p(m) is odd.
a(n) = (2/n)*Sum_{k = 0..n} k*p(k)*p(n-k) for n >= 1.
Conjecture: : a(n) is divisible by 5 when n is congruent to 2, 3 or 4 modulo 5. (End)
Conjecture is proved in Hammond and Lewis. - Yen-chi R. Lin, Jun 24 2024
G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
With the convention that a(n) = 0 for n < 0 we have the recurrence a(n) = g(n) + Sum_{k >= 1} (-1)^(k+1)*(2*k + 1)*a(n - k*(k + 1)/2), where g(n) = (-1)^m if n = m*(3*m - 1)/2 is a generalized pentagonal number (A001318) else g(n) = 0. For example, n = 7 = -2*(3*(-2) - 1)/2 is a pentagonal number, g(7) = 1, and so a(7) = 1 + 3*a(6) - 5*a(4) + 7*a(1) = 1 + 195 - 100 + 14 = 110. - Peter Bala, Apr 06 2022
a(n) = p(n/2) + Sum_{k \in Z, k != 0} (-1)^{k-1} a(n-k^2), here p(n) = A000041(n) and p(x) = 0 when x is not an integer. - Yen-chi R. Lin, Jun 24 2024
Conjecture: a(25*n + 23) is divisible by 25 (checked for n < 400). - Peter Bala, Jan 13 2025

More terms from Joe Keane (jgk(AT)jgk.org), Nov 17 2001
More terms from Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 15 2004
Definition rewritten by N. J. A. Sloane, Apr 02 2022

A266477 Triangle read by rows in which T(n,k) is the number of partitions of n with product of multiplicities of parts equal to k; n>=0, 1<=k<=A266480(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1, 4, 2, 2, 2, 0, 1, 5, 4, 2, 1, 1, 1, 1, 6, 6, 2, 3, 1, 2, 0, 2, 8, 7, 4, 4, 1, 2, 1, 0, 2, 1, 10, 8, 6, 6, 3, 2, 1, 3, 0, 1, 0, 2, 12, 13, 6, 6, 3, 7, 1, 2, 1, 1, 1, 1, 0, 1, 1, 15, 15, 9, 11, 3, 6, 2, 5, 3, 3, 0

Offset: 0

Emeric Deutsch and Alois P. Heinz, Dec 29 2015

Sum of entries in row n = A000041(n) = number of partitions of n.
T(n,1) = A000009(n) = number of partitions of n into distinct parts.
T(n,2) = A090858(n).
T(n,3) = A265251(n).
Smallest row m >= 0 with T(m,n) > 0 is A266325(n).
T(n,A266480(n)) gives A266871(n).

			Row 4 is [2,2,0,1]. Indeed, the products of the multiplicities of the parts in the partitions [4], [1,3], [2,2], [1,1,2], [1,1,1,1] are 1, 1, 2, 2, 4, respectively.
Triangle T(n,k) begins:
00 :  1;
01 :  1;
02 :  1,  1;
03 :  2,  0, 1;
04 :  2,  2, 0,  1;
05 :  3,  2, 1,  0, 1;
06 :  4,  2, 2,  2, 0, 1;
07 :  5,  4, 2,  1, 1, 1, 1;
08 :  6,  6, 2,  3, 1, 2, 0, 2;
09 :  8,  7, 4,  4, 1, 2, 1, 0, 2, 1;
10 : 10,  8, 6,  6, 3, 2, 1, 3, 0, 1, 0, 2;
11 : 12, 13, 6,  6, 3, 7, 1, 2, 1, 1, 1, 1, 0, 1, 1;
12 : 15, 15, 9, 11, 3, 6, 2, 5, 3, 3, 0, 2, 0, 0, 0, 2, 0, 1;

Alois P. Heinz, Rows n = 0..50, flattened

Columns k=1-10 give: A000009, A090858, A265251, A266687, A266688, A266689, A266690, A266691, A266692, A266693.
Main diagonal gives A266499.
Row lengths give A266480.
Cf. A000041, A077285, A266325, A266871.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      x^max(p, p*n), add(b(n-i*j, i-1, max(p, p*j)), j=0..n/i))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2, 1)):
seq(T(n), n=0..16);

