A191238 - OEIS

%I #31 Oct 20 2024 02:31:53

%S 1,0,1,1,0,1,0,2,0,1,1,0,3,0,1,0,3,0,4,0,1,0,0,6,0,5,0,1,0,2,0,10,0,6,

%T 0,1,0,0,7,0,15,0,7,0,1,0,1,0,16,0,21,0,8,0,1,0,0,6,0,30,0,28,0,9,0,1,

%U 0,0,0,19,0,50,0,36,0,10,0,1,0,0,3,0,45,0,77,0,45,0,11,0,1,0,0,0,16,0,90,0,112,0,55,0,12,0,1

%N Triangle T(n,k) = coefficient of x^n in expansion of (x+x^3+x^5)^k.

%C 1. Riordan Array (1,x+x^3+x^5) without first column.

%C 2. Riordan Array (1+x^2+x^4,x+x^3+x^5) numbering triangle (0,0).

%C 3. For the g.f. 1/(1-x-x^3-x^5) we have a(n)=sum(k=1..n, T(n,k)) (see A060961).

%C 4. For the e.g.f. exp(1-x-x^3-x^5) we have a(n)=n!*sum(k=1..n, T(n,k)/k!) (see A191237).

%C 5. Bell Polynomial of second kind B(n,k){1,0,6,0,120,0,0,...,0}=n!/k!*T(n,k).

%C For more formulas see preprints.

%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.

%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], 2011.

%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.

%F T(n,k) = Sum_{j=0..k} binomial(j,((n-k-2*j)/2))*binomial(k,j)*((-1)^(n-k)+1)/2.

%e Triangle begins:

%e   1,

%e   0,1,

%e   1,0,1,

%e   0,2,0,1,

%e   1,0,3,0,1,

%e   0,3,0,4,0,1,

%e   0,0,6,0,5,0,1,

%e   0,2,0,10,0,6,0,1,

%e   0,0,7,0,15,0,7,0,1,

%e   0,1,0,16,0,21,0,8,0,1

%p A191238 := proc(n,k)

%p     add(binomial(j,((n-k-2*j)/2))*binomial(k,j)*((-1)^(n-k)+1),j=0..k)/2 ;

%p end proc:

%p seq(seq(A191238(n,m),m=1..n),n=1..10) ;# _R. J. Mathar_, Dec 16 2015

%o (Maxima)

%o T(n,k):=sum(binomial(j,((n-k-2*j)/2))*binomial(k,j)*((-1)^(n-k)+1),j,0,k)/2;

%Y Cf. A060961 (row sums).

%K nonn,tabl

%O 1,8

%A _Vladimir Kruchinin_, May 27 2011