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 A210381 #23 Feb 20 2022 06:05:02
%S A210381 1,0,2,0,1,3,0,1,3,4,0,1,4,6,5,0,1,5,10,10,6,0,1,6,15,20,15,7,0,1,7,
%T A210381 21,35,35,21,8,0,1,8,28,56,70,56,28,9,0,1,9,36,84,126,126,84,36,10,0,
%U A210381 1,10,45,120,210,252,210,120,45,11
%N A210381 Triangle by rows, derived from the beheaded Pascal's triangle, A074909.
%C A210381 Row sums of the triangle = 2^n.
%C A210381 Let the triangle = an infinite lower triangular matrix, M. Then M * The Bernoulli numbers, A027641/A027642 as a vector V = [1, -1, 0, 0, 0,...]. M * the Bernoulli sequence variant starting [1, 1/2, 1/6,...] = [1, 1, 1,...]. M * 2^n: [1, 2, 4, 8,...] = A027649. M * 3^n = A255463; while M * [1, 2, 3,...] = A047859, and M * A027649 = A027650.
%C A210381 Row sums of powers of the triangle generate the Poly-Bernoulli number sequences shown in the array of A099594. - _Gary W. Adamson_, Mar 21 2012
%C A210381 Triangle T(n,k) given by (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 25 2012
%D A210381 Konrad Knopp, Elements of the Theory of Functions, Dover, 1952,pp 117-118.
%F A210381 Partial differences of the beheaded Pascal's triangle A074909 starting from the top, by columns.
%F A210381 G.f.: (1-x)/(1-x-2*y*x+y*x^2+y^2*x^2). - _Philippe Deléham_, Mar 25 2012
%F A210381 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(2,1) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 2, T(2,2) = 3 and T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Mar 25 2012
%e A210381 {1},
%e A210381 {0, 2},
%e A210381 {0, 1, 3},
%e A210381 {0, 1, 3, 4},
%e A210381 {0, 1, 4, 6, 5},
%e A210381 {0, 1, 5, 10, 10, 6},
%e A210381 {0, 1, 6, 15, 20, 15, 7},
%e A210381 {0, 1, 7, 21, 35, 35, 21, 8},
%e A210381 {0, 1, 8, 28, 56, 70, 56, 28, 9},
%e A210381 {0, 1, 9, 36, 84, 126, 126, 84, 36, 10},
%e A210381 {0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 11}
%e A210381 ...
%t A210381 t2[n_, m_] = If[m - 1 <= n, Binomial[n, m - 1], 0];
%t A210381 O2 = Table[Table[If[n == m, t2[n, m] + 1, t2[n, m]], {m, 0, n}], {n, 0, 10}];
%t A210381 Flatten[O2]
%Y A210381 Cf. A074909, A255463, A026749, A047859, A027650
%Y A210381 Cf. A099594.
%K A210381 nonn,tabl
%O A210381 0,3
%A A210381 _Roger L. Bagula_ and _Gary W. Adamson_, Mar 20 2012