Map[Table[Length@ Position[#, k], {k, Max@ #}] &, #] &@ Table[Map[Times @@ Map[Last, Tally@ #] &, IntegerPartitions@ n], {n, 12}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, x^Max[p, p*n], Sum[b[n - i*j, i - 1, Max[p, p*j]], {j, 0, n/i}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p,x]}]][ b[n, n, 1]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

Sum_{k>=1} k*T(n,k) = A077285(n).
G.f. of column p if p is prime: Sum_{k>0} x^(p*k)/(1+x^k) * Product_{i>0} (1+x^i), giving the number of partitions of n such that there is exactly one part which occurs p times, while all other parts occur only once.
If p is prime then column p is asymptotic to 3^(1/4) * c(p) * exp(Pi*sqrt(n/3)) / (2*Pi*n^(1/4)), where c(p) = Sum_{j>=0} (-1)^j/(j+p) = (PolyGamma((p+1)/2) - PolyGamma(p/2))/2. - Vaclav Kotesovec, May 24 2018

A258210 Expansion of f(-q) * f(-q^2) * chi(-q^3) in powers of q where chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, -2, 0, 1, 4, 0, 0, -2, -4, 2, 0, 0, -2, 0, 0, 1, 4, 4, 0, -4, 0, 0, 0, 0, -3, -4, 0, 0, 4, 0, 0, -2, 0, 2, 0, 4, -2, 0, 0, 2, 4, 0, 0, 0, -8, 0, 0, 0, -1, -6, 0, 2, 4, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 1, 8, 0, 0, -4, 0, 0, 0, 4, -2, -4, 0, 0, 0, 0, 0, -4

Offset: 0

Michael Somos, May 23 2015

sign

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

Denoted by a_6(n) in Kassel and Reutenauer 2015. - Michael Somos, Jun 04 2015

			G.f. = 1 - q - 2*q^2 + q^4 + 4*q^5 - 2*q^8 - 4*q^9 + 2*q^10 - 2*q^13 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015 , see page 31 7.2(d). [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016, see p. 13 paragraph 3.3.4.
Christian Kassel, Christophe Reutenauer, The Fourier expansion of eta(z)eta(2z)eta(3z)/eta(6z), arXiv:1603.06357 [math.NT], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002175, A077285, A104794, A121444, A258228, A258277, A258278, A290217.

For the square of this series see A252650.

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 / (QPochhammer[ q, q^6] QPochhammer[ q^5, q^6]), {q, 0, n}];
a[ n_] := SeriesCoefficient[ (1/2) EllipticThetaPrime[ 1, 0, q^(1/2)] / EllipticTheta[ 1, Pi/6, q^(1/2)], {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^3 + A) / eta(x^6 + A), n))};

{a(n) = if( n<1, n==0, (-1)^n * (1 - (n%3==2)*3) * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))}; /* Michael Somos, Jun 04 2015 */

Expansion of f(-q)^2 * f(-q^6) / f(-q, -q^5) in powers of q where f(,) is Ramanujan's general theta function.
Expansion of eta(q) * eta(q^2) * eta(q^3) / eta(q^6) in powers of q.
Euler transform of period 6 sequence [ -1, -2, -2, -2, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 12 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A121444.
G.f.: Product_{k>0} (1 - x^k) * (1 - x^2*k) / (1 + x^(3*k)).
a(n) = (-1)^n * A258228(n). Convolution inverse of A077285.
a(4*n + 3) = 0. a(6*n + 2) = -2 * A122865(n). a(6*n + 4) = A122856(n). a(12*n + 1) = -1 * A002175(n).
a(9*n + 3) = a(9*n + 6) = 0. a(9*n) = A104794(n). a(3*n + 1) = -A258277(n). a(3*n + 2) = -2*A258278(n). - Michael Somos, May 01 2016
G.f.: Product_{i>0} 1/(1 + Sum_{j>0} j*x^(j*i)). - Seiichi Manyama, Oct 08 2017

