A219554 -id:A219554 - cp's OEIS frontend

A107742 G.f.: Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

1, 1, 2, 4, 6, 10, 17, 25, 38, 59, 86, 125, 184, 260, 369, 524, 726, 1005, 1391, 1894, 2576, 3493, 4687, 6272, 8373, 11090, 14647, 19294, 25265, 32991, 42974, 55705, 72025, 92895, 119349, 152965, 195592, 249280, 316991, 402215, 508932, 642598, 809739, 1017850, 1276959, 1599015, 1997943, 2491874, 3102477, 3855165, 4782408, 5922954

Offset: 0

Vladeta Jovovic, Jun 11 2005

From Gus Wiseman, Sep 13 2022: (Start)
Also the number of multiset partitions of integer partitions of n into intervals, where an interval is a set of positive integers with all differences of adjacent elements equal to 1. For example, the a(1) = 1 through a(4) = 6 multiset partitions are:
  {{1}}  {{2}}      {{3}}          {{4}}
         {{1},{1}}  {{1,2}}        {{1},{3}}
                    {{1},{2}}      {{2},{2}}
                    {{1},{1},{1}}  {{1},{1,2}}
                                   {{1},{1},{2}}
                                   {{1},{1},{1},{1}}
Intervals are counted by A001227, ranked by A073485.
The initial version is A007294.
The strict version is A327731.
The version for gapless multisets instead of intervals is A356941.
The case of strict partitions is A356957.
Also the number of multiset partitions of integer partitions of n into distinct constant blocks. For example, the a(1) = 1 through a(4) = 6 multiset partitions are:
  {{1}}  {{2}}    {{3}}        {{4}}
         {{1,1}}  {{1,1,1}}    {{2,2}}
                  {{1},{2}}    {{1},{3}}
                  {{1},{1,1}}  {{1,1,1,1}}
                               {{2},{1,1}}
                               {{1},{1,1,1}}
Constant multisets are counted by A000005, ranked by A000961.
The non-strict version is A006171.
The unlabeled version is A089259.
The non-constant block version is A261049.
The version for twice-partitions is A279786, factorizations A296131.
Also the number of multiset partitions of integer partitions of n into constant blocks of odd length. For example, a(1) = 1 through a(4) = 6 multiset partitions are:
  {{1}}  {{2}}      {{3}}          {{4}}
         {{1},{1}}  {{1,1,1}}      {{1},{3}}
                    {{1},{2}}      {{2},{2}}
                    {{1},{1},{1}}  {{1},{1,1,1}}
                                   {{1},{1},{2}}
                                   {{1},{1},{1},{1}}
The strict version is A327731 (also).
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
N. J. A. Sloane, Transforms

Cf. A006171, A109386, A219554, A280473, A280486, A288007.
Product_{k>=1} (1 + x^k)^sigma_m(k): this sequence (m=0), A192065 (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).
A000041 counts integer partitions, strict A000009.
A000110 counts set partitions.
A072233 counts partitions by sum and length.
Cf. A001055, A001970, A011782, A055887, A061260, A270995, A279784, A294617, A304969.

nmax = 50; CoefficientList[Series[Product[(1+x^(i*j)), {i, 1, nmax}, {j, 1, nmax/i}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 04 2017 *)
nmax = 50; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 23 2018 *)
nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k], j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Aug 28 2018 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And@@chQ/@#&]],{n,0,5}] (* Gus Wiseman, Sep 13 2022 *)

a(n)=polcoeff(prod(k=1,n,prod(j=1,n\k,1+x^(j*k)+x*O(x^n))),n) /* Paul D. Hanna */

N=66;  x='x+O('x^N); gf=1/prod(j=0,N, eta(x^(2*j+1))); gf=prod(j=1,N,(1+x^j)^numdiv(j)); Vec(gf) /* Joerg Arndt, May 03 2008 */

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x^(2*m)+x*O(x^n))/m)),n))} /* Paul D. Hanna, Mar 28 2009 */

Euler transform of A001227.
Weigh transform of A000005.
G.f. satisfies: log(A(x)) = Sum_{n>=1} A109386(n)/n*x^n, where A109386(n) = Sum_{d|n} d*Sum_{m|d} (m mod 2). - Paul D. Hanna, Jun 26 2005
G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1-x^(2n)) /n ). - Paul D. Hanna, Mar 28 2009
G.f.: Product_{n>=1} Q(x^n) where Q(x) is the g.f. of A000009. - Joerg Arndt, Feb 27 2014
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A109386(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 04 2017
Conjecture: log(a(n)) ~ Pi*sqrt(n*log(n)/6). - Vaclav Kotesovec, Aug 29 2018

More terms from Paul D. Hanna, Jun 26 2005

A002774 Number of bipartite partitions of n white objects and n black ones.

