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1, 1, 1, 2, 2, 1, 2, 4, 3, 1, 3, 6, 7, 4, 1, 3, 9, 13, 11, 5, 1, 4, 12, 22, 24, 16, 6, 1, 4, 16, 34, 46, 40, 22, 7, 1, 5, 20, 50, 80, 86, 62, 29, 8, 1, 5, 25, 70, 130, 166, 148, 91, 37, 9, 1, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1
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OFFSET
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1,4
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COMMENTS
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Row Sums = A000975: (1, 2, 5, 10, 21, 42, 85, 170, ...).
Riordan array ( 1/((1 - x^2)*(1 - x)), x/(1 - x) ).
Let B_n be the set of length n nonzero binary words ending in an even number (possibly 0) of 0's. Then T(n,k) is the number of words in B_n having k 1's. An example is given below. (End)
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LINKS
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FORMULA
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A128174 * A007318 (Pascal's triangle), as infinite lower triangular matrices.
Working with a row and column offset of 0 we have T(n,k) = Sum_{i = 0..floor(n/2)} binomial(n - 2*i,k).
O.g.f.: 1/( (1 - z^2)*(1 - z*(1 + x)) ) = Sum_{n >= 0} R(n,x)*z^n = 1 + (1 + x)*z + (2 + 2*x + x^2)*z^2 + ....
The row polynomials satisfy R(n+2,x) - R(n,x) = (1 + x)^(n+1). (End)
Using offset 0, the triangle has the Pascal Triangle recursion pattern:
T(n, 0) = 1 + floor(n/2) and T(n, n) = 1, for n >= 0;
T(n, k) = T(n-1, k-1) + T(n-1, k) for n > 0 and 0 < k < n. (End)
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EXAMPLE
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First few rows of the triangle are:
1;
1, 1;
2, 2, 1;
2, 4, 3, 1;
3, 6, 7, 4, 1;
3, 9, 13, 11, 5, 1;
4, 12, 22, 24, 16, 6, 1;
4, 16, 34, 46, 40, 22, 7, 1;
...
Row 4: [2,4,3,1].
k Binary words in B_4 with k 1's Number
- - - - - - - - - - - - - - - - - - - - - - - - - -
1 0001, 0100 2
2 0011, 0101, 1001, 1100 4
3 0111, 1011, 1101 3
4 1111 1
- - - - - - - - - - - - - - - - - - - - - - - - - -
The infinitesimal generator matrix begins
0
1 0
1 2 0
-1 1 3 0
1 -1 1 4 0
-1 1 -1 1 5 0
...
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MATHEMATICA
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(* Dot product of two lower triangular matrices *)
dotRow[r_, s_, n_] := Map[Sum[r[n, k] s[k, #], {k, #, n}]&, Range[0, n]]
dotTriangle[r_, s_, n_] := Map[dotRow[r, s, #]&, Range[0, n]]
(* The pure function in the first argument computes A128174 *)
a128176[r_] := dotTriangle[If[EvenQ[#1 + #2], 1, 0]&, Binomial, r]
TableForm[a128176[7]] (* triangle *)
T[n_, n_] := 1; T[n_, 0] := 1 + Floor[n/2]; T[n_, k_] := T[n, k] = T[n - 1, k - 1] + T[n - 1, k]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Sep 30 2017 *)
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PROG
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(PARI) for(n=0, 10, for(k=0, n, print1(sum(i=0, floor(n/2), binomial(n - 2*i, k)), ", "))) \\ G. C. Greubel, Sep 30 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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