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A110681
A convolution triangle of numbers based on A071356.
1
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: A393390 A094527 A054335 * A117852 A080245 A080247
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Sep 14 2005
STATUS
approved