A118898 -id:A118898 - cp's OEIS frontend

A118897 Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0000 (n,k>=0).

1, 2, 4, 8, 15, 1, 29, 2, 1, 56, 5, 2, 1, 108, 12, 5, 2, 1, 208, 28, 12, 5, 2, 1, 401, 62, 29, 12, 5, 2, 1, 773, 136, 65, 30, 12, 5, 2, 1, 1490, 294, 145, 68, 31, 12, 5, 2, 1, 2872, 628, 319, 154, 71, 32, 12, 5, 2, 1, 5536, 1328, 694, 344, 163, 74, 33, 12, 5, 2, 1, 10671, 2787

Offset: 0

Emeric Deutsch, May 04 2006

Row n has n-2 terms (n>=3). Sum of entries in row n is 2^n (A000079). T(n,0) = A000078(n+4) (tetranacci numbers). T(n,1) = A118898(n). Sum(k*T(n,k),n>=0) = (n-3)*2^(n-4) (A001787).

			T(7,2) = 5 because we have 0000010, 0000011, 0100000, 1100000 and 1000001.
Triangle starts:
    1;
    2;
    4;
    8;
   15,  1;
   29,  2, 1;
   56,  5, 2, 1;
  108, 12, 5, 2, 1;
  ...

Alois P. Heinz, Rows n = 0..143, flattened

Cf. A000079, A000078, A118898, A001787, A076791, A118390.

G:=(1+(1-t)*(z+z^2+z^3))/(1-(1+t)*z-(1-t)*(z^2+z^3+z^4)): Gser:=simplify(series(G,z=0,17)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(coeff(Gser,z^n)) od: 1;2;4;8; for n from 4 to 14 do seq(coeff(P[n],t,j),j=0..n-3) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, 1,
      expand(b(n-1, min(3, t+1))*`if`(t>2, x, 1))+b(n-1, 0))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..14);  # Alois P. Heinz, Sep 17 2019

nn=15;a=x^3/(1-y x)+x+x^2;b=1/(1-x);f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[b (1+a)/(1-a x/(1-x)) ,{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Nov 18 2012 *)

G.f.: G(t,z) = [1+(1-t)(z+z^2+z^3)]/[1-(1+t)z-(1-t)(z^2+z^3+z^4)].

A202193 Triangle read by rows: T(n,m) = coefficient of x^n in expansion of (x/(1 - x - x^2 - x^3 - x^4))^m.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 8, 12, 9, 4, 1, 15, 28, 25, 14, 5, 1, 29, 62, 66, 44, 20, 6, 1, 56, 136, 165, 129, 70, 27, 7, 1, 108, 294, 401, 356, 225, 104, 35, 8, 1, 208, 628, 951, 944, 676, 363, 147, 44, 9, 1, 401, 1328, 2211, 2424, 1935, 1176, 553, 200, 54, 10, 1

Offset: 1

Vladimir Kruchinin, Dec 14 2011

From Philippe Deléham, Feb 16 2014: (Start)
As a Riordan array, this is (1/(1 - x - x^2 - x^3 - x^4), x/(1 - x - x^2 - x^3 - x^4)).
T(n,0) = A000078(n+3); T(n+1,1) = A118898(n+4).
Row sums are A103142(n).
Diagonal sums are A077926(n)*(-1)^n.
Tetranacci convolution triangle. (End)

			Triangle begins:
   1;
   1,  1;
   2,  2,  1;
   4,  5,  3,  1;
   8, 12,  9,  4,  1;
  15, 28, 25, 14,  5,  1;
  29, 62, 66, 44, 20,  6,  1;

Similar sequences : A037027 (Fibonacci convolution triangle), A104580 (tribonacci convolution triangle). - Philippe Deléham, Feb 16 2014

T(n,m):=if n=m then 1 else sum(sum((-1)^i*binomial(k,k-i)*binomial(n-m-4*i-1,k-1),i,0,(n-m-k)/4)*binomial(k+m-1,m-1),k,1,n-m);

T(n,m) = Sum_{k=1..n-m} (Sum_{i=0..floor((n-m-k)/4)} (-1)^i*binomial(k,k-i)*binomial(n-m-4*i-1,k-1))*binomial(k+m-1,m-1), n > m, T(n,n)=1.
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-3,k) + T(n-4,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Feb 16 2014
G.f. for column m: (x/(1 - x - x^2 - x^3 - x^4))^m. - Jason Yuen, Feb 17 2025

cp's OEIS Frontend

A118897 Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0000 (n,k>=0).

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