A126348 Limit of reversed rows of triangle A126347, in which row sums equal Bell numbers (A000110).

1, 1, 2, 4, 7, 12, 20, 33, 53, 84, 131, 202, 308, 465, 695, 1030, 1514, 2209, 3201, 4609, 6596, 9386, 13284, 18705, 26211, 36561, 50776, 70226, 96742, 132765, 181540, 247369, 335940, 454756, 613689, 825698, 1107755, 1482038, 1977465, 2631664

Offset: 0

Author

Paul D. Hanna, Dec 31 2006

Keywords

• nonn

Comments

In triangle A126347, row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).

Row sums of A253830. a(n) equals the number of colored compositions of n, as defined in A253830, whose associated color partition has distinct parts. An example is given below. - Peter Bala, Jan 20 2015

Examples

			a(5) = 12: The colored compositions (defined in A253830) of 5 whose color partitions have distinct parts are
5(c1), 5(c2), 5(c3), 5(c4), 5(c5),
1(c1) + 4(c2), 1(c1) + 4(c3), 1(c1) + 4(c4),
3(c1) + 2(c2),
2(c1) + 3(c2), 2(c1) + 3(c3), 2(c2) + 3(c3). - _Peter Bala_, Jan 20 2015
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..500 from Seiichi Manyama)

Crossrefs

Cf. A126347, A126349; factorial variant: A126471. A253830, A307599, A307601, A307602.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x + x^k)/(1 - x), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2019 *)

PARI

{B(n,q)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*B(k,q)*q^k))}
{a(n)=Vec(B(n+1,'q)+O('q^(n*(n-1)/2+1)))[n*(n-1)/2+1]}

PARI

{a(n) = local(t); if( n<0, 0, t = 1; polcoeff( sum(k=1, (sqrtint(8*n + 1) - 1)\2, t = t * x^k / (1 - x) / (1 - x^k) + x * O(x^n), 1), n))} /* Michael Somos, Aug 17 2008 */

Formula

1 + Sum_{k>0} x^(k * (k + 1) / 2) / ((1 - x)^k * (1 - x) * (1 - x^2) ... (1 - x^k)). - Michael Somos, Aug 17 2008

G.f.: Product_{k>0} (1+x^k/(1-x)). - Vladeta Jovovic, Oct 05 2008

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (-1)^(d+1)/(d*(1 - x)^d)). - Ilya Gutkovskiy, Apr 19 2019