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%I M1601 #102 May 28 2021 06:03:03
%S 1,2,6,14,33,70,149,298,591,1132,2139,3948,7199,12894,22836,39894,
%T 68982,117948,199852,335426,558429,922112,1511610,2460208,3977963,
%U 6390942,10206862,16207444,25596941,40214896,62868772,97814358
%N Expansion of 1 / Product_{k>=1} (1-x^k)^(k+1).
%C Also, a(n) = number of partitions of the integer n where there are k+1 different kinds of part k for k = 1, 2, 3, ....
%C Also, a(n) = number of partitions of n objects of 2 colors. These are set partitions, the n objects are not labeled but colored, using two colors. For each subset of size k there are k+1 different possibilities, i=0..k white and k-i black objects.
%C Also, a(n) = number of simple unlabeled graphs with n nodes of 2 colors whose components are complete graphs. - _Geoffrey Critzer_, Sep 27 2012
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Exercise 7.99, p. 484 and pp. 548-549.
%H Alois P. Heinz, <a href="/A005380/b005380.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H P. J. Cameron, <a href="http://dx.doi.org/10.1016/0012-365X(89)90081-2">Some sequences of integers</a>, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
%H Carlos A. A. Florentino, <a href="https://arxiv.org/abs/2105.13049">Plethystic exponential calculus and permutation polynomials</a>, arXiv:2105.13049 [math.CO], 2021. Mentions this sequence.
%H Vaclav Kotesovec, <a href="/A005380/a005380.jpg">Graph - The asymptotic ratio</a>
%H P. A. MacMahon, <a href="http://www.jstor.org/stable/90574">Memoir on symmetric functions of the roots of systems of equations</a>, Phil. Trans. Royal Soc. London, 181 (1890), 481-536; Coll. Papers II, 32-87.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H R. P. Stanley, <a href="http://www-math.mit.edu/~rstan//pubs/pubfiles/12-2.pdf">Theory and Applications of Plane Partitions: Part 2</a>, Studies in Appl. Math., 1 (1971), 259-279.
%H R. P. Stanley, <a href="http://dx.doi.org/10.1016/0097-3165(73)90063-0">The conjugate trace and trace of a plane partition</a>, J. Combin. Theory, vol. A14 53-65 1973, esp. p. 64.
%F EULER transform of b(n) = n+1.
%F a(n) ~ Zeta(3)^(13/36) * 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 * 2^(23/36) * 3^(1/2) * Pi * n^(31/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - _Vaclav Kotesovec_, Mar 07 2015
%F a(n) = A089353(n+m, m), n >= 1, for each m >= n. a(0) =1. See the Stanley reference, Exercise 7.99. - _Wolfdieter Lang_, Mar 09 2015
%F G.f.: exp(Sum_{k>=1} (sigma_1(k) + sigma_2(k))*x^k/k). - _Ilya Gutkovskiy_, Aug 11 2018
%e We represent each summand, k, in a partition of n as k identical objects. Then we color each object. We have no regard for the order of the colored objects.
%e a(3) = 14 because we have: www; wwb; wbb; bbb; ww + w; ww + b; wb + w; wb + b; bb + w; bb + b; w + w + w; w + w + b; w + b + b; b + b + b, where the 2 colors are black b and white w. - _Geoffrey Critzer_, Sep 27 2012
%e a(3) = 14 because we have: 3; 3'; 3''; 3'''; 2 + 1; 2 + 1'; 2' + 1; 2' + 1'; 2'' + 1; 2'' + 1'; 1 + 1 + 1; 1 + 1 + 1'; 1 + 1' + 1'; 1' + 1' + 1', where a part k of different sorts is given as k, k', k'', etc. - _Joerg Arndt_, Mar 09 2015
%e From _Alois P. Heinz_, Mar 09 2015: (Start)
%e The a(4) = 33 = 5 + 9 + 6 + 8 + 5 partitions of 4 objects of 2 colors are:
%e 5 partitions for the integer partition of 4 = 1 + 1 + 1 + 1:
%e 01: {{b}, {b}, {b}, {b}}
%e 02: {{b}, {b}, {b}, {w}}
%e 03: {{b}, {b}, {w}, {w}}
%e 04: {{b}, {w}, {w}, {w}}
%e 05: {{w}, {w}, {w}, {w}}
%e 9 partitions for the integer partition of 4 = 1 + 1 + 2:
%e 06: {{b}, {b}, {b,b}}
%e 07: {{b}, {w}, {b,b}}
%e 08: {{w}, {w}, {b,b}}
%e 09: {{b}, {b}, {w,b}}
%e 10: {{b}, {w}, {w,b}}
%e 11: {{w}, {w}, {w,b}}
%e 12: {{b}, {b}, {w,w}}
%e 13: {{b}, {w}, {w,w}}
%e 14: {{w}, {w}, {w,w}}
%e 6 partitions for the integer partition of 4 = 2 + 2:
%e 15: {{b,b}, {b,b}}
%e 16: {{b,b}, {w,b}}
%e 17: {{b,b}, {w,w}}
%e 18: {{w,b}, {w,b}}
%e 19: {{w,b}, {w,w}}
%e 20: {{w,w}, {w,w}}
%e 8 partitions for the integer partition of 4 = 1 + 3:
%e 21: {{b}, {b,b,b}}
%e 22: {{w}, {b,b,b}}
%e 23: {{b}, {w,b,b}}
%e 24: {{w}, {w,b,b}}
%e 25: {{b}, {w,w,b}}
%e 26: {{w}, {w,w,b}}
%e 27: {{b}, {w,w,w}}
%e 28: {{w}, {w,w,w}}
%e 5 partitions for the integer partition of 4 = 4:
%e 29: {{b,b,b,b}}
%e 30: {{w,b,b,b}}
%e 31: {{w,w,b,b}}
%e 32: {{w,w,w,b}}
%e 33: {{w,w,w,w}}
%e Some see number partitions, others see set partitions, ...
