A054225 -id:A054225 - cp's OEIS frontend

A002757 Number of bipartite partitions of n white objects and 8 black ones.

22, 67, 181, 401, 831, 1576, 2876, 4987, 8406, 13715, 21893, 34134, 52327, 78785, 116982, 171259, 247826, 354482, 502090, 704265, 979528, 1351109, 1849932, 2514723, 3396262, 4557867, 6081466, 8068930, 10650479, 13987419, 18283999

Offset: 0

N. J. A. Sloane

nonn

Number of ways to factor p^n*q^8 where p and q are distinct primes.

a(n) is the number of multiset partitions of the multiset {r^n, s^8}. - 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
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 8 of A054225.

Cf. A005380.

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]}]]; a[n_] := b[p^n*q^8, p^n*q^8]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[(22 + 23*x + 25*x^2 + 16*x^3 + 4*x^4 - 14*x^5 - 34*x^6 - 50*x^7 - 65*x^8 - 52*x^9 - 32*x^10 + 5*x^11 + 27*x^12 + 57*x^13 + 67*x^14 + 65*x^15 + 42*x^16 + 15*x^17 - 14*x^18 - 34*x^19 - 40*x^20 - 46*x^21 - 26*x^22 - 8*x^23 + 8*x^24 + 11*x^25 + 18*x^26 + 14*x^27 + 9*x^28 + 3*x^29 - 7*x^30 - 7*x^31 - 6*x^32 + 3*x^33 + 3*x^34 - x^35)/((1-x) * (1-x^2) * (1-x^3) * (1-x^4) * (1-x^5) * (1-x^6) * (1-x^7) * (1-x^8)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)

a(n) = if n <= 8 then A054225(8,n) else A054225(n,8). - Reinhard Zumkeller, Nov 30 2011

a(n) ~ 3*sqrt(3) * n^3 * exp(Pi*sqrt(2*n/3)) / (1120*Pi^8). - Vaclav Kotesovec, Feb 01 2016

Edited by Christian G. Bower, Jan 08 2004

A002758 Number of bipartite partitions of n white objects and 9 black ones.

Original entry on oeis.org

30, 97, 267, 608, 1279, 2472, 4571, 8043, 13715, 22652, 36535, 57568, 89079, 135384, 202747, 299344, 436597, 629364, 897970, 1268634, 1776562, 2466961, 3399463, 4650218, 6318429, 8529869, 11446563, 15272827, 20269135, 26762094

Offset: 0

N. J. A. Sloane

nonn

Number of ways to factor p^n*q^9 where p and q are distinct primes.

a(n) is the number of multiset partitions of the multiset {r^n, s^9}. - 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. 2.
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
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 9 of A054225.

Cf. A005380.

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]}]]; a[n_] := b[p^n*q^9, p^n*q^9]; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[(30 + 37*x + 43*x^2 + 37*x^3 + 20*x^4 - 3*x^5 - 35*x^6 - 65*x^7 - 97*x^8 - 119*x^9 - 109*x^10 - 69*x^11 - 26*x^12 + 37*x^13 + 89*x^14 + 131*x^15 + 142*x^16 + 141*x^17 + 97*x^18 + 44*x^19 - 18*x^20 - 72*x^21 - 100*x^22 - 108*x^23 - 96*x^24 - 69*x^25 - 25*x^26 + 12*x^27 + 42*x^28 + 52*x^29 + 54*x^30 + 35*x^31 + 14*x^32 + 2*x^33 - 4*x^34 - 20*x^35 - 19*x^36 - 14*x^37 - 8*x^38 + 7*x^39 + 8*x^40 + 8*x^41 - 2*x^42 - 4*x^43 + x^44)/((1-x) * (1-x^2) * (1-x^3) * (1-x^4) * (1-x^5) * (1-x^6) * (1-x^7) * (1-x^8) * (1-x^9)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)

a(n) = if n <= 9 then A054225(9,n) else A054225(n,9). - Reinhard Zumkeller, Nov 30 2011

a(n) ~ n^(7/2) * exp(Pi*sqrt(2*n/3)) / (560*sqrt(2)*Pi^9). - Vaclav Kotesovec, Feb 01 2016

Edited by Christian G. Bower, Jan 08 2004

A002759 Number of bipartite partitions of n white objects and 10 black ones.

Original entry on oeis.org

42, 139, 392, 907, 1941, 3804, 7128, 12693, 21893, 36535, 59521, 94664, 147794, 226524, 342006, 508866, 747753, 1085635, 1559725, 2218272, 3126541, 4368724, 6056705, 8333955, 11388614, 15460291, 20859497, 27979454, 37324367, 49529018

Offset: 0

N. J. A. Sloane

nonn

Number of ways to factor p^n*q^10 where p and q are distinct primes.

a(n) is the number of multiset partitions of the multiset {r^n, s^10}. - 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. 2.
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..10000 (terms 0..200 from Alois P. Heinz)
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 10 of A054225.

Cf. A005380.

