A002774 -id:A002774 - cp's OEIS frontend

A000491 Number of bipartite partitions of n white objects and 5 black ones.

7, 19, 47, 97, 189, 339, 589, 975, 1576, 2472, 3804, 5727, 8498, 12400, 17874, 25433, 35818, 49908, 68939, 94378, 128234, 172917, 231630, 308240, 407804, 536412, 701910, 913773, 1184022, 1527165, 1961432, 2508762, 3196473, 4057403, 5132066

Offset: 0

N. J. A. Sloane

nonn

Number of ways to factor p^n*q^5 where p and q are distinct primes.

a(n) is the number of multiset partitions of the multiset {r^n, s^5}. - Joerg Arndt, Jan 01 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, p. 1.
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..1000
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.
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)

Column 5 of A054225.

Cf. A005380.

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(243*2^n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 27 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]], {d, DeleteCases[Divisors[n], 1|n]}]]; a[n_] := b[3^5*2^n, 3^5*2^n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[(7 + 5*x + 2*x^2 - 2*x^3 - 7*x^4 - 9*x^5 - 6*x^6 + x^7 + 4*x^8 + 6*x^9 + 3*x^10 + x^11 - 3*x^12 - 2*x^13 + x^14)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)

a(n) = if n <= 5 then A054225(5,n) else A054225(n,5). - Reinhard Zumkeller, Nov 30 2011

a(n) ~ 3*n^(3/2) * exp(Pi*sqrt(2*n/3)) / (20*sqrt(2)*Pi^5). - Vaclav Kotesovec, Feb 01 2016

Edited by Christian G. Bower, Jan 08 2004

A201376 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of partitions of (n,k) into a sum of pairs.

Original entry on oeis.org

1, 1, 2, 2, 4, 9, 3, 7, 16, 31, 5, 12, 29, 57, 109, 7, 19, 47, 97, 189, 339, 11, 30, 77, 162, 323, 589, 1043, 15, 45, 118, 257, 522, 975, 1752, 2998, 22, 67, 181, 401, 831, 1576, 2876, 4987, 8406, 30, 97, 267, 608, 1279, 2472, 4571, 8043, 13715, 22652, 42, 139, 392, 907, 1941, 3804, 7128, 12693, 21893, 36535, 59521

Offset: 0

Reinhard Zumkeller, Nov 30 2011

By analogy with ordinary partitions (A000041). The empty partition gives T(0,0)=1 by definition. A201377 and A054225 give partitions of pairs into sums of 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 (3,1) into positive pairs, T(3,1) = 7:
(3,1),
(3,0) + (0,1),
(2,1) + (1,0),
(2,0) + (1,1),
(2,0) + (1,0) + (0,1),
(1,1) + (1,0) + (1,0),
(1,0) + (1,0) + (1,0) + (0,1).
First ten rows of triangle:
0:                      1
1:                    1  2
2:                  2  4  9
3:                3  7  16  31
4:              5  12  29  57  109
5:            7  19  47  97  189  339
6:          11  30  77  162  323  589  1043
7:        15  45  118  257  522  975  1752  2998
8:      22  67  181  401  831  1576  2876  4987  8406
9:    30  97  267  608  1279  2472  4571  8043  13715  22652
X:  42  139  392  907  1941  3804  7128  12693  21893  36535  59521

Alois P. Heinz, Rows n = 0..140, flattened
Reinhard Zumkeller, Haskell programs for A054225, A054242, A201376, A201377

T(n,0) = A000041(n);
T(1,k) = A000070(k),  k <= 1;  T(n,1) = A000070(n),  n > 1;
T(2,k) = A000291(k),  k <= 2;  T(n,2) = A000291(n),  n > 2;
T(3,k) = A000412(k),  k <= 3;  T(n,3) = A000412(n),  n > 3;
T(4,k) = A000465(k),  k <= 4;  T(n,4) = A000465(n),  n > 4;
T(5,k) = A000491(k),  k <= 5;  T(n,5) = A000491(n),  n > 5;
T(6,k) = A002755(k),  k <= 6;  T(n,6) = A002755(n),  n > 6;
T(7,k) = A002756(k),  k <= 7;  T(n,7) = A002756(n),  n > 7;
T(8,k) = A002757(k),  k <= 8;  T(n,8) = A002757(n),  n > 8;
T(9,k) = A002758(k),  k <= 9;  T(n,9) = A002758(n),  n > 9;
T(10,k) = A002759(n), k <= 10; T(n,10) = A002759(n), n > 10;
T(n,n) = A002774(n).
See A054225 for another version.
Cf. A000041, A054242, A201377.

