This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A124860 #29 Feb 17 2023 10:07:21
%S A124860 1,1,1,3,6,3,5,15,15,5,11,44,66,44,11,21,105,210,210,105,21,43,258,
%T A124860 645,860,645,258,43,85,595,1785,2975,2975,1785,595,85,171,1368,4788,
%U A124860 9576,11970,9576,4788,1368,171,341,3069,12276,28644,42966,42966,28644,12276,3069,341
%N A124860 A Jacobsthal-Pascal triangle.
%C A124860 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 A124860 G. C. Greubel, <a href="/A124860/b124860.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A124860 G.f.: 1/(1 - x*(1+y) - 2*x^2*(1+y)^2).
%F A124860 T(n, k) = J(n+1) * C(n, k), where J(n) = A001045(n).
%F A124860 T(n, 0) = T(n, n) = A001045(n+1).
%F A124860 T(2*n, n) = A124862(n).
%F A124860 Sum_{k=0..n} T(n, k) = A003683(n+1).
%F A124860 Sum_{k=0..floor(n/2)} T(n-k, k) = A124861(n).
%F A124860 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 A124860 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 A124860 From _G. C. Greubel_, Feb 17 2023: (Start)
%F A124860 T(n, n-k) = T(n, k).
%F A124860 T(n, 1) = A193449(n).
%F A124860 Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)
%e A124860 Triangle begins
%e A124860 1;
%e A124860 1, 1;
%e A124860 3, 6, 3;
%e A124860 5, 15, 15, 5;
%e A124860 11, 44, 66, 44, 11;
%e A124860 21, 105, 210, 210, 105, 21;
%e A124860 43, 258, 645, 860, 645, 258, 43;
%p A124860 A := proc(n,k) ## n >= 0 and k = 0 .. n
%p A124860 ((-1)^n+2^(n+1))/3*binomial(n, k)
%p A124860 end proc: # _Yu-Sheng Chang_, Jan 15 2020
%t A124860 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 A124860 (Magma)
%o A124860 A124860:= func< n,k | Binomial(n,k)*(2^(n+1) - (-1)^(n+1))/3 >;
%o A124860 [A124860(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 17 2023
%o A124860 (SageMath)
%o A124860 def A124860(n,k): return binomial(n,k)*(2^(n+1) - (-1)^(n+1))/3
%o A124860 flatten([[A124860(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Feb 17 2023
%Y A124860 Cf. A001045, A003683 (row sums), A016095, A084938, A124862 (diagonal sums), A193449.
%K A124860 easy,nonn,tabl
%O A124860 0,4
%A A124860 _Paul Barry_, Nov 10 2006