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 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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,((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: # 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

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Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)