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 A153861 #30 Jul 10 2022 13:23:47
%S A153861 1,1,1,2,3,1,3,6,4,1,4,10,10,5,1,5,15,20,15,6,1,6,21,35,35,21,7,1,7,
%T A153861 28,56,70,56,28,8,1,8,36,84,126,126,84,36,9,1,9,45,120,210,252,210,
%U A153861 120,45,10,1,10,55,165,330,462,462,330,165,55,11,1
%N A153861 Triangle read by rows, binomial transform of triangle A153860.
%C A153861 Row sums = A095121: (1, 2, 6, 14, 30, 62, 126,...).
%C A153861 Triangle T(n,k), 0<=k<=n, read by rows, given by [1,1,-1,1,0,0,0,0,0,0,0,...] DELTA [1,0,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jan 03 2009
%C A153861 A123110*A007318 as infinite lower triangular matrices. - _Philippe Deléham_, Jan 06 2009
%C A153861 A153861 is the fusion of polynomial sequences p(n,x)=x^n+x^(n-1)+...+x+1 and q(n,x)=(x+1)^n; see A193722 for the definition of fusion. - _Clark Kimberling_, Aug 06 2011
%H A153861 G. C. Greubel, <a href="/A153861/b153861.txt">Table of n, a(n) for the first 46 rows</a>
%F A153861 Triangle read by rows, A007318 * A153860. Remove left two columns of Pascal's triangle and append (1, 1, 2, 3, 4, 5,...).
%F A153861 As a recursive operation by way of example, row 3 = (3, 6, 4, 1) =
%F A153861 [1, 1, 1, 0] * (flipped Pascal's triangle matrix) = [1, 3, 3, 1]
%F A153861 [1, 2, 1, 0]
%F A153861 [1, 1, 0, 0]
%F A153861 [1, 0, 0, 0].
%F A153861 (Cf. analogous operation in A130405, but in A153861 the linear multiplier = [1,1,1,...,0].)
%F A153861 T(n,k) = 2*T(n-1,k)+T(n-1,k-1)-T(n-2,k)-T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0)=2, T(2,1)=3, T(n,k)=0 if k<0 or if k>n. - _Philippe Deléham_, Dec 15 2013
%F A153861 G.f.: (1-x+x^2+x^2*y)/((x-1)*(-1+x+x*y)). - _R. J. Mathar_, Aug 11 2015
%e A153861 First few rows of the triangle are:
%e A153861 1;
%e A153861 1, 1;
%e A153861 2, 3, 1;
%e A153861 3, 6, 4, 1;
%e A153861 4, 10, 10, 5, 1;
%e A153861 5, 15, 20, 15, 6, 1;
%e A153861 6, 21, 35, 35, 21, 7, 1;
%e A153861 7, 28, 56, 70, 56, 28, 8, 1;
%e A153861 8, 36, 84, 126, 126, 84, 36, 9, 1;
%e A153861 9, 45, 120, 210, 252, 210, 120, 45, 10, 1;
%e A153861 ...
%t A153861 z = 10; c = 1; d = 1;
%t A153861 p[0, x_] := 1
%t A153861 p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;
%t A153861 q[n_, x_] := (c*x + d)^n
%t A153861 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A153861 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A153861 g[n_] := CoefficientList[w[n, x], {x}]
%t A153861 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A153861 Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193815 *)
%t A153861 TableForm[Table[g[n], {n, -1, z}]]
%t A153861 Flatten[Table[g[n], {n, -1, z}]] (* A153861 *)
%t A153861 (* _Clark Kimberling_, Aug 06 2011 *)
%Y A153861 Cf. A153860, A095121, A130405.
%Y A153861 This is A137396 without the initial column and without signs.
%K A153861 nonn,tabl
%O A153861 0,4
%A A153861 _Gary W. Adamson_, Jan 03 2009