A060244 -id:A060244 - cp's OEIS frontend

A291553 Column 3 of A060244.

0, 0, 1, 2, 4, 8, 13, 22, 35, 54, 81, 121, 174, 250, 352, 491, 675, 924, 1246, 1674, 2226, 2944, 3862, 5046, 6541, 8449, 10846, 13869, 17641, 22365, 28214, 35485, 44443, 55494, 69036, 85650, 105894, 130594, 160561, 196923, 240847, 293907, 357722, 434477, 526448

Offset: 1

Vaclav Kotesovec, Aug 26 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A060244, A024786.

nmax = 50; col = 3; Flatten[{0, 0, CoefficientList[Coefficient[Normal[Series[Product[Product[1/(1 - x^(i - j)*y^j), {j, 0, i}], {i, 2, nmax + col}], {x, 0, col}, {y, 0, nmax}]], x^col], y]}]
nmax = 50; Rest[CoefficientList[Series[(x^3 * (1 + x - x^4))/((1-x)^2 * (1+x) * (1 + x + x^2)) / QPochhammer[x], {x, 0, nmax}], x]]
Table[Sum[(Floor[k/2] - Floor[(k-1)/3]) * PartitionsP[n-k], {k, 3, n}], {n, 1, 50}]

G.f.: x^3 * (1 + x - x^4) / ((1 - x)^2 * (1 + x) * (1 + x + x^2)) * Product_{k>=1} 1/(1 - x^k).
a(n) = Sum_{k=3..n} (floor(k/2) - floor((k-1)/3)) * A000041(n-k).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*Pi^2).
a(n) ~ n * A000041(n) / Pi^2.

A291589 Column 4 of A060244.

Original entry on oeis.org

0, 0, 0, 2, 3, 8, 13, 26, 40, 69, 104, 165, 241, 363, 517, 750, 1046, 1473, 2018, 2779, 3746, 5063, 6733, 8959, 11769, 15454, 20082, 26068, 33549, 43108, 54997, 70037, 88645, 111979, 140714, 176462, 220280, 274418, 340480, 421593, 520154, 640481, 786104, 962976

Offset: 1

Vaclav Kotesovec, Aug 27 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A060244.

nmax = 50; col = 4; Flatten[{0, 0, 0, CoefficientList[Coefficient[Normal[Series[Product[Product[1/(1 - x^(i - j)*y^j), {j, 0, i}], {i, 2, nmax + col}], {x, 0, col}, {y, 0, nmax}]], x^col], y]}]
Rest[CoefficientList[Series[x^4*(2 + x + x^2 - x^3 - x^4 - x^5 - x^6 + x^7) / ((1 - x)^3 * (1 + x)^2 * (1 + x^2) * (1 + x + x^2)) / QPochhammer[x], {x, 0, 100}], x]]
Table[Sum[(7*k^2/48 + 47*(k-4)/48 + Floor[(k-3)/4]/2 - (2*k + 19)*Floor[(k-3)/2]/8 + Floor[(k-2)/3]/3 - Floor[k/3]/3) * PartitionsP[n-k], {k, 4, n}], {n, 1, 50}]

G.f.: x^4 * (2 + x + x^2 - x^3 - x^4 - x^5 - x^6 + x^7) / ((1 - x)^3 * (1 + x)^2 * (1 + x^2) * (1 + x + x^2)) * Product_{k>=1} 1/(1 - x^k).
a(n) ~ exp(Pi*sqrt(2*n/3)) * sqrt(n) / (8*sqrt(2)*Pi^3).
a(n) ~ sqrt(3) * n^(3/2) * A000041(n) / (2^(3/2) * Pi^3).

A291590 Column 5 of A060244.

