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

0,4

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

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A351101 A134548 A151865 * A182412 A038138 A010704

Adjacent sequences: A124857 A124858 A124859 * A124861 A124862 A124863

Keyword

easy,nonn,tabl

Author

Paul Barry, Nov 10 2006

Status

approved