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
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Feb 09 2014
EXTENSIONS
More terms from Philippe Deléham, Feb 09 2014
STATUS
approved