A198321 - OEIS

%I #24 Oct 23 2024 09:27:05

%S 1,0,1,0,1,2,0,1,3,3,0,1,4,6,4,0,1,5,10,10,5,0,1,6,15,20,15,6,0,1,7,

%T 21,35,35,21,7,0,1,8,28,56,70,56,28,8,0,1,9,36,84,126,126,84,36,9,0,1,

%U 10,45,120,210,252,210,120,45,10,0,1,11,55,165,330

%N Triangle read by rows: T(n, k) = binomial(n, k-1) for 1 <= k <= n, and T(n, 0) = 0^n.

%H Paolo Xausa, <a href="/A198321/b198321.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).

%F T(n, k) is given by (0,1,0,0,0,0,0,0,0,0,0,...) DELTA (1,1,-1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

%F Sum_{k=0..n} T(n, k)*x^k = x*((x+1)^n - x^n) for n > 0.

%F G.f.: (1 - (1+y)*x + y*(1+y)*x^2)/((1 - (1+y)*x)*(1-y*x)).

%F T(n, k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 1, T(2,0) = 0, T(2,1) = 1, T(2,2) = 2, T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Feb 12 2014

%e Triangle begins :

%e 1

%e 0, 1

%e 0, 1, 2

%e 0, 1, 3, 3

%e 0, 1, 4, 6, 4

%e 0, 1, 5, 10, 10, 5

%e 0, 1, 6, 15, 20, 15, 6

%t A198321[n_, k_] := If[k == 0, Boole[n == 0], Binomial[n, k - 1]];

%t Table[A198321[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Oct 23 2024 *)

%Y Variant of A074909, A135278.

%Y Cf. A007318.

%K easy,nonn,tabl

%O 0,6

%A _Philippe Deléham_, Nov 01 2011

%E New name using a formula of the author by _Peter Luschny_, Oct 23 2024