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A Jacobsthal-Pascal triangle.
4

%I #29 Feb 17 2023 10:07:21

%S 1,1,1,3,6,3,5,15,15,5,11,44,66,44,11,21,105,210,210,105,21,43,258,

%T 645,860,645,258,43,85,595,1785,2975,2975,1785,595,85,171,1368,4788,

%U 9576,11970,9576,4788,1368,171,341,3069,12276,28644,42966,42966,28644,12276,3069,341

%N A Jacobsthal-Pascal triangle.

%C 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

%H G. C. Greubel, <a href="/A124860/b124860.txt">Rows n = 0..50 of the triangle, flattened</a>

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

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

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

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

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

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

%F 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

%F 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

%F From _G. C. Greubel_, Feb 17 2023: (Start)

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

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

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

%e Triangle begins

%e 1;

%e 1, 1;

%e 3, 6, 3;

%e 5, 15, 15, 5;

%e 11, 44, 66, 44, 11;

%e 21, 105, 210, 210, 105, 21;

%e 43, 258, 645, 860, 645, 258, 43;

%p A := proc(n,k) ## n >= 0 and k = 0 .. n

%p ((-1)^n+2^(n+1))/3*binomial(n, k)

%p end proc: # _Yu-Sheng Chang_, Jan 15 2020

%t 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 *)

%o (Magma)

%o A124860:= func< n,k | Binomial(n,k)*(2^(n+1) - (-1)^(n+1))/3 >;

%o [A124860(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 17 2023

%o (SageMath)

%o def A124860(n,k): return binomial(n,k)*(2^(n+1) - (-1)^(n+1))/3

%o flatten([[A124860(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Feb 17 2023

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

%K easy,nonn,tabl

%O 0,4

%A _Paul Barry_, Nov 10 2006