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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 October 21 07:06 EDT 2018. Contains 316405 sequences. (Running on oeis4.)