Original entry on oeis.org

1, 2, 9, 31, 109, 339, 1043, 2998, 8406, 22652, 59521, 151958, 379693, 927622, 2224235, 5236586, 12130780, 27669593, 62229990, 138095696, 302673029, 655627975, 1404599867, 2977831389, 6251060785, 12999299705, 26791990052, 54750235190, 110977389012

Offset: 0

N. J. A. Sloane

nonn

Number of ways to factor p^n * q^n where p and q are distinct primes.

M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 1.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, see p(n,n), page 778. - N. J. A. Sloane, Dec 30 2018
A. Murthy, Generalization of partition function, introducing Smarandache factor partitions. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
A. Murthy, Program for finding out the number of Smarandache factor partitions. (To be published in Smarandache Notions Journal).
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.14.
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 and Vaclav Kotesovec, Table of n, a(n) for n = 0..400 (terms 0..100 from Alois P. Heinz)
F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953). 72-83.
F. C. Auluck, On partitions of bipartite numbers, annotated scan of a few pages.
F. C. Auluck, On partitions of bipartite numbers, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 49, Issue 01, January 1953, pp. 72-83. (full article)
Katherine Ormeño Bastías, Paul Martin, and Steen Ryom-Hansen, On the spherical partition algebra, arXiv:2402.01890 [math.RT], 2024.
M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956 (Annotated scanned pages from, plus a review)
Edward Pearce-Crump, Permutation Equivariant Neural Networks for Symmetric Tensors, arXiv:2503.11276 [cs.LG], 2025. See p. 20.

Cf. A005380.
Cf. A219554. Column k=2 of A219727. - Alois P. Heinz, Nov 26 2012
Cf. also A000041, A000070, A000291, A000412, A000465, A000491, A002755, A002756, A002757, A002758, A002759, A277239.
Main diagonal of A054225 if that entry is drawn as a square array. - N. J. A. Sloane, Dec 30 2018

with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b(6^n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 27 2013

max = 26; se = Series[ Sum[ Log[1 - x^(n-k)*y^k], {n, 1, 2max}, {k, 0, n}], {x, 0, 2max}, {y, 0, 2max}]; coes = CoefficientList[ Series[ Exp[-se], {x, 0, 2max}, {y, 0, 2max}], {x, y}]; a[n_] := coes[[n+1, n+1]]; Table[a[n], {n, 0, max} ](* Jean-François Alcover, Dec 06 2011 *)

a(n) = A054225(2n, n) = A091437(2n).
a(n) ~ Zeta(3)^(19/36) * exp(3*Zeta(3)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (6*Zeta(3)^(1/3)) + Zeta'(-1) - Pi^4/(432*Zeta(3))) / (sqrt(3) * (2*Pi)^(3/2) * n^(55/36)). - Vaclav Kotesovec, Jan 30 2016
Formula (25) in the article by Auluck is incorrect. The correct formula is: p(n,n) ~ c^(19/12) * exp(3*c*n^(2/3) + 3*d*n^(1/3) + Zeta'(-1) - 3*d^2/(4*c)) / (sqrt(3) * (2*Pi)^(3/2) * n^(55/36)), where c = Zeta(3)^(1/3), d = Zeta(2)/(3*c). Also formula (24) is incorrect. - Vaclav Kotesovec, Jan 30 2016
From Vaclav Kotesovec, Feb 04 2016: (Start)
The correct formula (24) is p(m,n) ~ c^(7/4)/(2*Pi*sqrt(3)) * exp(3*c*(m*n)^(1/3) + 3*d*(m+n)/(2*(m*n)^(1/3)) - 19*log(m*n)/24 - ((m/n - 2*n/m)*log(m) + (n/m - 2*m/n)*log(n))/36 - (m/n + n/m)*(log(c)/12 + Zeta'(-1) - 1/12 + 3*d^2/(4*c)) + 3*d^2/(4*c) - 3*log(2*Pi)/4 + fi((n/m)^(1/2))),
where m and n are of the same order, c = Zeta(3)^(1/3), d = Zeta(2)/(3*c) and fi(alfa) = Integral_{t=0..infinity} (1/t)*(1/(exp(alfa*t)-1)/(exp(t/alfa)-1) - (alfa/t)/(exp(alfa*t)-1) - ((1/alfa)/t)/(exp(t/alfa)-1) + 1/t^2 + (1/4)/(exp(alfa*t)-1) + (1/4)/(exp(t/alfa)-1) - (alfa/4)/t - ((1/4)/alfa)/t).
If m = n then alfa = 1 and fi(1) = 3*Zeta'(-1) + log(2*Pi)/4 - 1/6.
For the asymptotic formula for fixed m see A054225.
(End)

