A339000 Triangle read by rows: T(n, k) = C(n, k)*Sum_{j=0..n} C(n, k-j)*C(n+j, j)/C(2*j, j).

1, 1, 2, 1, 7, 5, 1, 15, 32, 13, 1, 26, 111, 123, 34, 1, 40, 285, 603, 429, 89, 1, 57, 610, 2094, 2748, 1408, 233, 1, 77, 1155, 5845, 12170, 11196, 4437, 610, 1, 100, 2002, 14014, 42355, 60686, 42255, 13587, 1597, 1, 126, 3246, 30030, 124137, 254756, 271961, 150951, 40736, 4181

Offset: 0

Author

Vladimir Kruchinin, Nov 18 2020

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,  2;
  1,  7,   5;
  1, 15,  32,   13;
  1, 26, 111,  123,   34;
  1, 40, 285,  603,  429,   89;
  1, 57, 610, 2094, 2748, 1408, 233;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000045 (Fibonacci), A001519, A008459, A046748 (row sums).

Programs

Magma

b:=Binomial;
A339000:= func< n,k | b(n,k)*(&+[b(n,k-j)*b(n+j,j)/b(2*j,j): j in [0..n]]) >;
[A339000(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 31 2024

Mathematica

T[n_, k_]:= With[{B=Binomial}, B[n,k]*Sum[B[n,k-j]*B[n+j,j]/B[2*j,j], {j,0,n}]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 31 2024 *)

Maxima

T(n,m):=(binomial(n,m))*sum(((binomial(n,m-k))*(binomial(n+k,k)) )/(binomial(2*k,k)),k,0,n);

SageMath

b=binomial
def A339000(n,k): return b(n,k)*sum(b(n,k-j)*b(n+j,j)//b(2*j,j) for j in range(n+1))
flatten([[A339000(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 31 2024

Formula

G.f.: A008459(x,y)/(1-x*y*A008459(x,y)^2).

T(n,n) = Fibonacci(2*n+1).