

A191238


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


1



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
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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/(1xx^3x^5) we have a(n)=sum(k=1..n, T(n,k)) (see A060961).
4. For the e.g.f. exp(1xx^3x^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.
Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582
Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3


FORMULA

T(n,k)=sum(j=0..k, binomial(j,((nk2*j)/2))*binomial(k,j)*((1)^(nk)+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, ((nk2*j)/2))*binomial(k, j)*((1)^(nk)+1), j=0..k)/2 ;
end proc: # R. J. Mathar, Dec 16 2015


PROG

(Maxima)
T(n, k):=sum(binomial(j, ((nk2*j)/2))*binomial(k, j)*((1)^(nk)+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



