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A128175
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Binomial transform of A128174.
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5
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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; internal format)
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OFFSET
| 1,4
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COMMENTS
| Row sums = A045623: (1, 2, 5, 12, 28, 64, 144,...). A128176 = A128174 * A007318
Riordan array ((1-x)/(1-2x),x/(1-x)). [From Paul Barry (pbarry(AT)wit.ie), Oct 02 2010]
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. [From Clark Kimberling, Aug 4 2011]
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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.
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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;
...
Contribution from Paul Barry (pbarry(AT)wit.ie), 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,...
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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
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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 *)
(* From Clark Kimberling, Aug 6 2011 *)
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CROSSREFS
| Cf. A045623, A128176, A007318.
Sequence in context: A107356 A124725 A106522 * A104040 A169629 A110664
Adjacent sequences: A128172 A128173 A128174 * A128176 A128177 A128178
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KEYWORD
| nonn,tabl
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 17 2007
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