Corrected using A000491.

Edited by Christian G. Bower, Jan 08 2004

A054242 Triangle read by rows: row n (n>=0) gives the number of partitions of (n,0), (n-1,1), (n-2,2), ..., (0,n) respectively into sums of distinct pairs.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 3, 2, 2, 5, 5, 5, 2, 3, 7, 9, 9, 7, 3, 4, 10, 14, 17, 14, 10, 4, 5, 14, 21, 27, 27, 21, 14, 5, 6, 19, 31, 42, 46, 42, 31, 19, 6, 8, 25, 44, 64, 74, 74, 64, 44, 25, 8, 10, 33, 61, 93, 116, 123, 116, 93, 61, 33, 10

Offset: 0

Marc LeBrun, Feb 08 2000 and Jul 01 2003

By analogy with ordinary partitions into distinct parts (A000009). The empty partition gives T(0,0)=1 by definition. A054225 and A201376 give pair partitions with repeats allowed.
Also number of partitions into pairs which are not both even.
In the paper by S. M. Luthra: "Partitions of bipartite numbers when the summands are unequal", the square table on page 370 contains an errors. In the formula (6, p. 372) for fixed m there should be factor 1/m!. The correct asymptotic formula is q(m, n) ~ (sqrt(12*n)/Pi)^m * exp(Pi*sqrt(n/3)) / (4*3^(1/4)*m!*n^(3/4)). The same error is also in article by F. C. Auluck (see A054225). - Vaclav Kotesovec, Feb 02 2016

			The second row (n=1) is 1,1 since (1,0) and (0,1) each have a single partition.
The third row (n=2) is 1, 2, 1 from (2,0), (1,1) or (1,0)+(0,1), (0,2).
In the fourth row, T(1,3)=5 from (1,3), (0,3)+(1,0), (0,2)+(1,1), (0,2)+(0,1)+(1,0), (0,1)+(1,2).
The triangle begins:
  1;
  1,  1;
  1,  2,  1;
  2,  3,  3,  2;
  2,  5,  5,  5,  2;
  3,  7,  9,  9,  7,  3;
  4, 10, 14, 17, 14, 10,  4;
  5, 14, 21, 27, 27, 21, 14,  5;
  6, 19, 31, 42, 46, 42, 31, 19,  6;
  8, 25, 44, 64, 74, 74, 64, 44, 25, 8;
  ...

Alois P. Heinz, Rows n = 0..75, flattened
S. M. Luthra, Partitions of bipartite numbers when the summands are unequal, Proceedings of the Indian National Science Academy, vol.23, 1957, issue 5A, p. 370-376. [broken link]
Reinhard Zumkeller, Haskell programs for A054225, A054242, A201376, A201377

See A201377 for the same triangle formatted in a different way.
The outer diagonals are T(n,0) = T(n,n) = A000009(n).
Cf. A054225.
T(2*n,n) = A219554(n). Row sums give A219555. - Alois P. Heinz, Nov 22 2012
Columns 0-5: A000009, A036469, A268345, A268346, A268347, A268348.

