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URL: https://oeis.org/A022629

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A022629
Expansion of Product_{m>=1} (1 + m*q^m).
77
1, 1, 2, 5, 7, 15, 25, 43, 64, 120, 186, 288, 463, 695, 1105, 1728, 2525, 3741, 5775, 8244, 12447, 18302, 26424, 37827, 54729, 78330, 111184, 159538, 225624, 315415, 444708, 618666, 858165, 1199701, 1646076, 2288961, 3150951, 4303995, 5870539, 8032571, 10881794, 14749051, 19992626
OFFSET
0,3
COMMENTS
Sum of products of terms in all partitions of n into distinct parts. - Vladeta Jovovic, Jan 19 2002
Number of partitions of n into distinct parts, when there are j sorts of part j. a(4) = 7: 4, 4', 4'', 4''', 31, 3'1, 3''1. - Alois P. Heinz, Aug 24 2015
FORMULA
Conjecture: log(a(n)) ~ sqrt(n/2) * (log(2*n) - 2). - Vaclav Kotesovec, May 08 2018
EXAMPLE
The partitions of 6 into distinct parts are 6, 1+5, 2+4, 1+2+3, the corresponding products are 6,5,8,6 and their sum is a(6) = 25.
MAPLE
b:= proc(n, i) option remember; local f, g;
if n=0 then [1, 1] elif i<1 then [0, 0]
else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i-1));
[f[1]+g[1], f[2]+g[2]*i]
fi
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..60); # Alois P. Heinz, Nov 02 2012
# Alternative:
b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, i*b(n-i, i-1))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..60); # Alois P. Heinz, Aug 24 2015
MATHEMATICA
nn=20; CoefficientList[Series[Product[1+i x^i, {i, 1, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Nov 02 2012 *)
nmax = 50; CoefficientList[Series[Exp[Sum[(-1)^(j+1)*PolyLog[-j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2015 *)
(* More efficient program: 10000 terms, 4 minutes, 100000 terms, 6 hours *) nmax = 40; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j+1]] += k*poly[[j-k+1]], {j, nmax, k, -1}]; , {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 06 2016 *)
PROG
(PARI) N=66; q='q+O('q^N); Vec(prod(n=1, N, (1+n*q^n) )) \\ Joerg Arndt, Oct 06 2012
(Magma) Coefficients(&*[(1+m*x^m):m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 16 2018
KEYWORD
nonn
STATUS
approved