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 A128175 Binomial transform of A128174. 5
 1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 8, 8, 7, 4, 1, 16, 16, 15, 11, 5, 1, 32, 32, 31, 26, 16, 6, 1, 64, 64, 63, 57, 42, 22, 7, 1, 128, 128, 127, 120, 99, 64, 29, 8, 1, 256, 256, 255, 247, 219, 163, 93, 37, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Row sums = A045623: (1, 2, 5, 12, 28, 64, 144, ...). A128176 = A128174 * A007318. Riordan array ((1-x)/(1-2x),x/(1-x)). - Paul Barry, Oct 02 2010 Fusion of polynomial sequences p(n,x) = (x+1)^n and q(n,x) = x^n + x^(n-1) + ... + x + 1; see A193722 for the definition of fusion. - Clark Kimberling, Aug 04 2011 LINKS Table of n, a(n) for n=1..55. FORMULA A007318 * A128174 as infinite lower triangular matrices. Antidiagonals of an array in which the first row = (1, 1, 2, 4, 8, 16, ...); and (n+1)-th row = partial sums of n-th row. exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(4 + 4*x + 3*x^2/2! + x^3/3!) = 4 + 8*x + 15*x^2/2! + 26*x^3/3! + 42*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014 EXAMPLE First few rows of the triangle: 1; 1, 1; 2, 2, 1; 4, 4, 3, 1; 8, 8, 7, 4, 1; 16, 16, 15, 11, 5, 1; 32, 32, 31, 26, 16, 6, 1; 64, 64, 63, 57, 42, 22, 7, 1; ... From Paul Barry, Oct 02 2010: (Start) Production matrix is 1, 1; 1, 1, 1; 0, 0, 1, 1; 0, 0, 0, 1, 1; 0, 0, 0, 0, 1, 1; 0, 0, 0, 0, 0, 1, 1; 0, 0, 0, 0, 0, 0, 1, 1; 0, 0, 0, 0, 0, 0, 0, 1, 1; 0, 0, 0, 0, 0, 0, 0, 0, 1, 1; ... Matrix logarithm is 0; 1, 0; 1, 2, 0; 1, 1, 3, 0; 1, 1, 1, 4, 0; 1, 1, 1, 1, 5, 0; 1, 1, 1, 1, 1, 6, 0; 1, 1, 1, 1, 1, 1, 7, 0; 1, 1, 1, 1, 1, 1, 1, 8, 0; 1, 1, 1, 1, 1, 1, 1, 1, 9, 0; 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 0; ... (End) . First few rows of the array: 1, 1, 2, 4, 8, 16, ... 1, 2, 4, 8, 16, 32, ... 1, 3, 7, 15, 31, 63, ... 1, 4, 11, 26, 57, 120, ... 1, 5, 16, 42, 99, 219, ... ... (End) MAPLE A193820 := (n, k) -> `if`(k=0 or n=0, 1, A193820(n-1, k-1)+A193820(n-1, k)); A128175 := (n, k) -> A193820(n-1, n-k); seq(print(seq(A128175(n, k), k=0..n)), n=0..10); # Peter Luschny, Jan 22 2012 MATHEMATICA z = 10; a = 1; b = 1; p[n_, x_] := (a*x + b)^n q[0, x_] := 1 q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0; t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0; w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1 g[n_] := CoefficientList[w[n, x], {x}] TableForm[Table[Reverse[g[n]], {n, -1, z}]] Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193820 *) TableForm[Table[g[n], {n, -1, z}]] Flatten[Table[g[n], {n, -1, z}]] (* A128175 *) (* Clark Kimberling, Aug 06 2011 *) (* function dotTriangle[] is defined in A128176 *) a128175[r_] := dotTriangle[Binomial, If[EvenQ[#1 + #2], 1, 0]&, r] TableForm[a128174[7]] (* triangle *) Flatten[a128174[9]] (* data *) (* Hartmut F. W. Hoft, Mar 15 2017 *) CROSSREFS Cf. A045623, A128176, A007318. Sequence in context: A329854 A124725 A106522 * A104040 A338131 A332601 Adjacent sequences: A128172 A128173 A128174 * A128176 A128177 A128178 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, Feb 17 2007 STATUS approved

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Last modified December 3 10:02 EST 2023. Contains 367539 sequences. (Running on oeis4.)