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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 12 05:52 EST 2019. Contains 329051 sequences. (Running on oeis4.)