A236918 Triangle read by rows: Catalan triangle of the k-Fibonacci sequence.

1, 1, 1, 1, 2, 3, 1, 3, 7, 8, 1, 4, 12, 22, 24, 1, 5, 18, 43, 73, 75, 1, 6, 25, 72, 156, 246, 243, 1, 7, 33, 110, 283, 564, 844, 808, 1, 8, 42, 158, 465, 1092, 2046, 2936, 2742, 1, 9, 52, 217, 714, 1906, 4178, 7449, 10334, 9458, 1, 10, 63, 288, 1043, 3096, 7670, 15904, 27231, 36736, 33062

Offset

1,5

Comments

Reversal of the Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A000958. - Philippe Deléham, Feb 10 2014

Row sums are in A109262. - Philippe Deléham, Feb 10 2014

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832.

Formula

T(n, k) = coefficient of [x^k]( p(n, x) ), where p(n, x) = Sum_{j=0..n} (j/(2*n-j))*binomial(2*n-j, n-j)*x^(n-1)*Fibonacci(j, 1/x) with p(0, x) = 1 and Fibonacci(n, x) are the Fibonacci polynomials. - G. C. Greubel, Jun 14 2022

Example

Triangle begins:
  1;
  1,  1;
  1,  2,  3;
  1,  3,  7,   8;
  1,  4, 12,  22,   24;
  1,  5, 18,  43,   73,   75;
  1,  6, 25,  72,  156,  246,  243;
  1,  7, 33, 110,  283,  564,  844,   808;
  1,  8, 42, 158,  465, 1092, 2046,  2936,  2742;
  1,  9, 52, 217,  714, 1906, 4178,  7449, 10334,  9458;
  1, 10, 63, 288, 1043, 3096, 7670, 15904, 27231, 36736, 33062;
  ... - Extended by Philippe Deléham, Feb 10 2014

Mathematica

P[n_, x_]:= P[n, x]= If[n==0, 1, Sum[(j/(2*n-j))*Binomial[2*n-j, n-j]*Fibonacci[j, 1/x] *x^(n-1), {j, 0, n}]];
T[n_, k_]:= Coefficient[P[n, x], x, k];
Table[T[n, k], {n, 10}, {k, 0, n-1}]//Flatten (* G. C. Greubel, Jun 14 2022 *)

Prog

(SageMath)
def f(n, x): return sum( binomial(n-j-1, j)*x^(n-2*j-1) for j in (0..(n-1)//2) )
def p(n, x):
    if (n==0): return 1
    else: return sum( (j/(2*n-j))*binomial(2*n-j, n-j)*x^(n-1)*f(j, 1/x) for j in (0..n) )
def A236918(n, k): return ( p(n, x) ).series(x, n+1).list()[k]
flatten([[A236918(n, k) for k in (0..n-1)] for n in (1..12)]) # G. C. Greubel, Jun 14 2022

Crossrefs

Diagonals give A000958, A114495.

Cf. A109262 (row sums).

Sequence in context: A059397 A209567 A208338 * A152821 A071943 A357329

Adjacent sequences: A236915 A236916 A236917 * A236919 A236920 A236921

Keyword

nonn,tabl

Author

N. J. A. Sloane, Feb 09 2014

Extensions

More terms from Philippe Deléham, Feb 09 2014

Status

approved