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

0,3

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

Links

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

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

Example

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

Mathematica

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

Prog

(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 */

Crossrefs

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

Sequence in context: A014968 A289115 A342528 * A006731 A222036 A000071

Adjacent sequences: A126345 A126346 A126347 * A126349 A126350 A126351

Keyword

nonn

Author

Paul D. Hanna, Dec 31 2006

Status

approved