%I #26 Nov 07 2024 08:30:34

%S 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,

%T 1,808,844,564,283,110,33,7,1,2742,2936,2046,1092,465,158,42,8,1,9458,

%U 10334,7449,4178,1906,714,217,52,9,1,33062,36736,27231,15904,7670,3096,1043,288,63,10,1

%N Convolution triangle of A000958(n+1).

%C Riordan array (f(x)/x, f(x)) where f(x) is the g.f. of A000958.

%C Reversal of A236918.

%C Row sums are A109262(n+1).

%C Diagonal sums are A033297(n+2).

%H G. C. Greubel, <a href="/A237596/b237596.txt">Rows n = 0..50 of the triangle, flattened</a>

%H Sergio Falcon, <a href="http://dx.doi.org/10.4134/CKMS.2013.28.4.827">Catalan transform of the K-Fibonacci sequence</a>, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832.

%F G.f. for the column k-1: ((1-sqrt(1-4*x))^k/(1+sqrt(1-4*x) + 2*x)^k)/x.

%F Sum_{k=0..n} T(n,k) = A109262(n+1).

%F From _G. C. Greubel_, Jun 14 2022: (Start)

%F 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.

%F T(n, k) = A236918(n, n-k). (End)

%e Triangle begins:

%e 1;

%e 1, 1;

%e 3, 2, 1;

%e 8, 7, 3, 1;

%e 24, 22, 12, 4, 1;

%e 75, 73, 43, 18, 5, 1;

%e 243, 246, 156, 72, 25, 6, 1;

%e 808, 844, 564, 283, 110, 33, 7, 1;

%e ...

%p # Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.

%p PMatrix(10, n -> A000958(n)); # _Peter Luschny_, Oct 19 2022

%t 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 T[n_, k_] := Coefficient[P[n+1, x], x, k];

%t Table[T[n, k], {n,0,13}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 14 2022 *)

%o (SageMath)

%o def f(n,x): return sum( binomial(n-j-1, j)*x^(n-2*j-1) for j in (0..(n-1)//2) )

%o def p(n,x):

%o if (n==0): return 1

%o else: return sum( (j/(2*n-j))*binomial(2*n-j, n-j)*f(j, x) for j in (0..n) )

%o def A237596(n,k): return ( p(n+1,x) ).series(x, n+1).list()[k]

%o flatten([[A237596(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 14 2022

%Y Columns give A000958, A114495.

%Y Cf. A033297 (diagonal sums), A109262 (row sums), A236918 (row reversal).

%K nonn,tabl,changed

%O 0,4

%A _Philippe Deléham_, Feb 09 2014