A385001 Irregular triangle read by rows: T(n,k) is the number of partitions of n with k designated summands, n >= 0, 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 3, 0, 4, 1, 0, 7, 3, 0, 6, 9, 0, 12, 15, 1, 0, 8, 30, 3, 0, 15, 45, 9, 0, 13, 67, 22, 0, 18, 99, 42, 1, 0, 12, 135, 81, 3, 0, 28, 175, 140, 9, 0, 14, 231, 231, 22, 0, 24, 306, 351, 51, 0, 24, 354, 551, 97, 1, 0, 31, 465, 783, 188, 3, 0, 18, 540, 1134, 330, 9

Offset: 0

Omar E. Pol, Jul 17 2025

The divisor function sigma_1(n) = A000203(n) is also the number of partitions of n with only one designated summand, n >= 1.
When part i has multiplicity j > 0 exactly one part i is "designated".
The length of the row n is A002024(n+1) = 1 + A003056(n), hence the first positive element in column k is in the row A000217(k).
Alternating row sums give A329157.
Columns converge to A000716.
This triangle equals A060043 with reversed rows and an additional column 0.

			Triangle begins:
--------------------------------------------
   n\k:   0    1     2     3     4    5   6
--------------------------------------------
   0 |    1;
   1 |    0,   1;
   2 |    0,   3;
   3 |    0,   4,    1;
   4 |    0,   7,    3;
   5 |    0,   6,    9;
   6 |    0,  12,   15,    1;
   7 |    0,   8,   30,    3;
   8 |    0,  15,   45,    9;
   9 |    0,  13,   67,   22;
  10 |    0,  18,   99,   42,    1;
  11 |    0,  12,  135,   81,    3;
  12 |    0,  28,  175,  140,    9;
  13 |    0,  14,  231,  231,   22;
  14 |    0,  24,  306,  351,   51;
  15 |    0,  24,  354,  551,   97,   1;
  16 |    0,  31,  465,  783,  188,   3;
  17 |    0,  18,  540, 1134,  330,   9;
  18 |    0,  39,  681, 1546,  568,  22;
  19 |    0,  20,  765, 2142,  918,  51;
  20 |    0,  42,  945, 2835, 1452, 108;
  21 |    0,  32, 1040, 3758, 2233, 208,  1;
  ...
For n = 6 and k = 1 there are 12 partitions of 6 with only one designated summand as shown below:
   6'
   3'+ 3
   3 + 3'
   2'+ 2 + 2
   2 + 2'+ 2
   2 + 2 + 2'
   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'
So T(6,1) = 12, the same as A000203(6) = 12.
.
For n = 6 and k = 2 there are 15 partitions of 6 with two designated summands as shown below:
   5'+ 1'
   4'+ 2'
   4'+ 1'+ 1
   4'+ 1 + 1'
   3'+ 1'+ 1 + 1
   3'+ 1 + 1'+ 1
   3'+ 1 + 1 + 1'
   2'+ 2 + 1'+ 1
   2'+ 2 + 1 + 1'
   2 + 2'+ 1'+ 1
   2 + 2'+ 1 + 1'
   2'+ 1'+ 1 + 1 + 1
   2'+ 1 + 1'+ 1 + 1
   2'+ 1 + 1 + 1'+ 1
   2'+ 1 + 1 + 1 + 1'
So T(6,2) = 15, the same as A002127(6) = 15.
.
For n = 6 and k = 3 there is only one partition of 6 with three designated summands as shown below:
   3'+ 2'+ 1'
So T(6,3) = 1, the same as A002128(6) = 1.
There are 28 partitions of 6 with designated summands, so A077285(6) = 28.
.

