A128176 A128174 * A007318.

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

Offset: 1

Author

Gary W. Adamson, Feb 17 2007

Keywords

• nonn
• tabl

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)

Examples

			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)
		

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.

Crossrefs

Cf. A000975, A128175, A007318.

Cf. A035317 (mirror). [Johannes W. Meijer, Jul 20 2011]

Programs

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 *)

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

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)