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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, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 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

Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Part IV, 4. Mitteilungen zur Lehre vom Transfiniten, VIII Nr. 13, Springer, Berlin, 1932. See p. 434.

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 A329069

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 January 26 19:15 EST 2022. Contains 350599 sequences. (Running on oeis4.)