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A191238
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Triangle T(n,k) = coefficient of x^n in expansion of (x+x^3+x^5)^k.
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1
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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
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1,8
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COMMENTS
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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.
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LINKS
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FORMULA
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T(n,k)} = Sum_{j=0..k binomial(j,((n-k-2*j)/2))*binomial(k,j)*((-1)^(n-k)+1))/2.
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EXAMPLE
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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
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MAPLE
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add(binomial(j, ((n-k-2*j)/2))*binomial(k, j)*((-1)^(n-k)+1), j=0..k)/2 ;
end proc:
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PROG
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(Maxima)
T(n, k):=sum(binomial(j, ((n-k-2*j)/2))*binomial(k, j)*((-1)^(n-k)+1), j, 0, k)/2;
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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