Original entry on oeis.org

0, 0, 0, 0, 2, 5, 11, 22, 40, 70, 116, 187, 292, 448, 670, 988, 1432, 2051, 2896, 4052, 5603, 7687, 10446, 14096, 18870, 25108, 33176, 43601, 56960, 74051, 95762, 123300, 158011, 201692, 256368, 324682, 409642, 515116, 645509, 806430, 1004292, 1247146, 1544237

Offset: 1

Vaclav Kotesovec, Aug 27 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A060244.

nmax = 30; col = 5; Flatten[{0, 0, 0, 0, CoefficientList[Coefficient[Normal[Series[Product[Product[1/(1 - x^(i - j)*y^j), {j, 0, i}], {i, 2, nmax + col}], {x, 0, col}, {y, 0, nmax}]], x^col], y]}]
Rest[CoefficientList[Series[x^5*(2 + 3*x + 2*x^2 + x^3 - 2*x^4 - 3*x^5 - 4*x^6 - 2*x^7 + 2*x^9 + 2*x^10 + x^11 - x^13)/((1 - x)^4 * (1 + x)^2 * (1 + x^2) * (1 + x + x^2) * (1 + x + x^2 + x^3 + x^4)) / QPochhammer[x], {x, 0, 100}], x]]
Table[Sum[(1 + (k-4)*(645 + 10*k + k^2)/720 - Floor[(k-4)/5]/5 - Floor[(k-4)/4]/4 + (k+1)*Floor[(k-4)/2]/8 - Floor[(k-3)/5]/5 - Floor[(k-3)/4]/4 - Floor[(k-3)/3]/3 - 3*Floor[(k-1)/5]/5) * PartitionsP[n-k], {k, 5, n}], {n, 1, 100}]

G.f.: x^5 * (2 + 3*x + 2*x^2 + x^3 - 2*x^4 - 3*x^5 - 4*x^6 - 2*x^7 + 2*x^9 + 2*x^10 + x^11 - x^13)/((1 - x)^4 * (1 + x)^2 * (1 + x^2) * (1 + x + x^2) * (1 + x + x^2 + x^3 + x^4)) * Product_{k>=1} 1/(1 - x^k).
a(n) ~ sqrt(3) * n * exp(Pi*sqrt(2*n/3)) / (40*Pi^4).
a(n) ~ 3*n^2 * A000041(n) / (10*Pi^4).

A291596 Column 6 of A060244.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 7, 19, 35, 69, 116, 204, 323, 523, 799, 1225, 1809, 2675, 3843, 5515, 7756, 10869, 14998, 20621, 27996, 37865, 50701, 67612, 89419, 117806, 154101, 200838, 260168, 335824, 431202, 551824, 702890, 892503, 1128577, 1422846, 1787183, 2238554

Offset: 1

Vaclav Kotesovec, Aug 27 2017

nonn

Conjecture: column k of A060244 is asymptotic to (6*n)^((k-1)/2) * A000041(n) / (Pi^(k-1) * k!) ~ 2^((k-5)/2) * 3^((k-2)/2) * n^((k-3)/2) * exp(Pi*sqrt(2*n/3)) / (Pi^(k-1) * k!).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A060244.

nmax = 30; col = 6; Flatten[{0, 0, 0, 0, 0, CoefficientList[Coefficient[Normal[Series[Product[Product[1/(1 - x^(i - j)*y^j), {j, 0, i}], {i, 2, nmax + col}], {x, 0, col}, {y, 0, nmax}]], x^col], y]}]
Rest[CoefficientList[Series[x^6 * (4 + 3*x + 4*x^2 + 2*x^3 - 4*x^5 - 6*x^6 - 6*x^7 - 5*x^8 - x^9 + x^10 + 5*x^11 + 3*x^12 + 3*x^13 + x^14 - x^15 - x^16 - x^17) / ((1 - x)^5 * (1 + x)^3 * (1 + x^2) * (1 - x + x^2) * (1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)) / QPochhammer[x], {x, 0, 100}], x]]
Table[Sum[(k^4/17280 + 101*k^3/8640 + 1661*k^2/4320 - 23017*k/17280 - 2563/576 + Floor[(k-5)/4]/8 - 7*Floor[(k-5)/3]/18 - (19/192 + 7*k/12 + k^2/96) * Floor[(k-5)/2] + Floor[(k-4)/6]/6 - Floor[(k-4)/4]/8 - (4/3 + k/18) * Floor[(k-4)/3] - Floor[(k-3)/5]/5 + Floor[(k-2)/5]/5) * PartitionsP[n-k], {k, 6, n}], {n, 1, 100}]

G.f.: x^6 * (4 + 3*x + 4*x^2 + 2*x^3 - 4*x^5 - 6*x^6 - 6*x^7 - 5*x^8 - x^9 + x^10 + 5*x^11 + 3*x^12 + 3*x^13 + x^14 - x^15 - x^16 - x^17) / ((1 - x)^5 * (1 + x)^3 * (1 + x^2) * (1 - x + x^2) * (1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)) * Product_{k>=1} 1/(1 - x^k).
a(n) ~ exp(Pi*sqrt(2*n/3)) * n^(3/2) / (40*sqrt(2)*Pi^5).
a(n) ~ sqrt(3/2) * n^(5/2) * A000041(n) / (10*Pi^5).

