A124860 A Jacobsthal-Pascal triangle.

1, 1, 1, 3, 6, 3, 5, 15, 15, 5, 11, 44, 66, 44, 11, 21, 105, 210, 210, 105, 21, 43, 258, 645, 860, 645, 258, 43, 85, 595, 1785, 2975, 2975, 1785, 595, 85, 171, 1368, 4788, 9576, 11970, 9576, 4788, 1368, 171, 341, 3069, 12276, 28644, 42966, 42966, 28644, 12276, 3069, 341

Offset: 0

Author

Paul Barry, Nov 10 2006

Keywords

• easy
• nonn
• tabl

Comments

Triangle T(n, k) read by rows given by [1, 2, -2, 0, 0, 0, ...] DELTA [1, 2, -2, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 11 2006

Examples

			Triangle begins
   1;
   1,   1;
   3,   6,   3;
   5,  15,  15,   5;
  11,  44,  66,  44,  11;
  21, 105, 210, 210, 105,  21;
  43, 258, 645, 860, 645, 258, 43;
		

Links

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

Crossrefs

Cf. A001045, A003683 (row sums), A016095, A084938, A124862 (diagonal sums), A193449.

Programs

Magma

A124860:= func< n,k | Binomial(n,k)*(2^(n+1) - (-1)^(n+1))/3 >;
[A124860(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2023

Maple

A := proc(n,k) ## n >= 0 and k = 0 .. n
    ((-1)^n+2^(n+1))/3*binomial(n, k)
end proc: # Yu-Sheng Chang, Jan 15 2020

Mathematica

jacobPascal[n_, k_]:= Binomial[n, k]*(2^(n+1) -(-1)^(n+1))/3; ColumnForm[Table[jacobPascal[n, k], {n,0,12}, {k,0,n}], Center] (* Alonso del Arte, Jan 16 2020 *)

SageMath

def A124860(n,k): return binomial(n,k)*(2^(n+1) - (-1)^(n+1))/3
flatten([[A124860(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 17 2023

Formula

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

T(n, k) = J(n+1) * C(n, k), where J(n) = A001045(n).

T(n, 0) = T(n, n) = A001045(n+1).

T(2*n, n) = A124862(n).

Sum_{k=0..n} T(n, k) = A003683(n+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = A124861(n).

T(n, k) = T(n-1, k-1) + T(n-1, k) + 2*T(n-2, k-2) + 4*T(n-2, k-1) + 2*T(n-2, k), T(0, 0) = 1, T(n, k) = 0 if k < 0 or if k > n . - Philippe Deléham, Nov 11 2006

G.f.: T(0)/2, where T(k) = 1 + 1/(1 - (2*k + 1 + 2*x*(1+y))*x*(1 + y)/((2*k + 2 + 2*x*(1+y))*x*(1+y) + 1/T(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013

From G. C. Greubel, Feb 17 2023: (Start)

T(n, n-k) = T(n, k).

T(n, 1) = A193449(n).

Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)