Haskell
```
see Zumkeller link.
```

max = 10; f[x_, y_] := Product[1 + x^n*y^k, {n, 0, max}, {k, 0, max}]/2; se = Series[f[x, y], {x, 0, max}, {y, 0, max}] ; coes = CoefficientList[ se, {x, y}]; t[n_, k_] := coes[[n-k+1, k+1]]; Flatten[ Table[ t[n, k], {n, 0, max}, {k, 0, n}]] (* Jean-François Alcover, Dec 06 2011 *)

G.f.: (1/2)*Product(1+x^i*y^j), i, j>=0.

Entry revised by N. J. A. Sloane, Nov 30 2011, to incorporate corrections provided by Reinhard Zumkeller, who also contributed the alternative version A201377.

A219585 Number A(n,k) of k-partite partitions of {n}^k into distinct k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 5, 2, 1, 1, 15, 40, 17, 2, 1, 1, 52, 457, 364, 46, 3, 1, 1, 203, 6995, 14595, 2897, 123, 4, 1, 1, 877, 136771, 937776, 407287, 21369, 323, 5, 1, 1, 4140, 3299218, 88507276, 107652681, 10200931, 148257, 809, 6, 1

Offset: 0

Alois P. Heinz, Nov 23 2012

A(n,k) is the number of factorizations of m^n into distinct factors where m is a product of k distinct primes. A(2,2) = 5: (2*3)^2 = 36 has 5 factorizations into distinct factors: 36, 3*12, 4*9, 2*18, 2*3*6.

			A(1,3) = 5: [(1,1,1)], [(1,1,0),(0,0,1)], [(1,0,1),(0,1,0)], [(1,0,0),(0,1,0),(0,0,1)], [(0,1,1),(1,0,0)].
A(3,2) = 17: [(3,3)], [(3,0),(0,3)], [(3,2),(0,1)], [(2,3),(1,0)], [(3,1),(0,2)], [(2,2),(1,1)], [(1,3),(2,0)], [(2,1),(1,2)], [(2,1),(1,1),(0,1)], [(3,0),(0,2),(0,1)], [(2,2),(1,0),(0,1)], [(2,1),(0,2),(1,0)], [(1,2),(2,0),(0,1)], [(1,2),(1,1),(1,0)], [(0,3),(2,0),(1,0)], [(2,0),(1,1),(0,2)], [(2,0),(0,2),(1,0),(0,1)].
Square array A(n,k) begins:
  1,  1,   1,      1,          1,            1,         1, ...
  1,  1,   2,      5,         15,           52,       203, ...
  1,  1,   5,     40,        457,         6995,    136771, ...
  1,  2,  17,    364,      14595,       937776,  88507276, ...
  1,  2,  46,   2897,     407287,    107652681,  ...
  1,  3, 123,  21369,   10200931,  10781201973,  ...
  1,  4, 323, 148257,  233051939,  ...
  1,  5, 809, 970246, 4909342744,  ...

Andrew Howroyd, Table of n, a(n) for n = 0..209

Columns k=0..5 give: A000012, A000009, A219554, A219560, A219561, A219565.
Rows n=0..3 give: A000012, A000110, A094574, A319591.
Cf. A188445, A219727 (partitions of {n}^k into k-tuples), A318286.

f[n_, k_] := f[n, k] = 1/2 Product[Sum[O[x[j]]^(n+1), {j, 1, k}]+1+ Product[x[j]^i[j], {j, 1, k}], Evaluate[Sequence @@ Table[{i[j], 0, n}, {j, 1, k}]]];
a[0, ] = a[, 0] = 1; a[n_, k_] := SeriesCoefficient[f[n, k], Sequence @@ Table[{x[j], 0, n}, {j, 1, k}]];
Table[Print[a[n-k, k]]; a[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 11 2013, updated Sep 16 2019 *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); EulerT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)} \\ Andrew Howroyd, Dec 16 2018

A(n,k) = [(Product_{j=1..k} x_j)^n] 1/2 * Product_{i_1,...,i_k>=0} (1+Product_{j=1..k} x_j^i_j).