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]}]]; a[n_] := b[p^n*q^10, p^n*q^10]; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[(42 + 55*x + 72*x^2 + 68*x^3 + 55*x^4 + 22*x^5 - 21*x^6 - 72*x^7 - 126*x^8 - 178*x^9 - 222*x^10 - 203*x^11 - 169*x^12 - 81*x^13 + 15*x^14 + 125*x^15 + 209*x^16 + 286*x^17 + 303*x^18 + 299*x^19 + 219*x^20 + 107*x^21 - 4*x^22 - 117*x^23 - 208*x^24 - 263*x^25 - 257*x^26 - 232*x^27 - 151*x^28 - 69*x^29 + 29*x^30 + 92*x^31 + 130*x^32 + 145*x^33 + 143*x^34 + 97*x^35 + 48*x^36 - 2*x^37 - 39*x^38 - 48*x^39 - 58*x^40 - 41*x^41 - 31*x^42 - 19*x^43 - 4*x^44 + 19*x^45 + 21*x^46 + 20*x^47 + 13*x^48 - 4*x^49 - 9*x^50 - 10*x^51 + 2*x^52 + 4*x^53 - x^54)/((1-x) * (1-x^2) * (1-x^3) * (1-x^4) * (1-x^5) * (1-x^6) * (1-x^7) * (1-x^8) * (1-x^9) * (1-x^10)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)

a(n) = if n <= 10 then A054225(10,n) else A054225(n,10). - Reinhard Zumkeller, Nov 30 2011

a(n) ~ sqrt(3) * n^4 * exp(Pi*sqrt(2*n/3)) / (5600*Pi^10). - Vaclav Kotesovec, Feb 01 2016

Edited by Christian G. Bower, Jan 08 2004

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 *)

A091438 Triangle a(n,k) of partitions of n objects of 2 colors, k of which are black and each part with at least one black object.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 6, 7, 7, 1, 4, 8, 12, 12, 11, 1, 4, 10, 16, 21, 19, 15, 1, 5, 12, 23, 31, 36, 30, 22, 1, 5, 15, 28, 45, 55, 58, 45, 30, 1, 6, 17, 37, 60, 84, 94, 92, 67, 42, 1, 6, 20, 44, 80, 115, 147, 153, 140, 97, 56, 1, 7, 23, 55, 101, 161, 211, 249, 244, 211, 139, 77

Offset: 1

Wouter Meeussen and Christian G. Bower, Jan 08 2004

Number of ways to factor p^(n-k)*q^k where p and q are distinct primes and each factor is a multiple of q.

			  1;
  1, 2;
  1, 2, 3;
  1, 3, 4, 5;
  1, 3, 6, 7, 7; ...

Alois P. Heinz, Rows n = 1..141, flattened

Row sums: A000219.
Main diagonal: A000041.
a(2n,n) gives A108457.
Cf. A054225.

b:= proc(n, i, j, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
      `if`(i<1 or k<1, 0, `if`(j<1, b(n, i-1, i-1, k),
       b(n, i, j-1, k)+`if`(i>n or j>k, 0, b(n-i, i, j, k-j)))))
    end:
a:= (n, k)->  b(n$2, k$2):
seq(seq(a(n,k), k=1..n), n=1..15);  # Alois P. Heinz, Mar 14 2015

b[n_, i_, j_, k_] := b[n, i, j, k] = If[n == 0, If[k == 0, 1, 0], If[i < 1 || k < 1, 0, If[j < 1, b[n, i - 1, i - 1, k], b[n, i, j - 1, k] + If[i > n || j > k, 0, b[n - i, i, j, k - j]]]]]; a[n_, k_] :=  b[n, n, k, k]; Table[a[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 10 2016, after Alois P. Heinz *)

G.f.: A(x,y) = Product_{i>=1, j=1..i} (1/(1-x^i*y^j)).

A060285 Number of partitions of n objects of 2 colors with parts size >1.

Original entry on oeis.org

1, 0, 3, 4, 11, 18, 42, 70, 144, 248, 466, 802, 1442, 2444, 4247, 7116, 12030, 19878, 32938, 53670, 87429, 140680, 225815, 359100, 569157, 895224, 1402941, 2184662, 3388915, 5228458, 8035921, 12291710, 18732318, 28425342, 42981877, 64740330

Offset: 0

Vladeta Jovovic, Mar 23 2001

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Vaclav Kotesovec)
N. J. A. Sloane, Transforms

Cf. (row sums of) A060244, A054225, A005380.

