A233295 - OEIS

%I #12 Dec 26 2014 04:04:57

%S 1,4,2,9,10,4,16,28,24,8,25,60,80,56,16,36,110,200,216,128,32,49,182,

%T 420,616,560,288,64,64,280,784,1456,1792,1408,640,128,81,408,1344,

%U 3024,4704,4992,3456,1408,256,100,570,2160,5712,10752,14400,13440,8320,3072,512

%N Riordan array ((1+x)/(1-x)^3, 2*x/(1-x)).

%C Subtriangle of the triangle in A208532.

%C Row sums are A060188(n+2).

%C Diagonal sums are A000295(n+2)=A125128(n+1)=A130103(n+2).

%F G.f. for the column k: 2^k*(1+x)/(1-x)^(k+3).

%F T(n,k) = 2^k*(binomial(n,k)+3*binomial(n,k+1)+2*binomial(n,k+2)), 0<=k<=n.

%F T(n,0) = 2*T(n-1,0)-T(n-2,0)+2, T(n,k)=2*T(n-1,k)+2*T(n-1,k-1)-2*T(n-2,k-1)-T(n-2,k) for k>=1, T(0,0)=1, T(1,0)=4, T(1,1)=2, T(n,k)=0 if k<0 or if k>n.

%F Sum_{k=0..n} T(n,k) = A060188(n+2).

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

%F T(n,k) = 2*sum_{j=1..n-k+1} T(n-j,k-1).

%F T(n,k) = 2^k*A125165(n,k).

%F T(n,n) = 2^n=A000079(n).

%F T(n,0) = (n+1)^2=A000290(n+1).

%F exp(2*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(2*x)*(16 + 28*x + 24*x^2/2! + 8*x^3/3!) = 16 + 60*x + 200*x^2/2! + 616*x^3/3! + 1792*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), 2*x/(1 - x) ). Cf. A125165. - _Peter Bala_, Dec 21 2014

%e Triangle begins :

%e 1

%e 4, 2

%e 9, 10, 4

%e 16, 28, 24, 8

%e 25, 60, 80, 56, 16

%e 36, 110, 200, 216, 128, 32

%e 49, 182, 420, 616, 560, 288, 64

%e 64, 280, 784, 1456, 1792, 1408, 640, 128

%e 81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256

%e 100, 570, 2160, 5712, 10752, 14400, 13440, 8320, 3072, 512

%Y Cf. Columns: A000290, A006331, A112742.

%Y Cf. Diagonal: A000079.

%K nonn,tabl

%O 0,2

%A _Philippe Deléham_, Dec 07 2013