Alois P. Heinz, Rows n = 0..1000, flattened

Columns: A000007 (k=0), A000203 (k=1), A002127 (k=2), A002128 (k=3), A365664 (k=4), A365665 (k=5), A384926 (k=6).
Row sums give A077285.
Cf. A000217, A002024, A003056, A060043, A196020, A252117, A329157, A365434, A384999.
Cf. A000096, A000716, A116608, A293421, A384998, A384999.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(expand(b(n-i*j, i-1)*j*x), j=1..n/i)))
    end:
T:= n-> (p-> seq(coeff(p,x,i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 18 2025

From Alois P. Heinz, Jul 18 2025: (Start)
Sum_{k>=1} k * T(n,k) = A293421(n).
T(A000096(n),n) = A000716(n). (End)
G.f.: Product_{i>0} 1 + (y*x^i)/(1 - x^i)^2. - John Tyler Rascoe, Jul 23 2025
Conjecture: For fixed k >= 1, Sum_{j=1..n} T(j,k) ~ Pi^(2*k) * n^(2*k) / ((2*k)! * (2*k+1)!). - Vaclav Kotesovec, Aug 01 2025

A053993 The number phi_2(n) of Frobenius partitions that allow up to 2 repetitions of an integer in a row.

Original entry on oeis.org

1, 1, 3, 5, 9, 14, 24, 35, 55, 81, 120, 171, 248, 345, 486, 669, 920, 1246, 1690, 2256, 3014, 3984, 5253, 6870, 8970, 11618, 15022, 19306, 24745, 31557, 40154, 50845, 64244, 80850, 101501, 126982, 158514, 197218, 244865, 303143, 374497, 461435

Offset: 0

James Sellers, Apr 04 2000

Sum of products of multiplicities of odd parts in all partitions of n (if there are no odd parts in a partition then product of multiplicities is considered to be 1). - Vladeta Jovovic, Feb 16 2005

The sequence A077285 is the same but with multiplicities of all parts.

			1 + x + 3*x^2 + 5*x^3 + 9*x^4 + 14*x^5 + 24*x^6 + 35*x^7 + 55*x^8 + ...
q^-1 + q^11 + 3*q^23 + 5*q^35 + 9*q^47 + 14*q^59 + 24*q^71 + 35*q^83 + ...
a(6) = 24 since the 5 partitions 6 = 5+1 = 4+2 = 3+2+1 = 2+2+2 each contribute 1, the 3 partitions 4+1+1 = 3+3 = 2+2+1+1 each contribute 2, the partition 3+1+1+1 contributes 3, the partition 2+1+1+1+1 contributes 4, and the partition 1+1+1+1+1+1 contributes 6 to give total 24 = 5*1 + 3*2 + 1*3 + 1*4 + 1*6. - _Michael Somos_, Mar 09 2011

George E. Andrews, Generalized Frobenius partitions, Memoirs of the American Mathematical Society, Number 301, May 1984.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953. See formula (18) on page 3944.

Cf. A077285, A052992, A097242, A247661, A247662.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)
      +add(b(n-i*j, i-1)*`if`(irem(i, 2)=1, j, 1), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 16 2013

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-1] * If[Mod[i, 2] == 1, j, 1], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)
QP = QPochhammer; s = QP[q^4] * (QP[q^6]^2 / (QP[q] * QP[q^2] * QP[q^3] * QP[q^12])) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015, adapted from PARI *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))} /* Michael Somos, Mar 09 2011 */

Expansion of q^(1/12) * eta(q^4) * eta(q^6)^2 / (eta(q) * eta(q^2) * eta(q^3) * eta(q^12)) in powers of q. - Michael Somos, Mar 09 2011
Euler transform of period 12 sequence [ 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, ...]. - Michael Somos, Mar 09 2011
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(12*k - 10)) * (1 - x^(12*k - 9)) * (1 - x^(12*k - 3)) * (1 - x^(12*k - 2)))^(-1). [Andrews, p. 10, equation (5.9)]
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (6*sqrt(2)*n). - Vaclav Kotesovec, Nov 28 2015

A102186 The PDO(n) function (Partitions with Designated summands in which all parts are Odd): the sum of products of multiplicities of parts in all partitions of n into odd parts.

