A110681 A convolution triangle of numbers based on A071356.

1, 2, 1, 6, 4, 1, 20, 16, 6, 1, 72, 64, 30, 8, 1, 272, 260, 140, 48, 10, 1, 1064, 1072, 636, 256, 70, 12, 1, 4272, 4480, 2856, 1288, 420, 96, 14, 1, 17504, 18944, 12768, 6272, 2320, 640, 126, 16, 1, 72896, 80928, 57024, 29952, 12192, 3852, 924, 160, 18, 1

Offset

0,2

Links

Michael De Vlieger, Table of n, a(n) for n = 0..11324 (rows 0 <= n <= 150).

G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), p. 32-33.

G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)

Formula

T(0, 0) = 1; T(n, k) = 0 if k<0 or if k>n; T(n, k) = T(n-1, k-1) + 2*T(n-1, k) + 2*T(n-1, k+1).

Sum_{k, k>=0} T(m, k)*T(n, k)*2^k = T(m+n, 0) = A071356(m+n).

Sum_{k, k>=0} T(n, k)*(2^(k+1) - 1) = 5^n.

Sum_{k, k>=0} (-1)^(n-k)*T(n, k)*(2^(k+1) - 1) = 1.

Example

Triangle starts:
   1;
   2,  1;
   6,  4,  1;
  20, 16,  6,  1;
  72, 64, 30,  8,  1;
  ...

Mathematica

T[n_, k_] := T[n, k] = Which[n == k == 0, 1, n == 0, 0, k == 0, 0, k > n, 0, True, T[n - 1, k - 1] + 2 T[n - 1, k] + 2 T[n - 1, k + 1]]; Table[T[n, k], {n, 0, 10}, {k, n}] // Flatten (* Michael De Vlieger, Nov 05 2017 *)

Crossrefs

Cf. A071356 (1st column), A071357 (2nd column).

Cf. A052705 (diagonal sums), A001003 (row sums).

Sequence in context: A125693 A094527 A054335 * A117852 A080245 A080247

Adjacent sequences: A110678 A110679 A110680 * A110682 A110683 A110684

Keyword

nonn,tabl

Author

Philippe Deléham, Sep 14 2005

Status

approved