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(k+1),{k,2,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 04 2015 *)

Euler transform of sequence [0, 3, 4, 5, 6, ...].
G.f.: Product_{k=2..infinity} 1/(1-x^k)^(k+1).
From Vaclav Kotesovec, Mar 09 2015: (Start)
For n>=2, a(n) = A005380(n-2) - 2*A005380(n-1) + A005380(n).
a(n) ~ 2^(1/36) * Zeta(3)^(37/36) * exp(1/12 - Pi^4/(432*Zeta(3)) + Pi^2 * n^(1/3) / (3*2^(4/3)*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * 3^(1/2) * Pi * n^(55/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... .
a(n) ~ (2*Zeta(3))^(2/3) * A005380(n) / n^(2/3).
(End)

Edited by Christian G. Bower, Jan 08 2004

A054241 Number of partitions of bit-interleaved numbers; partitions of n in base sqrt(2).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 7, 2, 4, 3, 7, 9, 16, 16, 31, 5, 7, 12, 19, 11, 15, 30, 45, 29, 47, 57, 97, 77, 118, 162, 257, 5, 12, 7, 19, 29, 57, 47, 97, 11, 30, 15, 45, 77, 162, 118, 257, 109, 189, 189, 339, 323, 522, 589, 975, 323, 589, 522, 975, 1043, 1752, 1752

Offset: 0

Marc LeBrun, Feb 07 2000

Rearrangement of A054225 via A054238. Also can be directly derived from A054240 (bit-interleaved addition table).

			a(6)=4 thus: {6, 4+2, 3+1, 2+1+1} all in base sqrt(2).
From _Sean A. Irvine_, Jan 26 2022: (Start)
a(12)=9 from {12, 9+1, 8+4, 8+1+1, 6+2, 4+2+2, 3+3, 3+2+1, 2+2+1+1}.
a(13)=16 from {13, 12+1, 9+4, 9+1+1, 8+5, 8+4+1, 8+1+1+1, 7+2, 6+3, 6+2+1, 5+2+2, 4+3+2, 4+2+2+1, 3+3+1, 3+2+1+1, 2+2+1+1+1}.
(End)

Cf. A054225, A054238, A054239, A054240.

a(n) = A054225(A054239(n)). - Sean A. Irvine, Jan 26 2022

Data corrected by Sean A. Irvine, Jan 26 2022

A060287 Triangle formed from coefficients in expansion of Product_{i=3..infinity, j=0..i} 1/(1-x^(i-j)*y^j).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 2, 2, 2, 3, 4, 5, 5, 4, 3, 2, 3, 4, 6, 7, 8, 7, 6, 4, 3, 4, 6, 9, 12, 13, 13, 12, 9, 6, 4, 5, 8, 13, 17, 21, 21, 21, 17, 13, 8, 5, 6, 11, 18, 25, 31, 34, 34, 31, 25, 18, 11, 6, 9, 15, 26, 37, 48, 53, 58, 53, 48, 37, 26

Offset: 0

Vladeta Jovovic, Mar 23 2001

			Series ends ... + 2*x^6 + 2*x^5*y + 3*x^4*y^2 + 3*x^3*y^3 + 3*x^2*y^4 + 2*x*y^5 + 2*y^6 + x^5 + x^4*y + x^3*y^2 + x^2*y^3 + x*y^4 + y^5 + x^4 + x^3*y + x^2*y^2 + x*y^3 + y^4 + x^3 + x^2*y + x*y^2 + y^3 + 1.
[1], [0, 0], [0, 0, 0], [1, 1, 1, 1], [1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1], [2, 2, 3, 3, 3, 2, 2], [2, 3, 4, 5, 5, 4, 3, 2], [3, 4, 6, 7, 8, 7, 6, 4, 3], ...

Cf. A054225, A060244.

A136099 Triangle read by rows: the number of ways to factor 52^(n-k)3^k, columns 0<=k<=n, rows n>=0.

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 7, 11, 11, 7, 12, 21, 26, 21, 12, 19, 38, 52, 52, 38, 19, 30, 64, 98, 109, 98, 64, 30, 45, 105, 171, 212, 212, 171, 105, 45, 67, 165, 289, 382, 424, 382, 289, 165, 67, 97, 254, 467, 662, 783, 783, 662, 467, 254, 97, 139, 381, 737, 1097, 1386, 1481, 1386, 1097, 737, 381, 139

Offset: 0

Alford Arnold, Dec 15 2007

Second in the series of arrays beginning with A054225.

			5*A036561(2,1) = 5*6 = 30 and there are five ways to factor 30.
Triangle begins:
   1;
   2,  2;
   4,  5,  4;
   7, 11, 11,   7;
  12, 21, 26,  21, 12;
  19, 38, 52,  52, 38, 19;
  30, 64, 98, 109, 98, 64, 30;
  ...

Cf. A001055, A036561, A054225, A129306, A131419.

T(n,k) = A001055(5*A036561(n,k)).

cp's OEIS Frontend

A002757 Number of bipartite partitions of n white objects and 8 black ones.

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A002758 Number of bipartite partitions of n white objects and 9 black ones.

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A002759 Number of bipartite partitions of n white objects and 10 black ones.

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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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A091438 Triangle a(n,k) of partitions of n objects of 2 colors, k of which are black and each part with at least one black object.

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Maple

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A060285 Number of partitions of n objects of 2 colors with parts size >1.

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A054241 Number of partitions of bit-interleaved numbers; partitions of n in base sqrt(2).

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A060287 Triangle formed from coefficients in expansion of Product_{i=3..infinity, j=0..i} 1/(1-x^(i-j)*y^j).

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A136099 Triangle read by rows: the number of ways to factor 52^(n-k)3^k, columns 0<=k<=n, rows n>=0.