A236843 Triangle read by rows related to the Catalan transform of the Fibonacci numbers.

1, 1, 1, 2, 3, 1, 5, 9, 4, 1, 14, 28, 14, 6, 1, 42, 90, 48, 27, 7, 1, 132, 297, 165, 110, 35, 9, 1, 429, 1001, 572, 429, 154, 54, 10, 1, 1430, 3432, 2002, 1638, 637, 273, 65, 12, 1, 4862, 11934, 7072, 6188, 2548, 1260, 350, 90, 13, 1, 16796, 41990, 25194, 23256, 9996, 5508, 1700, 544, 104, 15, 1

Offset: 0

Author

Philippe Deléham, Feb 01 2014

Keywords

• nonn
• tabl

Comments

Row sums are A109262(n+1).

Examples

			Triangle begins:
    1;
    1,   1;
    2,   3,   1;
    5,   9,   4,   1;
   14,  28,  14,   6,  1;
   42,  90,  48,  27,  7, 1;
  132, 297, 165, 110, 35, 9, 1;
Production matrix is:
  1...1
  1...2...1
  0...1...1...1
  0...1...1...2...1
  0...0...0...1...1...1
  0...0...0...1...1...2...1
  0...0...0...0...0...1...1...1
  0...0...0...0...0...1...1...2...1
  0...0...0...0...0...0...0...1...1...1
  0...0...0...0...0...0...0...1...1...2...1
  0...0...0...0...0...0...0...0...0...1...1...1
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

Columns: A000108 (k=0), A000245 (k=1), A002057 (k=2), A003517 (k=3), A000588 (k=4), A001392 (k=5), A003519 (k=6), A090749 (k=7), A000590 (k=8).

Cf. A026674, A026726, A032766, A109262.

Cf. A000045, A039599, A106566.

Programs

Magma

F:=Factorial;
A236843:= func< n,k | (1/4)*(6*k+5-(-1)^k)*F(2*n-Floor(k/2))/(F(n-k)*F(n+Floor((k+1)/2)+1)) >;
[A236843(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 13 2022

Mathematica

T[n_, k_]:= (1/4)*(6*k+5-(-1)^k)*(2*n-Floor[k/2])!/((n-k)!*(n+Floor[(k+1)/2]+1)!);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 13 2022 *)

PARI

T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - (k\2))!/((n-k)!*(n + (k+1)\2 + 1)!) \\ Andrew Howroyd, Jan 04 2023

SageMath

F=factorial
def A236843(n,k): return (1/2)*(3*k+2+(k%2))*F(2*n-(k//2))/(F(n-k)*F(n+((k+1)//2)+1))
flatten([[A236843(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 13 2022

Formula

G.f. for the column k (with zeros omitted): C(x)^A032766(k+1) where C(x) is g.f. for Catalan numbers (A000108).

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

Sum_{k=0..n} T(n+k,2k) = A026726(n).

Sum_{k=0..n} T(n+1+k,2k+1) = A026674(n+1).

T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - floor(k/2))!/((n-k)!*(n + floor((k+1)/2) + 1)!). - G. C. Greubel, Jun 13 2022