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 A117919 Triangle, row sums = the Pell sequence. 3
 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, 42, 70, 140, 84, 56, 8, 1, 8, 56, 112, 280, 224, 224, 64, 16 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row terms sums of the triangle = the Pell sequence A000129: (1, 2, 5, 12, 29...). Right border of the triangle = inverse binomial transform of the Pell sequence: (A016116): (1, 1, 2, 2, 4, 4,...). A117919 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)x and v(n,x)=2x*u(n-1,x)+v(n-1,x).  See the Mathematica section. - Clark Kimberling, Feb 26 2012 LINKS FORMULA The 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.. 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 EXAMPLE First few rows of the generating array are: 1, 1, 1, 1, 1,... 1, 2, 3, 4, 5,... 1, 2, 5, 10, 17,... 1, 2, 5, 12, 25,... 1, 2, 5, 12, 29,... ... Taking difference terms of the columns, we get the triangle A117919. First few rows are: 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, 42, 70, 140, 84, 56, 8; ... MATHEMATICA u[1, x_] := 1; v[1, x_] := 1; z = 13; u[n_, x_] := u[n - 1, x] + x*v[n - 1, x]; v[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x]; Table[Expand[u[n, x]], {n, 1, z/2}] Table[Expand[v[n, x]], {n, 1, z/2}] cu = Table[CoefficientList[u[n, x], x], {n, 1, z}]; TableForm[cu] Flatten[%]    (* A117919 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]    (* A135837 *) CROSSREFS Cf. A000129, A016116, A135837. Sequence in context: A309495 A080955 A125231 * A309106 A160014 A068956 Adjacent sequences:  A117916 A117917 A117918 * A117920 A117921 A117922 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, Apr 02 2006 STATUS approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)