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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 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 are: 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,... 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: A107356 A124725 A106522 * A104040 A182222 A225639 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 October 13 18:57 EDT 2019. Contains 327981 sequences. (Running on oeis4.)