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A237596
Convolution triangle of A000958(n+1).
1
1, 1, 1, 3, 2, 1, 8, 7, 3, 1, 24, 22, 12, 4, 1, 75, 73, 43, 18, 5, 1, 243, 246, 156, 72, 25, 6, 1, 808, 844, 564, 283, 110, 33, 7, 1, 2742, 2936, 2046, 1092, 465, 158, 42, 8, 1, 9458, 10334, 7449, 4178, 1906, 714, 217, 52, 9, 1, 33062, 36736, 27231, 15904, 7670, 3096, 1043, 288, 63, 10, 1
OFFSET
0,4
COMMENTS
Riordan array (f(x)/x, f(x)) where f(x) is the g.f. of A000958.
Reversal of A236918.
Row sums are A109262(n+1).
Diagonal sums are A033297(n+2).
LINKS
Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832.
FORMULA
G.f.: for the column k-1: ((1-sqrt((1-4*x))^k/(1+sqrt(1-4*x) + 2*x)^k)/x.
Sum_{k=0..n} T(n,k) = A109262(n+1).
From G. C. Greubel, Jun 14 2022: (Start)
T(n, k) = coefficient of [x^k]( p(n+1, x) ), where p(n, x) = Sum_{j=0..n} (j/(2*n-j))*binomial(2*n-j, n-j)*Fibonacci(j, x) with p(0, x) = 1 and Fibonacci(n, x) are the Fibonacci polynomials.
T(n, k) = A236918(n, n-k). (End)
EXAMPLE
Triangle begins:
1;
1, 1;
3, 2, 1;
8, 7, 3, 1;
24, 22, 12, 4, 1;
75, 73, 43, 18, 5, 1;
243, 246, 156, 72, 25, 6, 1;
808, 844, 564, 283, 110, 33, 7, 1;
...
MAPLE
# Uses function PMatrix from A357368. Adds column 1, 0, 0, 0, ... to the left.
PMatrix(10, n -> A000958(n)); # Peter Luschny, Oct 19 2022
MATHEMATICA
P[n_, x_]:= P[n, x]= If[n==0, 1, Sum[(j/(2*n-j))*Binomial[2*n-j, n-j]*Fibonacci[j, x], {j, 0, n}]];
T[n_, k_] := Coefficient[P[n+1, x], x, k];
Table[T[n, k], {n, 0, 13}, {k, 0, n}]//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)*f(j, x) for j in (0..n) )
def A237596(n, k): return ( p(n+1, x) ).series(x, n+1).list()[k]
flatten([[A237596(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 14 2022
CROSSREFS
Columns give A000958, A114495.
Cf. A033297 (diagonal sums), A109262 (row sums), A236918 (row reversal).
Sequence in context: A110439 A327917 A065602 * A292898 A198498 A016648
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Feb 09 2014
STATUS
approved