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A039692
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Jabotinsky-triangle related to A039647.
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37
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1, 3, 1, 8, 9, 1, 42, 59, 18, 1, 264, 450, 215, 30, 1, 2160, 4114, 2475, 565, 45, 1, 20880, 43512, 30814, 9345, 1225, 63, 1, 236880, 528492, 420756, 154609, 27720, 2338, 84, 1, 3064320, 7235568, 6316316, 2673972, 594489, 69552, 4074, 108, 1
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OFFSET
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1,2
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COMMENTS
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Triangle gives the nonvanishing entries of the Jabotinsky matrix for F(z)= A(z)/z = 1/(1-z-z^2) where A(z) is the g.f. of the Fibonacci numbers A000045. (Notation of F(z) as in Knuth's paper).
E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x)=1, are exponential convolution polynomials: E(n,x+y) = sum(binomial(n,k)*E(k,x)*E(n-k,y),k=0..n) (cf. Knuth's paper with E(n,x)= n!*F(n,x)).
E.g.f. for E(n,x): (1-z-z^2)^(-x).
Explicit a(n,m) formula: see Knuth's paper for f(n,m) formula with f(k)= A039647(n).
E.g.f. for the m-th column sequence: ((-log(1-z-z^2))^m)/m!.
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REFERENCES
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D. E. Knuth, Convolution polynomials, The Mathematica J., 2.1 (1992) 67-78.
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LINKS
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Table of n, a(n) for n=1..45.
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FORMULA
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a(n, 1)= A039647(n)=(n-1)!*L(n), L(n) := A000032(n) (Lucas); a(n, m) = sum(binomial(n-1, j-1)*A039647(j)*a(n-j, m-1), j=1..n-m+1), n >= m >= 2.
Conjectured row sums: sum_{m=1..n} a(n,m) = A005442(n). [From R. J. Mathar, Jun 01 2009]
T(n,m) = n!*sum(k=m..n, (stirling1(k,m)*binomial(k,n-k))*(-1)^(k+m)/k!). [Vladimir Kruchinin, Mar 26 2013]
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EXAMPLE
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1;
3, 1;
8, 9, 1;
42, 59, 18, 1;
264, 450, 215, 30, 1;
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MAPLE
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A000032 := proc(n) option remember; coeftayl( (2-x)/(1-x-x^2), x=0, n) ; end: A039647 := proc(n) (n-1)!*A000032(n) ; end: A039692 := proc(n, m) option remember ; if m = 1 then A039647(n) ; else add( binomial(n-1, j-1)*A039647(j)*procname(n-j, m-1), j=1..n-m+1) ; fi; end: [From R. J. Mathar, Jun 01 2009]
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PROG
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(Maxima) T(n, m) := n!*sum((stirling1(k, m)*binomial(k, n-k))*(-1)^(k+m)/k!, k, m, n); [Vladimir Kruchinin, Mar 26 2013]
(PARI)
T(n, m) = n!*sum(k=m, n, (stirling(k, m, 1)*binomial(k, n-k))*(-1)^(k+m)/k!);
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print());
/* Joerg Arndt, Mar 27 2013 */
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CROSSREFS
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Cf. A039647, A000032, A000045. Another version of this triangle is in A194938.
Sequence in context: A176103 A076238 A008298 * A071815 A178301 A120236
Adjacent sequences: A039689 A039690 A039691 * A039693 A039694 A039695
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KEYWORD
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nonn,tabl
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AUTHOR
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Wolfdieter Lang
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STATUS
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approved
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