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

max = 10; 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}]; t[n_, k_] := coes[[n+1, k+1]]; Flatten[ Table[ t[n, k], {n, 0, max}, {k, 0, n}]] (* Jean-François Alcover, Dec 05 2011 *)
p = 2; q = 3; b[n_, k_] := b[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, b[n/d, d]], {d, DeleteCases[Divisors[n] , 1|n]}]]; t[n_, k_] := b[p^n*q^k, p^n*q^k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)

For references, programs and g.f. see A054225.

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

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

A091437 Number of bipartite partitions of ceiling(n/2) white objects and floor(n/2) black ones.

Original entry on oeis.org

1, 1, 2, 4, 9, 16, 31, 57, 109, 189, 339, 589, 1043, 1752, 2998, 4987, 8406, 13715, 22652, 36535, 59521, 94664, 151958, 239241, 379693, 591271, 927622, 1431608, 2224235, 3402259, 5236586, 7947530, 12130780, 18272221, 27669593, 41393154

Offset: 0

Christian G. Bower, Jan 08 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

a(n) = A054225(n, [n/2]). Cf. A002774, A005380.

max = 35; 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[[ Ceiling[(n+2)/2], Floor[(n+2)/2] ]]; Table[a[n], {n, 0, max} ](* Jean-François Alcover, Dec 06 2011 *)

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

A254686 Number of ways to put n red and n blue balls into n indistinguishable boxes.

Original entry on oeis.org

1, 1, 5, 19, 74, 248, 814, 2457, 7168, 19928, 53688, 139820, 354987, 878434, 2128102, 5052010, 11781881, 27019758, 61035671, 135928105, 298784144, 648726349, 1392474574, 2956730910, 6214668074, 12937060340, 26686392239, 54572423946, 110680119454, 222710856175, 444776676764

Offset: 0

Brian Chen, Feb 08 2015

nonn

See a comment on A254811 about multiset partitions and the Knuth reference. - Wolfdieter Lang, Mar 26 2015

			For n = 2 the a(2) = 5 ways to put 2 red balls and 2 blue balls into 2 indistinguishable boxes are (RRBB)(), (RRB)(B), (RBB)(R), (RR)(BB), (RB)(RB).

Brian Chen, Table of n, a(n) for n = 0..64

Cf. A002774, A060150, A254811.
Column k=2 of A256384.
Main diagonal of A277239.

with(numtheory):
b:= proc(n, k, i) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n) or i<2, 0, add(
      `if`(d>k, 0, b(n/d, d, i-1)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b(6^n$2,n):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 26 2015

b[n_, k_, i_] := b[n, k, i] = If[n > k, 0, 1] + If[PrimeQ[n] || i < 2, 0, Sum[If[d > k, 0, b[n/d, d, i - 1]], {d, Divisors[n] [[2 ;; -2]]}]]; a[n_] := b[6^n, 6^n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

A108462 Number of factorizations of (n,n) into pairs (i,j) with i,j >= 1, not both 1.

Original entry on oeis.org

1, 2, 2, 9, 2, 15, 2, 31, 9, 15, 2, 92, 2, 15, 15, 109, 2, 92, 2, 92, 15, 15, 2, 444, 9, 15, 31, 92, 2, 203, 2, 339, 15, 15, 15, 712, 2, 15, 15, 444, 2, 203, 2, 92, 92, 15, 2, 1903, 9, 92, 15, 92, 2, 444, 15, 444, 15, 15, 2, 1663, 2, 15, 92, 1043, 15, 203, 2, 92, 15, 203, 2

Offset: 1

Christian G. Bower, Jun 03 2005

nonn

The rule of building products is (a,b)*(x,y) = (a*x,b*y).

