A025564 Triangular array, read by rows: pairwise sums of trinomial array A027907.

1, 1, 2, 1, 1, 3, 4, 3, 1, 1, 4, 8, 10, 8, 4, 1, 1, 5, 13, 22, 26, 22, 13, 5, 1, 1, 6, 19, 40, 61, 70, 61, 40, 19, 6, 1, 1, 7, 26, 65, 120, 171, 192, 171, 120, 65, 26, 7, 1, 1, 8, 34, 98, 211, 356, 483, 534, 483, 356, 211, 98, 34, 8, 1, 1, 9, 43, 140, 343, 665, 1050, 1373

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabf
• easy

Comments

Counting the top row as row 0, T(n,k) is the number of strings of nonnegative integers "s(1)s(2)s(3)...s(k)" such that s(1)+s(2)+s(3)+...+s(k) = n and the string does not contain the substring "00". E.g., T(3,5) = 8 because the valid strings are 02010, 01020, 11010, 10110, 10101, 01110, 01101 and 01011. T(4,3) = 13, counting 040, 311, 301, 130, 031, 103, 013, 220, 202, 022, 211, 121 and 112. - Jose Luis Arregui (arregui(AT)unizar.es), Dec 05 2007

Examples

			                  1
              1   2   1
          1   3   4   3   1
      1   4   8  10   8   4   1
  1   5  13  22  26  22  13   5   1
		

Crossrefs

Columns include A025565, A025566, A025567, A025568.

Cf. A025177.

Programs

Mathematica

T[, 0] = 1; T[1, 1] = 2; T[n, k_] /; 0 <= k <= 2n := T[n, k] = T[n-1, k-2] + T[n-1, k-1] + T[n-1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, 2n}] // Flatten (* Jean-François Alcover, Jul 22 2018 *)

PARI

{T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, if( n==1, [1,2,1][k+1], if( n==2, [1,3,4,3,1][k+1], T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)))))};

PARI

T(n,k)=polcoeff(Ser(polcoeff(Ser((1+y*z)/(1-z*(1+y+y^2)),y),k,y),z),n,z)

PARI

{T(n, k) = if( k<0 || k>2*n, 0, if(n==0, 1, polcoeff( (1 + x + x^2)^n, k)+ polcoeff( (1 + x + x^2)^(n-1), k-1)))};

Formula

T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 2, 1], [1, 3, 4, 3, 1].

G.f.: (1+yz)/[1-z(1+y+y^2)].

Extensions

Edited by Ralf Stephan, Jan 09 2005

Edited by Clark Kimberling, Jun 20 2012