A201377 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of partitions of (n,k) into a sum of distinct pairs.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 2, 5, 9, 17, 2, 7, 14, 27, 46, 3, 10, 21, 42, 74, 123, 4, 14, 31, 64, 116, 197, 323, 5, 19, 44, 93, 174, 303, 506, 809, 6, 25, 61, 132, 254, 452, 769, 1251, 1966, 8, 33, 83, 185, 363, 659, 1141, 1885, 3006, 4660

Offset: 0

Reinhard Zumkeller, Nov 30 2011

By analogy with ordinary partitions into distinct parts (A000009). The empty partition gives T(0,0)=1 by definition. A201376 and A054242 give partitions of pairs into sums of not necessarily distinct pairs.

Parts (i,j) are "positive" in the sense that min {i,j} >= 0 and max {i,j} >0. The empty partition of (0,0) is counted as 1.

			Partitions of (2,1) into distinct positive pairs, T(2,1) = 3:
(2,1),
(2,0) + (0,1),
(1,1) + (1,0);
Partitions of (2,2) into distinct positive pairs, T(2,2) = 5:
(2,2),
(2,1) + (0,1),
(2,0) + (0,2),
(1,2) + (1,0),
(1,1) + (1,0) + (0,1).
First ten rows of triangle:
0:                      1
1:                    1  2
2:                  1  3  5
3:                2  5  9  17
4:              2  7  14  27  46
5:            3  10  21  42  74  123
6:          4  14  31  64  116  197  323
7:        5  19  44  93  174  303  506  809
8:      6  25  61  132  254  452  769  1251  1966
9:    8  33  83  185  363  659  1141  1885  3006  4660

Alois P. Heinz, Rows n = 0..80, flattened
Reinhard Zumkeller, Haskell programs for A054225, A054242, A201376, A201377

T(n,0) = A000009(n);
T(1,0) = A036469(0);  T(n,1) = A036469(n) for n > 0.
See A054242 for another version.
Cf. A000009, A054225, A201376.
T(n,n) = A219554(n). Row sums give: A219557. - Alois P. Heinz, Nov 22 2012

Haskell
```
-- see link.
```

nmax = 10;
f[x_, y_] := Product[1 + x^n y^k, {n, 0, nmax}, {k, 0, nmax}]/2;
se = Series[f[x, y], {x, 0, nmax}, {y, 0, nmax}];
coes = CoefficientList[se, {x, y}];
t[n_ /; n >= 0, k_] /; 0 <= k <= n := coes[[n-k+1, k+1]];
T[n_, k_] := t[n+k, k];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 08 2021 *)

For g.f. see A054242.

Entry revised by N. J. A. Sloane, Nov 30 2011

A219560 Number of tripartite partitions of (n,n,n) into distinct triples.

Original entry on oeis.org

1, 5, 40, 364, 2897, 21369, 148257, 970246, 6032341, 35850410, 204646488, 1126463948, 5999145787, 30999381232, 155798366059, 763194776551, 3650648583934, 17079277343463, 78262895082681, 351708874155894, 1551843168854346

Offset: 0

Alois P. Heinz, Nov 23 2012

Number of factorizations of (p*q*r)^n into distinct factors where p, q, r are distinct primes.

			a(0) = 1: [].
a(1) = 5: [(1,1,1)], [(1,1,0),(0,0,1)], [(1,0,1),(0,1,0)], [(0,1,1),(1,0,0)], [(0,0,1),(0,1,0),(1,0,0)].

Column k=3 of A219585.

Cf. A002774, A219554, A219561, A219565, A219678.

with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b(30^n$2):
seq(a(n), n=0..10);  # Alois P. Heinz, May 26 2013

b[n_, k_] := b[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, b[n/d, d - 1]], {d, Divisors[n][[2 ;; -2]]}]]; a[0] = 1; a[n_] := b[30^n, 30^n];  Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)

a(n) = [(x*y*z)^n] 1/2 * Product_{i,j,k>=0} (1+x^i*y^j*z^k).

a(16) from Alois P. Heinz, May 26 2013
a(17) from Alois P. Heinz, Sep 24 2014
More terms from Jean-François Alcover, Jan 15 2016

A219561 Number of 4-partite partitions of (n,n,n,n) into distinct quadruples.

