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A128175 Binomial transform of A128174.

%I A128175 #43 Aug 16 2025 06:20:05
%S A128175 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,
%T A128175 64,63,57,42,22,7,1,128,128,127,120,99,64,29,8,1,256,256,255,247,219,
%U A128175 163,93,37,9,1
%N A128175 Binomial transform of A128174.
%C A128175 Row sums = A045623: (1, 2, 5, 12, 28, 64, 144, ...).
%C A128175 A128176 = A128174 * A007318.
%C A128175 Riordan array ((1-x)/(1-2x),x/(1-x)). - _Paul Barry_, Oct 02 2010
%C A128175 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
%F A128175 A007318 * A128174 as infinite lower triangular matrices.
%F A128175 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.
%F A128175 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
%F A128175 T(n, k) = Sum_{i=0..floor((n-k)/2)} binomial(n-1, k-1+2*i). - _Werner Schulte_, Mar 05 2025
%F A128175 T(n, k) = binomial(n-1, k-1)*hypergeom([1, (k-n)/2, (1+k-n)/2], [(1+k)/2, k/2], 1). - _Stefano Spezia_, Mar 07 2025
%e A128175 First few rows of the triangle:
%e A128175    1;
%e A128175    1,  1;
%e A128175    2,  2,  1;
%e A128175    4,  4,  3,  1;
%e A128175    8,  8,  7,  4,  1;
%e A128175   16, 16, 15, 11,  5,  1;
%e A128175   32, 32, 31, 26, 16,  6,  1;
%e A128175   64, 64, 63, 57, 42, 22,  7,  1;
%e A128175   ...
%e A128175 From _Paul Barry_, Oct 02 2010: (Start)
%e A128175 Production matrix is
%e A128175   1, 1;
%e A128175   1, 1, 1;
%e A128175   0, 0, 1, 1;
%e A128175   0, 0, 0, 1, 1;
%e A128175   0, 0, 0, 0, 1, 1;
%e A128175   0, 0, 0, 0, 0, 1, 1;
%e A128175   0, 0, 0, 0, 0, 0, 1, 1;
%e A128175   0, 0, 0, 0, 0, 0, 0, 1, 1;
%e A128175   0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
%e A128175   ...
%e A128175 Matrix logarithm is
%e A128175   0;
%e A128175   1, 0;
%e A128175   1, 2, 0;
%e A128175   1, 1, 3, 0;
%e A128175   1, 1, 1, 4, 0;
%e A128175   1, 1, 1, 1, 5, 0;
%e A128175   1, 1, 1, 1, 1, 6, 0;
%e A128175   1, 1, 1, 1, 1, 1, 7, 0;
%e A128175   1, 1, 1, 1, 1, 1, 1, 8, 0;
%e A128175   1, 1, 1, 1, 1, 1, 1, 1, 9,  0;
%e A128175   1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 0;
%e A128175   ... (End)
%e A128175 .
%e A128175 First few rows of the array:
%e A128175   1, 1,  2,  4,  8,  16, ...
%e A128175   1, 2,  4,  8, 16,  32, ...
%e A128175   1, 3,  7, 15, 31,  63, ...
%e A128175   1, 4, 11, 26, 57, 120, ...
%e A128175   1, 5, 16, 42, 99, 219, ...
%e A128175   ...
%p A128175 A193820 := (n,k) -> `if`(k=0 or n=0, 1, A193820(n-1,k-1)+A193820(n-1,k));
%p A128175 A128175 := (n,k) -> A193820(n-1,n-k);
%p A128175 seq(print(seq(A128175(n,k),k=0..n)),n=0..10); # _Peter Luschny_, Jan 22 2012
%t A128175 z = 10; a = 1; b = 1;
%t A128175 p[n_, x_] := (a*x + b)^n
%t A128175 q[0, x_] := 1
%t A128175 q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;
%t A128175 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A128175 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A128175 g[n_] := CoefficientList[w[n, x], {x}]
%t A128175 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A128175 Flatten[Table[Reverse[g[n]], {n, -1, z}]]   (* A193820 *)
%t A128175 TableForm[Table[g[n], {n, -1, z}]]
%t A128175 Flatten[Table[g[n], {n, -1, z}]]  (* A128175 *)
%t A128175 (* _Clark Kimberling_, Aug 06 2011 *)
%t A128175 (* function dotTriangle[] is defined in A128176 *)
%t A128175 a128175[r_] := dotTriangle[Binomial, If[EvenQ[#1 + #2], 1, 0]&, r]
%t A128175 TableForm[a128174[7]] (* triangle *)
%t A128175 Flatten[a128174[9]] (* data *) (* _Hartmut F. W. Hoft_, Mar 15 2017 *)
%Y A128175 Cf. A045623, A128176, A007318.
%K A128175 nonn,tabl
%O A128175 1,4
%A A128175 _Gary W. Adamson_, Feb 17 2007