Original entry on oeis.org

1, 1, 2, 4, 5, 8, 12, 16, 22, 32, 42, 56, 76, 98, 128, 168, 213, 272, 348, 436, 548, 688, 852, 1056, 1308, 1603, 1964, 2404, 2920, 3544, 4296, 5176, 6230, 7488, 8958, 10704, 12772, 15182, 18024, 21368, 25254, 29808, 35136, 41308, 48504, 56880, 66552, 77776

Offset: 0

Vladeta Jovovic, Feb 16 2005

			a(8)=22 because in the six partitions of 8 into odd parts, namely, 71,53,5111,3311,311111,11111111, the multiplicities of the parts are (1,1),(1,1),(1,3),(2,2),(1,5),(8) with products 1,1,3,4,5,8, having sum 22.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
G. E. Andrews, R. P. Lewis, and J. Lovejoy, Partitions with designated summands, Acta Arith. 105 (2002), no. 1, 51-66.
Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See pp. 4, 10, 12.
Nayandeep Deka Baruah and Kanan Kumari Ojah, Partitions with designated summands in which all parts are odd, INTEGERS 15 (2015), #A9.
Shishuo Fu and James Sellers, A refined view of a curious identity for partitions into odd parts with designated summands, arXiv:2505.21111 [math.CO], 2025.

Cf. A077285 (partitions with designated summands).

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-2) +add(b(n-i*j, i-2)*j, j=1..n/i)))
    end:
a:= n-> b(n, iquo(1+n,2)*2-1):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 26 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 2] + Sum[b[n - i*j, i - 2]*j, {j, 1, n/i}]]]; a[n_] := b[n, Quotient[1 + n, 2]*2 - 1]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)
nmax=60; CoefficientList[Series[Product[(1-x^(4*k)) * (1+x^(3*k)) / ((1-x^k) * (1+x^(6*k))), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Nov 28 2015 *)
Table[Total[l = Tally /@ Select[IntegerPartitions@n, VectorQ[#, OddQ] &];
  Table[x = l[[i]]; Product[x[[j, 2]], {j, Length[x]}], {i, Length[l]}]], {n, 0, 47}] (* Robert Price, Jun 08 2020 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^4+A)*eta(x^6+A)^2/ eta(x+A)/eta(x^3+A)/eta(x^12+A), n))} /* Michael Somos, Jul 30 2006 */

Euler transform of period 12 sequence [1, 1, 2, 0, 1, 0, 1, 0, 2, 1, 1, 0, ...].
a(n) ~ 5^(1/4) * exp(sqrt(5*n)*Pi/3) / (2^(5/2)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 28 2015
G.f.: Product_{k>=1} (1 + Sum_{j>=1} j * x^(j*(2*k - 1))). - Ilya Gutkovskiy, Nov 06 2019

More terms from Emeric Deutsch, Mar 28 2005

Name expanded by N. J. A. Sloane, Nov 21 2015

cp's OEIS Frontend

A000712 Generating function = Product_{m>=1} 1/(1 - x^m)^2; a(n) = number of partitions of n into parts of 2 kinds.

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A266477 Triangle read by rows in which T(n,k) is the number of partitions of n with product of multiplicities of parts equal to k; n>=0, 1<=k<=A266480(n).

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A258210 Expansion of f(-q) * f(-q^2) * chi(-q^3) in powers of q where chi(), f() are Ramanujan theta functions.

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A385001 Irregular triangle read by rows: T(n,k) is the number of partitions of n with k designated summands, n >= 0, 0 <= k <= A003056(n).

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A053993 The number phi_2(n) of Frobenius partitions that allow up to 2 repetitions of an integer in a row.

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A102186 The PDO(n) function (Partitions with Designated summands in which all parts are Odd): the sum of products of multiplicities of parts in all partitions of n into odd parts.

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A290216 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(ji)).

A293426 Expansion of Product_{k>0} ((1 - q^(3k))^3(1 - q^(6k))^3)/((1 - q^k)^5(1 - q^(2*k))^3).