Original entry on oeis.org

1, 15, 457, 14595, 407287, 10200931, 233051939, 4909342744, 96272310302, 1771597038279, 30795582025352, 508466832109216, 8011287089600483, 120926718707154007, 1754672912487450236, 24547188914867491083, 331937179344717327559, 4348524173437743243649, 55300773426746984710983

Offset: 0

Alois P. Heinz, Nov 23 2012

nonn

Number of factorizations of (p*q*r*s)^n into distinct factors where p, q, r, s are distinct primes.

			a(0) = 1: [].
a(1) = 15: [(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)], [(0,0,1,1),(1,0,0,0),(0,1,0,0)], [(0,1,0,1),(1,0,0,0),(0,0,1,0)], [(0,1,1,0),(1,0,0,0),(0,0,0,1)], [(1,0,0,1),(0,1,0,0),(0,0,1,0)], [(1,0,0,1),(0,1,1,0)], [(1,0,1,0),(0,1,0,0),(0,0,0,1)], [(1,0,1,0),(0,1,0,1)], [(1,1,0,0),(0,0,1,0),(0,0,0,1)], [(1,1,0,0),(0,0,1,1)], [(0,1,1,1),(1,0,0,0)], [(1,0,1,1),(0,1,0,0)], [(1,1,0,1),(0,0,1,0)], [(1,1,1,0),(0,0,0,1)], [(1,1,1,1)].

Column k=4 of A219585.

Cf. A002774, A219554, A219560, A219565.

a[n_] := If[n == 0, 1, (1/2) Coefficient[Product[O[w]^(n+1) + O[x]^(n+1) + O[y]^(n+1) + O[z]^(n+1) + (1 + w^i x^j y^k z^m), {i, 0, n}, {j, 0, n}, {k, 0, n}, {m, 0, n}] // Normal, (w x y z)^n]];
Table[Print[n]; a[n], {n, 0, 12}] (* Jean-François Alcover, Sep 16 2019 *)

a(n) = [(w*x*y*z)^n] 1/2 * Product_{i,j,k,m>=0} (1+w^i*x^j*y^k*z^m).

a(9) from Alois P. Heinz, Oct 15 2014

a(10)-a(18) from Andrew Howroyd, Dec 17 2018

A219565 Number of 5-partite partitions of (n,n,n,n,n) into distinct quintuples.

Original entry on oeis.org

1, 52, 6995, 937776, 107652681, 10781201973, 958919976957, 76861542428397, 5620227129073491, 378709513816248475, 23713852762539359688, 1389561695379881634055, 76647024053735036288641, 3999799865715906390697377, 198328846122797866982616805, 9379277765981012067789260214

Offset: 0

Alois P. Heinz, Nov 23 2012

nonn

Number of factorizations of (p*q*r*s*t)^n into distinct factors where p, q, r, s, t are distinct primes.

Column k=5 of A219585.

Cf. A002774, A219554, A219560, A219561.

a[n_] := a[n] = If[n == 0, 1, (1/2) Coefficient[Product[O[v]^(n+1) + O[w]^(n+1) + O[x]^(n+1) + O[y]^(n+1) + O[z]^(n+1) + (1 + v^i w^j x^k y^l z^m), {i, 0, n}, {j, 0, n}, {k, 0, n}, {l, 0, n}, {m, 0, n}] // Normal, (v w x y z)^n]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 7}] (* Jean-François Alcover, Sep 24 2019 *)

a(n) = [(v*w*x*y*z)^n] 1/2 * Product_{h,i,j,k,m>=0} (1+v^h*w^i*x^j*y^k*z^m).

a(6) from Alois P. Heinz, Sep 25 2014

a(7)-a(15) from Andrew Howroyd, Dec 16 2018

A331317 Number of nonnegative integer matrices with 2 distinct columns and any number of distinct nonzero rows with each column sum being n and rows in decreasing lexicographic order.

Original entry on oeis.org

0, 1, 4, 15, 44, 120, 319, 804, 1960, 4652, 10782, 24435, 54329, 118663, 254969, 539825, 1127247, 2323811, 4733634, 9535025, 19005218, 37507726, 73333405, 142112298, 273092198, 520612163, 984943887, 1849920530, 3450475858, 6393203485, 11770416017, 21538245911, 39181212114

Offset: 0

Andrew Howroyd, Jan 13 2020

nonn

The condition that the rows be in decreasing order is equivalent to considering nonequivalent matrices with distinct rows up to permutation of rows.