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).

			From _Alois P. Heinz_ and _Antti Karttunen_, Nov 24 2017: (Start)
a(4) = 9 because for pair (4,4) there are nine factorizations:
  (4,4)
  (1,4)*(4,1)
  (1,2)*(4,2)
  (2,1)*(2,4)
  (2,2)*(2,2)
  (1,2)*(2,1)*(2,2)
  (1,4)*(2,1)*(2,1)
  (4,1)*(1,2)*(1,2)
  (1,2)*(1,2)*(2,1)*(2,1)
(End)
a(pq) = 15 for primes p<>q: (pq,pq); (p,1)(q,pq); (p,1)(q,1)(1,pq); (p,1)(q,1)(1,p)(1,q); (p,1)(q,q)(1,p); (p,1)(q,p)(1,q); (p,q)(q,p); (p,q)(q,1)(1,p); (p,p)(q,q) ; (p,p)(q,1)(1,q); (p,pq)(q,1); (pq,1)(1,pq); (pq,1)(1,p)(1,q); (pq,q)(1,p); (pq,p)(1,q). - _R. J. Mathar_, Nov 30 2017

Main diagonal of A108461.

Cf. A002774, A020557, A108463.

a(n) = if(n==1, return(1)); my(b, c, r, x, y, v=List([]), w=List([[n]])); while(#w>r, c++; for(k=r+1, r=#w, y=w[k]; if(!isprime(x=y[c]), fordiv(x, d, if(d!=1&&d!=x, listput(w, concat([y[1..c-1], d, x/d]))))))); for(i=1, #w, x=w[i]; r=#x; for(j=1, #w, y=w[j]; for(k=0, 2^r-1, b=concat(b=binary(k), vector(r-#b)); if(#y>=t=vecsum(b), c=0; listput(v, vecsort(vector(r+#y-t, m, if(m>r, [1, y[m-r+t]], if(b[m], [x[m], y[c++]], [x[m], 1]))))))))); #Set(v); \\ Jinyuan Wang, Jan 17 2022

a(A025487(n)) = A108463(n).
a(p^k) = A002774(k).
a(A002110(n)) = A020557(n).
a(n) = A108461(n,n).

A331197 Number of nonnegative integer matrices with 2 distinct columns and any number of nonzero rows with each column sum being n and rows in nonincreasing lexicographic order.

Original entry on oeis.org

0, 1, 7, 28, 104, 332, 1032, 2983, 8384, 22622, 59479, 151902, 379616, 927521, 2224100, 5236410, 12130549, 27669296, 62229605, 138095206, 302672402, 655627183, 1404598865, 2977830134, 6251059210, 12999297747, 26791987616, 54750232180, 110977385294, 223204454700, 445590973235

Offset: 0

Andrew Howroyd, Jan 11 2020

nonn

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

			The a(2) = 7 matrices are:
   [2 1]   [2 0]   [1 2]   [1 1]   [2 0]   [1 0]   [1 0]
   [0 1]   [0 2]   [1 0]   [1 0]   [0 1]   [1 0]   [1 0]
                           [0 1]   [0 1]   [0 2]   [0 1]
                                                   [0 1]
See the example in A331197 for the a(3) = 28 case.

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

Column k=2 of A331161.

Cf. A000041, A002774, A331197.

a(n) = A002774(n) - A000041(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

A000491 Number of bipartite partitions of n white objects and 5 black ones.

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A201376 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of partitions of (n,k) into a sum of pairs.

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

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A091437 Number of bipartite partitions of ceiling(n/2) white objects and floor(n/2) black ones.

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Mathematica

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

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A254686 Number of ways to put n red and n blue balls into n indistinguishable boxes.

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Maple

Mathematica

A108462 Number of factorizations of (n,n) into pairs (i,j) with i,j >= 1, not both 1.

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A331197 Number of nonnegative integer matrices with 2 distinct columns and any number of nonzero rows with each column sum being n and rows in nonincreasing lexicographic order.

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