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Search: id:A054446
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| A054446 |
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Triangle of partial row sums of triangle A037027(n,m), n >= m >= 0 (Fibonacci convolution triangle). |
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3
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| 1, 2, 1, 5, 3, 1, 12, 9, 4, 1, 29, 24, 14, 5, 1, 70, 62, 42, 20, 6, 1, 169, 156, 118, 67, 27, 7, 1, 408, 387, 316, 205, 100, 35, 8, 1, 985, 951, 821, 588, 332, 142, 44, 9, 1, 2378, 2323, 2088, 1614, 1020, 509, 194, 54, 10, 1, 5741, 5652, 5232, 4290, 2966, 1671, 747, 257
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is Pell(z)/(1-x*z*Fib(z)) with Pell(x)=1/(1-2*x-x^2) = g.f. for A000129(n+1) (Pell numbers without 0) and Fib(x)=1/(1-x-x^2) = g.f. for A000045(n+1) (Fibonacci numbers without 0).
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FORMULA
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a(n, m)=sum(A037027(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n<m, (sequence of partial row sums in columns m).
Column m recursion: a(n, m)= sum(a(j-1, m)*A037027(n-j, 0), j=m..n) + A037027(n, m), n >= m >= 0, a(n, m) := 0 if n<m.
G.f. for column m: Pell(x)*(x*Fib(x))^m, m >= 0, with Fib(x) = g.f. A000045(n+1) and Pell(x) = g.f. A000129(n+1).
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EXAMPLE
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{1}; {2,1}; {5,3,1}; {12,9,4,1};...
Fourth row polynomial (n=3): p(3,x)= 12+9*x+4*x^2+x^3
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CROSSREFS
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Cf. A037027, A000045, A000129. Row sums: A054447(n).
Sequence in context: A120095 A130197 A106513 this_sequence A047858 A125171 A048472
Adjacent sequences: A054443 A054444 A054445 this_sequence A054447 A054448 A054449
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 27 2000 and May 08 2000
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