A024786 Number of 2's in all partitions of n.

Original entry on oeis.org

0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 112, 160, 213, 295, 389, 526, 686, 911, 1176, 1538, 1968, 2540, 3223, 4115, 5181, 6551, 8191, 10269, 12756, 15873, 19598, 24222, 29741, 36532, 44624, 54509, 66261, 80524, 97446, 117862, 142029, 171036, 205290, 246211

Offset: 1

Clark Kimberling

nonn

Also number of partitions of n-1 with a distinguished part different from all the others. [Comment corrected by Emeric Deutsch, Aug 13 2008]
In general the number of times that j appears in the partitions of n equals Sum_{kA024787, ..., A024794, for j = 2,...,10; it generalizes the formula given for A000070 for j=1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002
Equals row sums of triangle A173238. - Gary W. Adamson, Feb 13 2010
The sums of two successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of second largest and the sum of third largest elements in all partitions of n. More generally, the number of occurrences of k in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+1)st largest elements in all partitions of n. And more generally, the sum of the number of occurrences of k, k+1, k+2..k+m in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+m+1)st largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
Number of singletons in all partitions of n-1. A singleton in a partition is a part that occurs exactly once. Example: a(5) = 4 because in the partitions of 4, namely [1,1,1,1], [1,1,2'], [2,2], [1',3'], [4'] we have 4 singletons (marked by '). - Emeric Deutsch, Sep 12 2016
a(n) is also the number of non-isomorphic vertex-transitive cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_{n-1}. See Table 1 in the Hoang/Mütze reference in the Links section. - Torsten Muetze, Nov 28 2019
Assuming a partition is in weakly decreasing order, a(n) is also the number of times -1 occurs in the differences of the partitions of n+1. - George Beck, Mar 28 2023

			From _Omar E. Pol_, Oct 25 2012: (Start)
For n = 7 we have:
--------------------------------------
.                             Number
Partitions of 7               of 2's
--------------------------------------
7 .............................. 0
4 + 3 .......................... 0
5 + 2 .......................... 1
3 + 2 + 2 ...................... 2
6 + 1 .......................... 0
3 + 3 + 1 ...................... 0
4 + 2 + 1 ...................... 1
2 + 2 + 2 + 1 .................. 3
5 + 1 + 1 ...................... 0
3 + 2 + 1 + 1 .................. 1
4 + 1 + 1 + 1 .................. 0
2 + 2 + 1 + 1 + 1 .............. 2
3 + 1 + 1 + 1 + 1 .............. 0
2 + 1 + 1 + 1 + 1 + 1 .......... 1
1 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0
------------------------------------
.  24 - 13 =                    11
.
The difference between the sum of the second column and the sum of the third column of the set of partitions of 7 is 24 - 13 = 11 and equals the number of 2's in all partitions of 7, so a(7) = 11.
(End)

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 184.

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
Philip Cuthbertson, Fixed hooks in arbitrary columns of partitions, Integers (2025) Vol. 25, Art. No. A28. See p. 3.
Manosij Ghosh Dastidar and Sourav Sen Gupta, Generalization of a few results in Integer Partitions, arXiv preprint arXiv:1111.0094 [cs.DM], 2011.
Emeric Deutsch et al., Problem 11237, Amer. Math. Monthly, 115 (No. 7, 2008), 666-667. [From _Emeric Deutsch_, Aug 13 2008]
Hung Phuc Hoang and Torsten Mütze, Combinatorial generation via permutation languages. II. Lattice congruences, arXiv:1911.12078 [math.CO], 2019.
Joseph Vandehey, Digital problems in the theory of partitions, Integers (2024) Vol. 24A, Art. No. A18. See p. 3.

