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 A128176 A128174 * A007318. 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Row Sums = A000975: (1, 2, 5, 10, 21, 42, 85, 170, ...) A007318 * A128174 = A128175. From Peter Bala, Aug 14 2014: (Start) 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) LINKS G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened FORMULA A128174 * A007318 (Pascal's triangle), as infinite lower triangular matrices. From Peter Bala, Aug 14 2014: (Start) 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) From Hartmut F. W. Hoft, Mar 15 2017: (Start) 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) EXAMPLE 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;   ... From Peter Bala, Aug 14 2014: (Start) 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   ... Cf. A132440. (End) MATHEMATICA (* 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 *) Flatten[a128176[9]] (* data *) (* Hartmut F. W. Hoft, Mar 15 2017 *) 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 *) PROG (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 CROSSREFS Cf. A000975, A128175, A007318. Cf. A035317 (mirror). [Johannes W. Meijer, Jul 20 2011] Sequence in context: A300667 A129687 A274742 * A144963 A305632 A035374 Adjacent sequences:  A128173 A128174 A128175 * A128177 A128178 A128179 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, Feb 17 2007 STATUS approved

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Last modified July 19 09:07 EDT 2019. Contains 325155 sequences. (Running on oeis4.)