A191238 Triangle T(n,k) = coefficient of x^n in expansion of (x+x^3+x^5)^k.

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, 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, 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

Offset

1,8

Comments

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

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

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).

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).

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

For more formulas see preprints.

Links

Table of n, a(n) for n=1..105.

Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.

Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065 [math.CO], 2011.

Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

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

Example

Triangle begins:
  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,
  0,2,0,10,0,6,0,1,
  0,0,7,0,15,0,7,0,1,
  0,1,0,16,0,21,0,8,0,1

Maple

A191238 := proc(n, k)
    add(binomial(j, ((n-k-2*j)/2))*binomial(k, j)*((-1)^(n-k)+1), j=0..k)/2 ;
end proc:
seq(seq(A191238(n, m), m=1..n), n=1..10) ; # R. J. Mathar, Dec 16 2015

Prog

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

Crossrefs

Cf. A060961 (row sums).

Sequence in context: A234954 A321201 A180649 * A049310 A168561 A253190

Adjacent sequences: A191235 A191236 A191237 * A191239 A191240 A191241

Keyword

nonn,tabl

Author

Vladimir Kruchinin, May 27 2011

Status

approved