Cf. A066633, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794, A173238.
Column 2 of A060244.
First differences of A000097.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0$2], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+`if`(i=2, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, May 18 2012

Table[ Count[ Flatten[ IntegerPartitions[n]], 2], {n, 1, 50} ]
(* Second program: *)
b[n_, i_] := b[n, i] = Module[{f, g}, If[n==0 || i==1, {1, 0}, f = b[n, i - 1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + If[i == 2, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 22 2015, after Alois P. Heinz *)
Join[{0}, (1/((1 - x^2) QPochhammer[x]) + O[x]^50)[[3]]] (* Vladimir Reshetnikov, Nov 22 2016 *)
Table[Sum[(1 + (-1)^k)/2 * PartitionsP[n-k], {k, 2, n}], {n, 1, 50}] (* Vaclav Kotesovec, Aug 27 2017 *)

from sympy import npartitions
def A024786(n): return sum(npartitions(n-(k<<1)) for k in range(1,(n>>1)+1)) # Chai Wah Wu, Oct 25 2023

a(n) = Sum_{k=1..floor(n/2)} A000041(n-2k). - Christian G. Bower, Jun 22 2000
a(n) = Sum_{kA000041, P(0) = 1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002
G.f.: (x^2/((1-x)*(1-x^2)^2))*Product_{j>=3} 1/(1-x^j) from Riordan reference second term, last eq.
a(n) = A006128(n-1) - A194452(n-1). - Omar E. Pol, Nov 20 2011
a(n) = A181187(n,2) - A181187(n,3). - Omar E. Pol, Oct 25 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2) * Pi * sqrt(n)) * (1 - 25*Pi/(24*sqrt(6*n)) + (25/48 + 433*Pi^2/6912)/n). - Vaclav Kotesovec, Mar 07 2016, extended Nov 05 2016
a(n) = Sum_{k} k * A116595(n-1,k). - Emeric Deutsch, Sep 12 2016
G.f.: x^2/((1 - x)*(1 - x^2)) * Sum_{n >= 0} x^(2*n)/( Product_{k = 1..n} 1 - x^k ); that is, convolution of A004526 (partitions into 2 parts, or, modulo offset differences, partitions into parts <= 2) and A002865 (partitions into parts >= 2). - Peter Bala, Jan 17 2021

A054225 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 pairs.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 3, 5, 7, 9, 7, 5, 7, 12, 16, 16, 12, 7, 11, 19, 29, 31, 29, 19, 11, 15, 30, 47, 57, 57, 47, 30, 15, 22, 45, 77, 97, 109, 97, 77, 45, 22, 30, 67, 118, 162, 189, 189, 162, 118, 67, 30, 42, 97, 181, 257, 323, 339, 323, 257, 181, 97, 42, 56, 139, 267, 401, 522, 589, 589, 522, 401, 267, 139, 56

Offset: 0

Marc LeBrun, Feb 04 2000

By analogy with ordinary partitions (A000041). The empty partition gives T(0,0)=1 by definition. A054225 and A201377 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.
Or, triangle T(n,k) of bipartite partitions of n objects, k of which are black.
Or, number of ways to factor p^(n-k)*q^k where p and q are distinct primes.
In the paper by F. C. Auluck: "On partitions of bipartite numbers", p.74, in the formula for fixed m there should be factor 1/m!. The correct asymptotic formula is p(m, n) ~ (sqrt(6*n)/Pi)^m * exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*m!*n). - Vaclav Kotesovec, Feb 01 2016
T(n,k)=T(n,k-n) is the number of multiset partitions of the multiset {1^k, 2^(n-k)}, see example link. - Joerg Arndt, Jan 01 2024
Let R be the ring of power series in two countably infinite sets of variables x_1,y_1,x_2,y_2,... that are invariant under the diagonal action (i.e, the group S of permutations of positive integers acts by w(x_i)=x_{w(i)} and w(y_i)=y_{w(i)}).  Then T(n,k) is the dimension of the (n,k)-bigraded piece of R, i.e., the bihomogeneous power series of degree n in the x-variables and k in the y-variables that are S-invariant. - Jeremy L. Martin, Nov 27 2024

