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 A185675 Riordan array ( (1+x)/(1-x-x^2), x*A000108(x) ). 2
 1, 2, 1, 3, 3, 1, 5, 7, 4, 1, 8, 17, 12, 5, 1, 13, 43, 35, 18, 6, 1, 21, 116, 103, 60, 25, 7, 1, 34, 333, 312, 196, 93, 33, 8, 1, 55, 1010, 976, 643, 331, 135, 42, 9, 1, 89, 3202, 3147, 2137, 1161, 518, 187, 52, 10, 1, 144, 10504, 10415, 7213, 4066, 1929, 768, 250, 63, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices and Riordan arrays, arXiv:math/0702638 [math.CO], 2007. FORMULA R(n,k) = k*Sum_{i=0..n-k}(Sum_{j=1..i+1}binomial(j,i+1-j))*binomial(2*(n-i)-k-1,n-i-1)/(n-i), k>0. R(n,0) = A000045(n+2). EXAMPLE Triangle begins: 1; 2, 1; 3, 3, 1; 5, 7, 4, 1; 8, 17, 12, 5, 1; 13, 43, 35, 18, 6, 1; 21, 116, 103, 60, 25, 7, 1; 34, 333, 312, 196, 93, 33, 8, 1; Production matrix begins: 2, 1; -1, 1, 1; 2, 1, 1, 1; -3, 1, 1, 1, 1; 5, 1, 1, 1, 1, 1; -8, 1, 1, 1, 1, 1, 1; 13, 1, 1, 1, 1, 1, 1, 1; -21, 1, 1, 1, 1, 1, 1, 1, 1; ... Philippe Deléham, Sep 21 2014 MAPLE A185675 := proc(n, k) if n = k then 1; elif k = 0 then combinat[fibonacci](n+2) ; else k*add(1/(n-i)*add(binomial(j, i+1-j)*binomial(2*n-2*i-k-1, n-i-1), j=1..i+1), i=0..n-k) ; end if; end proc: seq(seq(A185675(n, k), k=0..n), n=0..15) ; # R. J. Mathar, Feb 10 2011 MATHEMATICA r[n_, k_] := k*Sum[Binomial[2*(n - i) - k - 1, n - i - 1]*Fibonacci[i + 2]/(n - i), {i, 0, n - k}]; r[n_, 0] := Fibonacci[n + 2]; r[n_, n_] := 1; Table[r[n, k], {n, 0, 3}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2013; modified by G. C. Greubel, Jul 10 2017 *) CROSSREFS Sequence in context: A144265 A209416 A122075 * A153341 A127119 A322265 Adjacent sequences: A185672 A185673 A185674 * A185676 A185677 A185678 KEYWORD nonn,tabl AUTHOR Vladimir Kruchinin, Feb 09 2011 STATUS approved

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Last modified November 28 22:03 EST 2023. Contains 367420 sequences. (Running on oeis4.)