A191239 - OEIS

%I #12 Oct 30 2022 11:05:15

%S 1,1,1,2,2,1,0,5,3,1,0,4,9,4,1,0,4,13,14,5,1,0,0,18,28,20,6,1,0,0,12,

%T 49,50,27,7,1,0,0,8,56,105,80,35,8,1,0,0,0,56,161,195,119,44,9,1,0,0,

%U 0,32,210,366,329,168,54,10,1,0,0,0,16,200,581,721,518,228,65,11,1,0,0,0,0,160,732,1337,1288,774,300,77,12,1,0,0,0,0,80,780,2045,2716,2142,1110,385,90,13,1

%N Triangle T(n,k) = coefficient of x^n in expansion of (x+x^2+2*x^3)^k.

%C 1. Riordan Array (1,x+x^2+2*x^3) without first column.

%C 2. Riordan Array (1+x+2*x^3,x+x^2+2*x^3) numbering triangle (0,0).

%C 3. Bell Polynomial of second kind B(n,k){1,2,12,0,0,0,...,0}=n!/k!*T(n,k).

%C 4. For the g.f. 1/(1-x-x^2-2*x^3) we have a(n)=sum(k=1..n, T(n,k)) (see A077947)

%C For more formulas see preprints.

%H Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.

%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1104.5065">Derivation of Bell Polynomials of the Second Kind</a>, arXiv:1104.5065 [math.CO], 2013.

%F T(n,k) = Sum_{j=0..k} binomial(j,n-3*k+2*j)*2^(k-j)*binomial(k,j).

%e Triangle begins:

%e   1,

%e   1,1,

%e   2,2,1,

%e   0,5,3,1,

%e   0,4,9,4,1,

%e   0,4,13,14,5,1,

%e   0,0,18,28,20,6,1,

%o (Maxima)

%o T(n,k):=sum(binomial(j,n-3*k+2*j)*2^(k-j)*binomial(k,j),j,0,k);

%K nonn,tabl

%O 1,4

%A _Vladimir Kruchinin_, May 27 2011