			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 2, 2, 2 from (2,0) = (1,0)+(1,0), (1,1) = (1,0)+(0,1), (0,2) = (0,1)+(0,1).
In the fourth row (n=3), T(2,1)=4 from (2,1) = (2,0)+(0,1) = (1,0)+(1,1) = (1,0)+(1,0)+(0,1).
The triangle begins:
   1;
   1,  1;
   2,  2,  2;
   3,  4,  4,  3;
   5,  7,  9,  7,   5;
   7, 12, 16, 16,  12,  7;
  11, 19, 29, 31,  29, 19, 11;
  15, 30, 47, 57,  57, 47, 30, 15;
  22, 45, 77, 97, 109, 97, 77, 45, 22;
  ...
A further example: T(2,2) = 9:
[(2,2)],
[(2,1),(0,1)],
[(2,0),(0,2)],
[(2,0),(0,1),(0,1)],
[(1,2),(1,0)],
[(1,1),(1,1)],
[(1,1),(1,0),(0,1)],
[(1,0),(1,0),(0,2)],
[(1,0),(1,0),(0,1),(0,1)].

M. S. Cheema, Tables of partitions of Gaussian integers, National Institute of Sciences of India, New Delhi, 1956.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018

Alois P. Heinz, Rows n = 0..200, flattened
Joerg Arndt, Multiset partitions of the multisets {1^r,2^s} for r+s=5.
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)
P. A. MacMahon, Memoir on symmetric functions of the roots of systems of equations, Phil. Trans. Royal Soc. London, 181 (1890), 481-536; Coll. Papers II, 32-87.
Reinhard Zumkeller, Haskell programs for A201376, A054225, A201377, A054242

See A201376 for the same triangle formatted in a different way.
Columns 0-10: A000041, A000070, A000291, A000412, A000465, A000491, A002755, A002756, A002757, A002758, A002759.
Row sums: A005380. a(2n, n): A002774. a(n, [n/2]): A091437. Cf. A060244.
The outer edges are T(n,0) = T(0,n) = A000041(n).
A054242 gives partitions into sums of distinct pairs.

Haskell
```
see Zumkeller link.
```

read transforms; t1 := mul( mul( 1/(1-x^(i-j)*y^j), j=0..i), i=1..11): SERIES2(t1,x,y,6);

rows = 11; se = Series[ Product[ 1/(1-x^(n-k)*y^k), {n, 1, rows}, {k, 0, n}], {x, 0, rows}, {y, 0, rows}]; coes = CoefficientList[ se, {x, y}]; Flatten[ Table[ coes[[n-k+1, k]], {n, 1, rows+1}, {k, 1, n}]] (* Jean-François Alcover, Nov 21 2011, after g.f. *)
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-k)*q^k, p^(n-k)*q^k]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)

{T(n, k) = if( n<0 || k<0, 0, polcoeff( polcoeff( prod( i=1, n, prod( j=0, i, 1 / (1 - x^i * y^j), 1 + x * O(x^n))),n),k))} /* Michael Somos, Apr 19 2005 */

G.f.: Product_{i>=1, j=0..i} 1/(1-x^(i-j)*y^j).

Series ends ... + 7*x^5 + 12*x^4*y + 16*x^3*y^2 + 16*x^2*y^3 + 12*x*y^4 + 7*y^5 + 5*x^4 + 7*x^3*y + 9*x^2*y^2 + 7*x*y^3 + 5*y^4 + 3*x^3 + 4*x^2*y + 4*x*y^2 + 3*y^3 + 2*x^2 + 2*x*y + 2*y^2 + x + y + 1.

Entry revised by N. J. A. Sloane, Nov 30 2011, to incorporate corrections provided by Reinhard Zumkeller, who also contributed the alternative version A201376. Once the errors were corrected, this sequence coincided with A060243, due to N. J. A. Sloane, Mar 22 2001, which included edits by Vladeta Jovovic, Mar 23 2001, and Christian G. Bower, Jan 08 2004. The two entries have now been merged.

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

cp's OEIS Frontend

A291553 Column 3 of A060244.

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A291589 Column 4 of A060244.

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A291590 Column 5 of A060244.

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A291596 Column 6 of A060244.

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A024786 Number of 2's in all partitions of n.

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A054225 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 pairs.

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

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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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