Andrew Howroyd, Table of n, a(n) for n = 0..400

Column k=2 of A331160.

Cf. A000009, A219554, A331197.

a(n)={my(p=prod(i=0, n, prod(j=0, n, 1 + x^i*y^j + O(x*x^n) + O(y*y^n))), q=prod(i=1, n, 1 + x^i + O(x*x^n))); polcoef(polcoef(p,n), n)/2 - polcoef(q,n)}

a(n) = A219554(n) - A000009(n).

A267862 Number of planar lattice convex polygonal lines joining the origin and the point (n,n).

Original entry on oeis.org

1, 2, 5, 13, 32, 77, 178, 399, 877, 1882, 3959, 8179, 16636, 33333, 65894, 128633, 248169, 473585, 894573, 1673704, 3103334, 5705383, 10405080, 18831761, 33836627, 60378964, 107035022, 188553965, 330166814, 574815804, 995229598, 1714004131, 2936857097

Offset: 0

Christoph Koutschan, Apr 07 2016

In other words, we are counting walks on the integer lattice N^2 that start at (0,0) and end at (n,n); they may take arbitrary steps, but the slopes of the steps in the walk must strictly increase. As a result, we obtain a convex polygon when joining the two endpoints of the walk with the point (0,n).

			The two walks for n = 1 are
(0,0) -> (1,1)
(0,0) -> (1,0) -> (1,1).
The five possibilities for n = 2 are
(0,0) -> (2,2)
(0,0) -> (1,0) -> (2,1) -> (2,2)
(0,0) -> (1,0) -> (2,2)
(0,0) -> (2,0) -> (2,2)
(0,0) -> (2,1) -> (2,2).

Vaclav Kotesovec, Table of n, a(n) for n = 0..100
J. Bureaux, N. Enriquez, On the number of lattice convex chains, arXiv:1603.09587 [math.PR], 2016.

Cf. A002774, A090806, A219554.

a[i_Integer, j_Integer, s_] := a[i, j, s] = If[i === 0, 1, Sum[a[i - x, j - y, y/x], {x, 1, i}, {y, Floor[s*x] + 1, j}]]; a[n_Integer] := a[n] = 1 + Sum[a[n - x, n - y, y/x], {x, 1, n}, {y, 0, x - 1}]; Flatten[{1, Table[a[n], {n, 30}]}]
nmax = 20; p = (1 - x)*(1 - y); Do[Do[p = Expand[p*If[GCD[i, j] == 1, (1 - x^i*y^j), 1]]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &], {i, 1, nmax}], {j, 1, nmax}]; p = Expand[Normal[Series[1/p, {x, 0, nmax}, {y, 0, nmax}]]]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Flatten[{1, Table[Coefficient[p, x^n*y^n], {n, 1, nmax}]}] (* Vaclav Kotesovec, Apr 08 2016 *)

a(n) = [x^n*y^n] 1/((1-x)*(1-y)*Product_{i>0,j>0,gcd(i,j)=1} (1-x^i*y^j)).

An asymptotic formula for a(n) is given by Bureaux and Enriquez: a(n) ~ e^(-2*zeta'(-1))/((2*Pi)^(7/6)*sqrt(3)*kappa^(1/18)*n^(17/18)) * e^(3*kappa^(1/3)*n^(2/3)+...) where kappa := zeta(3)/zeta(2) and zeta denotes the Riemann zeta function.

cp's OEIS Frontend

A107742 G.f.: Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

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A002774 Number of bipartite partitions of n white objects and n black ones.

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A054242 Triangle read by rows: row n (n>=0) gives the number of partitions of (n,0), (n-1,1), (n-2,2), ..., (0,n) respectively into sums of distinct pairs.

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A219585 Number A(n,k) of k-partite partitions of {n}^k into distinct k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A201377 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of partitions of (n,k) into a sum of distinct pairs.

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A219560 Number of tripartite partitions of (n,n,n) into distinct triples.

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A219561 Number of 4-partite partitions of (n,n,n,n) into distinct quadruples.

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