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 A117919 #26 Aug 16 2025 06:19:40
%S A117919 1,1,1,1,2,2,1,3,6,2,1,4,12,8,4,1,5,20,20,20,4,1,6,30,40,60,24,8,1,7,
%T A117919 42,70,140,84,56,8,1,8,56,112,280,224,224,64,16,1,9,72,168,504,504,
%U A117919 672,288,144,16,1,10,90,240,840,1008,1680,960,720,160,32,1,11,110,330,1320,1848,3696,2640,2640,880,352,32
%N A117919 Triangle read by rows: T(n, k) = 2^floor((k-1)/2)*binomial(n-1, k-1).
%C A117919 Row sums are the Pell sequence A000129.
%C A117919 Right border = inverse binomial transform of the Pell sequence: (A016116).
%C A117919 This triangle = difference terms of columns from an array generated from binomial transforms of (1,0,0,0...); (1,1,0,0,0...); (1,1,2,2...); (1,1,2,2,4,...); where (1, 1, 2, 2, 4, 4,...) = A016116, the inverse binomial transform of the Pell sequence A000129.
%C A117919 Triangle read by rows, iterates of X * [1,0,0,0,...] where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (1,2,1,2,1,2,...) in the subdiagonal, with the rest zeros. - _Gary W. Adamson_, May 10 2008
%C A117919 This sequence is jointly generated with A135837 as a triangular array of coefficients of polynomials u(n,x): initially, u(1,x) = v(1,x) = 1; for n>1, u(n,x) = u(n-1,x) + x*v(n-1) and v(n,x) = 2*x*u(n-1,x) + v(n-1,x). See the Mathematica section. - _Clark Kimberling_, Feb 26 2012
%H A117919 G. C. Greubel, <a href="/A117919/b117919.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A117919 From _G. C. Greubel_, Oct 23 2021: (Start)
%F A117919 T(n, k) = 2^floor((k-1)/2)*binomial(n-1, k-1).
%F A117919 Sum_{k=0..n} T(n, k) = A000129(n). (End)
%e A117919 First few rows of the generating array are:
%e A117919 1, 1, 1, 1, 1, ...
%e A117919 1, 2, 3, 4, 5, ...
%e A117919 1, 2, 5, 10, 17, ...
%e A117919 1, 2, 5, 12, 25, ...
%e A117919 1, 2, 5, 12, 29, ...
%e A117919 ...
%e A117919 Taking difference terms of the columns, we get this triangle. First few rows are:
%e A117919 1;
%e A117919 1, 1;
%e A117919 1, 2, 2;
%e A117919 1, 3, 6, 2;
%e A117919 1, 4, 12, 8, 4;
%e A117919 1, 5, 20, 20, 20, 4;
%e A117919 1, 6, 30, 40, 60, 24, 8;
%e A117919 1, 7, 42, 70, 140, 84, 56, 8;
%e A117919 ...
%t A117919 (* First program *)
%t A117919 u[1, x_]:= 1; v[1, x_]:= 1; z = 13;
%t A117919 u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
%t A117919 v[n_, x_]:= 2*x*u[n-1, x] + v[n-1, x];
%t A117919 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A117919 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A117919 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A117919 TableForm[cu]
%t A117919 Flatten[%] (* A117919 *)
%t A117919 Table[Expand[v[n, x]], {n, 1, z}]
%t A117919 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A117919 TableForm[cv]
%t A117919 Flatten[%] (* A135837 *)
%t A117919 (* Second program *)
%t A117919 Table[2^Floor[(k-1)/2]*Binomial[n-1, k-1], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Oct 23 2021 *)
%o A117919 (Magma) [2^Floor((k-1)/2)*Binomial(n-1, k-1): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Oct 23 2021
%o A117919 (Sage) flatten([[2^((k-1)//2)*binomial(n-1,k-1) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Oct 23 2021
%Y A117919 Cf. A000129, A016116, A135837.
%K A117919 nonn,tabl
%O A117919 1,5
%A A117919 _Gary W. Adamson_, Apr 02 2006
%E A117919 Name changed and more terms added by _G. C. Greubel_, Oct 23 2021