|
%I #27 Nov 06 2017 02:43:19
%S 1,2,1,6,4,1,20,16,6,1,72,64,30,8,1,272,260,140,48,10,1,1064,1072,636,
%T 256,70,12,1,4272,4480,2856,1288,420,96,14,1,17504,18944,12768,6272,
%U 2320,640,126,16,1,72896,80928,57024,29952,12192,3852,924,160,18,1
%N A convolution triangle of numbers based on A071356.
%H Michael De Vlieger, <a href="/A110681/b110681.txt">Table of n, a(n) for n = 0..11324</a> (rows 0 <= n <= 150).
%H G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1973__20__3_0">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), p. 32-33.
%H G. Kreweras, <a href="/A001844/a001844.pdf">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
%F 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).
%F Sum_{k, k>=0} T(m, k)*T(n, k)*2^k = T(m+n, 0) = A071356(m+n).
%F Sum_{k, k>=0} T(n, k)*(2^(k+1) - 1) = 5^n.
%F Sum_{k, k>=0} (-1)^(n-k)*T(n, k)*(2^(k+1) - 1) = 1.
%e Triangle starts:
%e 1;
%e 2, 1;
%e 6, 4, 1;
%e 20, 16, 6, 1;
%e 72, 64, 30, 8, 1;
%e ...
%t 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 *)
%Y Cf. A071356 (1st column), A071357 (2nd column).
%Y Cf. A052705 (diagonal sums), A001003 (row sums).
%K nonn,tabl
%O 0,2
%A _Philippe Deléham_, Sep 14 2005
|
