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

0,3

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

Links

Table of n, a(n) for n=0..71.

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)].

Example

                  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

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 *)

Prog

(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)))};

Crossrefs

Columns include A025565, A025566, A025567, A025568.

Cf. A025177.

Sequence in context: A159933 A305431 A128314 * A052265 A306565 A055068

Adjacent sequences: A025561 A025562 A025563 * A025565 A025566 A025567

Keyword

nonn,tabf,easy

Author

Clark Kimberling

Extensions

Edited by Ralf Stephan, Jan 09 2005

Edited by Clark Kimberling, Jun 20 2012

Status

approved