0,4

Euler transform of sequence [0,1,2,3,...]. - Michael Somos, Jul 02 2004

Number of partitions of n objects of 2 colors, where each part must contain at least one of each color. - Franklin T. Adams-Watters, Jan 23 2006

Number of partitions of n without 1s, one kind of 2s, two kinds of 3s, etc. - Joerg Arndt, Jul 31 2011

From Vaclav Kotesovec, Oct 17 2015: (Start)

In general, if v>=0 and g.f. = Product_{k>=1} 1/(1-x^(k+v))^k, then a(n) ~ d1(v) * n^(v^2/6 - 25/36) * exp(-Pi^4 * v^2 / (432*Zeta(3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(2/3) - v * Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * 2^(v^2/6 + 11/36) * Zeta(3)^(v^2/6 - 7/36)), where Zeta(3) = A002117.

d1(v) = exp(Integral_{x=0..infinity} (1/(x*exp((v-1)*x) * (exp(x)-1)^2) - (6*v^2-1) / (12*x*exp(x)) + v/x^2 - 1/x^3) dx).

d1(v) = (exp(Zeta'(-1) - v*Zeta'(0))) / Product_{j=0..v-1} j!, where Zeta'(0) = -A075700, Zeta'(-1) = A084448 and Product_{j=0..v-1} j! = A000178(v-1).

d1(v) = exp(1/12) * (2*Pi)^(v/2) / (A * G(v+1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.

(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 815

Vaclav Kotesovec, Graph - The asymptotic ratio

a(n) = 1/n*Sum_{k=1..n} (sigma[2](k)-sigma[1](k))*a(n-k).

G.f.: exp( Sum_{k>0} ( x^k / (1 - x^k) )^2 / k ).

G.f.: exp( sum(n>=0, (sigma[2](n)-sigma[1](n)) *x^n/n ) ). - Joerg Arndt, Jul 31 2011

a(n) ~ 2^(1/36) * Zeta(3)^(1/36) * 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) * n^(19/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 07 2015

1 + x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 12*x^6 + 18*x^7 + 33*x^8 + 52*x^9 + ...

From Gus Wiseman, Jan 22 2019: (Start)

The partitions described in Franklin T. Adams-Watters's comment are (n = 2 through 6):

  {{12}}  {{112}}  {{1112}}    {{11112}}    {{111112}}

          {{122}}  {{1122}}    {{11122}}    {{111122}}

                   {{1222}}    {{11222}}    {{111222}}

                   {{12}{12}}  {{12222}}    {{112222}}

                               {{12}{112}}  {{122222}}

                               {{12}{122}}  {{112}{112}}

                                            {{112}{122}}

                                            {{12}{1112}}

                                            {{12}{1122}}

                                            {{12}{1222}}

                                            {{122}{122}}

                                            {{12}{12}{12}}

(End)

spec := [S, {B=Sequence(Z, 1 <= card), C=Prod(B, B), S= Set(C)}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

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..50); # Vaclav Kotesovec, Mar 04 2015 after Alois P. Heinz

Clear[a]; a[n_]:= a[n] = 1/n*Sum[(DivisorSigma[2, k]-DivisorSigma[1, k])*a[n-k], {k, 1, n}]; a[0]=1; Table[a[n], {n, 0, 100}] (* Vaclav Kotesovec, Mar 04 2015 *)

nmax = 40; CoefficientList[Series[Product[1/(1-x^(k+1))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 16 2015 *)

(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, (1 - x^k + x*O(x^n))^(k-1)), n))}

Cf. A005380, A000203, A001157, A052812.

Cf. A000219 (v=0), A052847 (v=1), A263358 (v=2), A263359 (v=3), A263360 (v=4), A263361 (v=5), A263362 (v=6), A263363 (v=7), A263364 (v=8).

Cf. A054974, A321407, A321760, A323654, A323655, A323656.

Sequence in context: A167777 A259941 A007436 * A331933 A052823 A063516

Adjacent sequences:  A052844 A052845 A052846 * A052848 A052849 A052850

easy,nonn

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Edited by Vladeta Jovovic, Sep 10 2002

approved