A236830 Riordan array (1/(1-x*C(x)^3), x*C(x)), C(x) the g.f. of A000108.

1, 1, 1, 4, 2, 1, 16, 7, 3, 1, 65, 27, 11, 4, 1, 267, 108, 43, 16, 5, 1, 1105, 440, 173, 65, 22, 6, 1, 4597, 1812, 707, 267, 94, 29, 7, 1, 19196, 7514, 2917, 1105, 398, 131, 37, 8, 1, 80380, 31307, 12111, 4597, 1680, 575, 177, 46, 9, 1, 337284, 130883, 50503, 19196, 7085, 2488, 808, 233, 56, 10, 1

Offset

0,4

Comments

T(n+3,n) = A011826(n+5).

Links

G. C. Greubel, Rows n= 0..100 of triangle, flattened

Formula

Sum_{k=0..n} T(n,k) = A026726(n).

G.f.: 1/((x^2*C(x)^4-x*C(x))*y-x*C(x)^3+1), where C(x) the g.f. of A000108. - Vladimir Kruchinin, Apr 22 2015

From Peter Bala, Feb 18 2018: (Start)

T(n,k) = Sum_{i = 0..n-k} Fibonacci(2*i-1)*binomial(2*n-2-k-i,n-k-i).

The n-th row polynomial of row reverse triangle is the n-th degree Taylor polynomial of the rational function (1 - 3*x + 2*x^2)/(1 - 3*x + x^2) * 1/(1 - x)^n about 0. For example, for n = 4, (1 - 3*x + 2*x^2)/(1 - 3*x + x^2) * 1/(1 - x)^4 = 1 + 4*x + 11*x^2 + 27*x^3 + 65*x^4 + O(x^5), giving row 4 as (65, 27, 11, 4, 1). (End)

Example

Triangle begins:
      1;
      1,    1;
      4,    2,    1;
     16,    7,    3,    1;
     65,   27,   11,    4,   1;
    267,  108,   43,   16,   5,   1;
   1105,  440,  173,   65,  22,   6,  1;
   4597, 1812,  707,  267,  94,  29,  7, 1;
  19196, 7514, 2917, 1105, 398, 131, 37, 8, 1;
Production matrix is:
   1  1
   3  1   1
   6  1   1   1
  10  1   1   1   1
  15  1   1   1   1   1
  21  1   1   1   1   1   1
  28  1   1   1   1   1   1   1
  36  1   1   1   1   1   1   1   1
  45  1   1   1   1   1   1   1   1   1
  55  1   1   1   1   1   1   1   1   1   1
  66  1   1   1   1   1   1   1   1   1   1   1
  78  1   1   1   1   1   1   1   1   1   1   1   1
  91  1   1   1   1   1   1   1   1   1   1   1   1   1
  ...

Maple

A236830 := (n, k) -> add(combinat:-fibonacci(2*i-1)*binomial(2*n-2-k-i, n-k-i), i = 0..n-k): seq(seq(A236830(n, k), k = 0..n), n = 0..10); # Peter Bala, Feb 18 2018

Mathematica

(* The function RiordanArray is defined in A256893. *)
c[x_] := (1 - Sqrt[1 - 4 x])/(2 x);
RiordanArray[1/(1 - # c[#]^3)&, # c[#]&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)
Table[Sum[Binomial[2*n-k-j-2, n-k-j]*Fibonacci[2*j-1], {j, 0, n-k}], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 18 2019 *)

Prog

(PARI) T(n, k) = sum(j=0, n-k, binomial(2*n-k-j-2, n-k-j)*fibonacci(2*j -1));
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 18 2019
(Magma) [(&+[Binomial(2*n-k-j-2, n-k-j)*Fibonacci(2*j-1): j in [0..n-k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 18 2019
(Sage) [[sum( binomial(2*n-k-j-2, n-k-j)*fibonacci(2*j -1) for j in (0..n-k) ) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 18 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Sum([0..n-k], j-> Binomial(2*n-k-j-2, n-k-j)*Fibonacci(2*j-1) )))); # G. C. Greubel, Jul 18 2019

Crossrefs

Cf. (columns): A165201, A026726, A026671, A026674, A026672, A026675, A026673, A026843, A026842 or A026846 or A026849, A026844, A026841 or A026848.

Sequence in context: A143777 A365566 A326659 * A269736 A264535 A256039

Adjacent sequences: A236827 A236828 A236829 * A236831 A236832 A236833

Keyword

nonn,tabl

Author

Philippe Deléham, Feb 01 2014

Status

approved