%e (End)
%e It is obvious from the example of _Alois P. Heinz_ that a(n) enumerates multi-set partitions of a multi-set of n elements of two kinds. In the case that there is only one kind, this reduces to the usual case of numerical partitions. In the case that all the n elements are distinct, then this reduces to the case of set partitions. - _Michael Somos_, Mar 09 2015
%e There are a(3) = 14 plane partitions of 6 with trace 3; of 7 with trace 4; of 8 with trace 5; etc. See a formula above with the Stanley Exercise 7.99. - _Wolfdieter Lang_, Mar 09 2015
%e From _Daniel Forgues_, Mar 09 2015: (Start)
%e The a(3) = 14 = 4 + 6 + 4 partitions of 3 objects of 2 colors are:
%e 4 partitions for the integer partition of 3 = 1 + 1 + 1:
%e 01: {{b}, {b}, {b}}
%e 02: {{b}, {b}, {w}}
%e 03: {{b}, {w}, {w}}
%e 04: {{w}, {w}, {w}}
%e 6 partitions for the integer partition of 3 = 1 + 2:
%e 05: {{b}, {b,b}}
%e 06: {{w}, {b,b}}
%e 07: {{b}, {w,b}}
%e 08: {{w}, {w,b}}
%e 09: {{b}, {w,w}}
%e 10: {{w}, {w,w}}
%e 4 partitions for the integer partition of 3 = 3:
%e 11: {{b,b,b}}
%e 12: {{w,b,b}}
%e 13: {{w,w,b}}
%e 14: {{w,w,w}}
%e The a(2) = 6 = 3 + 3 partitions of 2 objects of 2 colors are:
%e 3 partitions for the integer partition of 2 = 1 + 1:
%e 01: {{b}, {b}}
%e 02: {{b}, {w}}
%e 03: {{w}, {w}}
%e 3 partitions for the integer partition of 2 = 2:
%e 04: {{b,b}}
%e 05: {{w,b}}
%e 06: {{w,w}}
%e The a(1) = 2 partitions of 1 object of 2 colors are:
%e 2 partitions for the integer partition of 1 = 1:
%e 01: {{b}}
%e 02: {{w}}
%e a(0) = 1: the empty partition, since empty sum is 0.
%e Triangle(sort of, since n_th row has p(n) = A000041 terms):
%e 1: 2
%e 2: 3, 3
%e 3: 4, 6, 4
%e 4: 5, 9, 6, 8, 5
%e 5: 6, ?, ?, ?, ?, ?, 6
%e 6: 7, ?, ?, ?, ?, ?, ?, ?, ?, ?, 7
%e Can we find a recurrence relation? (End)
%p mul( (1-x^i)^(-i-1),i=1..80); series(%,x,80); seriestolist(%);
%p # second Maple program:
%p with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> n+1): seq(a(n), n=0..40); # _Alois P. Heinz_, Sep 08 2008
%t max = 31; f[x_] = Product[ 1/(1-x^k)^(k+1), {k, 1, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* _Jean-François Alcover_, Nov 08 2011, after g.f. *)
%t etr[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, If[n==0, 1, Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]]; b]; a = etr[#+1&]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Nov 23 2015, after _Alois P. Heinz_ *)
%o (PARI) a(n)=polcoeff(prod(i=1,n,(1-x^i+x*O(x^n))^-(i+1)),n)
%Y Row sums of A054225.
%Y Column k=2 of A075196.
%Y Cf. A000219, A219555, A217093, A255050, A255052, A089353, A298988.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E Edited by _Christian G. Bower_, Sep 07 2002
%E New name from _Joerg Arndt_, Mar 09 2015
%E Restored 1995 name. - _N. J. A. Sloane_, Mar 09 2015
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