login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A022629 Expansion of Product_{m>=1} (1 + m*q^m). 48
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 (list; graph; refs; listen; history; text; internal format)
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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

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

# second Maple program:

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

CROSSREFS

Cf. A000009, A006906, A022661, A022693, A265758, A266891, A285222, A304043.

Cf. A092484, A265840, A265841, A265842.

Sequence in context: A215513 A111328 A244962 * A032216 A285290 A032141

Adjacent sequences:  A022626 A022627 A022628 * A022630 A022631 A022632

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 19 05:13 EDT 2018. Contains 313844 sequences. (Running on oeis4.)