A096811 - OEIS

%I #12 May 11 2019 10:27:36

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,2,1,1,1,1,1,2,3,2,

%T 1,1,1,1,1,2,3,3,2,1,1,1,1,1,2,4,4,3,2,1,1,1,1,1,2,4,5,4,4,2,1,1,1,1,

%U 1,2,4,6,6,5,4,2,1,1,1,1,1,2,4,7,7,8,6,4,2,1,1,1,1,1,2,4,7,9,10,9,7,4,2,1,1,1,1,1,2,4,8,10,12,12,11,7,4,2,1,1,1,1,1,2,4,8,12,14,16,15,12,7,4,2,1,1,1,1,1,2,4,8,13,17,18,21,17,13,8,4,2,1,1,1,1,1,2,4,8,14,19,23,25,24,20,14,8,4,2,1,1,1,1,1,2,4,8,15,22,27,32,30,29,23,15,8,4,2,1,1

%N Triangle, read by rows, such that T(n,k) equals the k-th term of the convolution of the two prior rows indexed by (n-k) and (k-2).

%C Two row convergents exist simultaneously. When the rows are read forwards, they converge to A096812. When the rows are read backwards, they converge to A096813. The row sums form A096814.

%H Paul D. Hanna, <a href="/A096811/b096811.txt">Table of n, a(n) for n = 0..5150</a>

%F T(n, k) = Sum_{j=1..min(n-k, k-1)} T(n-k, j)*T(k-2, k-j-1), for n>=k>=1, with T(n, 0)=T(n+1, 1)=T(n, n)=1 for n>=0.

%e T(11,5) = 6 = 5th term of convolution of row (11-5) with row (5-2) =

%e T(6,1)*T(3,3) + T(6,2)*T(3,2) + T(6,3)*T(3,1) + T(6,4)*T(3,0).

%e Rows begin with n=0:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 1, 1, 1;

%e 1, 1, 1, 1, 1;

%e 1, 1, 1, 2, 1, 1;

%e 1, 1, 1, 2, 2, 1, 1;

%e 1, 1, 1, 2, 3, 2, 1, 1;

%e 1, 1, 1, 2, 3, 3, 2, 1, 1;

%e 1, 1, 1, 2, 4, 4, 3, 2, 1, 1;

%e 1, 1, 1, 2, 4, 5, 4, 4, 2, 1, 1;

%e 1, 1, 1, 2, 4, 6, 6, 5, 4, 2, 1, 1;

%e 1, 1, 1, 2, 4, 7, 7, 8, 6, 4, 2, 1, 1;

%e 1, 1, 1, 2, 4, 7, 9, 10, 9, 7, 4, 2, 1, 1;

%e 1, 1, 1, 2, 4, 8, 10, 12, 12, 11, 7, 4, 2, 1, 1;

%e 1, 1, 1, 2, 4, 8, 12, 14, 16, 15, 12, 7, 4, 2, 1, 1;

%e 1, 1, 1, 2, 4, 8, 13, 17, 18, 21, 17, 13, 8, 4, 2, 1, 1;

%e 1, 1, 1, 2, 4, 8, 14, 19, 23, 25, 24, 20, 14, 8, 4, 2, 1, 1;

%e 1, 1, 1, 2, 4, 8, 15, 22, 27, 32, 30, 29, 23, 15, 8, 4, 2, 1, 1;

%e 1, 1, 1, 2, 4, 8, 15, 24, 31, 38, 40, 38, 35, 25, 16, 8, 4, 2, 1, 1;

%e 1, 1, 1, 2, 4, 8, 16, 26, 36, 45, 48, 52, 46, 40, 28, 17, 8, 4, 2, 1, 1;

%e 1, 1, 1, 2, 4, 8, 16, 28, 40, 53, 59, 66, 64, 55, 45, 30, 17, 8, 4, 2, 1, 1;

%e 1, 1, 1, 2, 4, 8, 16, 30, 44, 60, 71, 83, 84, 78, 66, 51, 32, 17, 8, 4, 2, 1, 1;

%e 1, 1, 1, 2, 4, 8, 16, 31, 48, 68, 83, 102, 108, 106, 95, 76, 55, 33, 18, 8, 4, 2, 1, 1; ...

%e Forwards row convergent forms A096812:

%e [1,1,1,2,4,8,16,34,72,156,336,746,1652,3696,...].

%e Backwards row convergent forms A096813:

%e [0,1,1,2,4,8,18,40,92,210,490,1178,2834,6908,...].

%o (PARI) {T(n,k) = if(n<k || k<0,0,if(k<=1 || k==n,1,sum(j=1,k-1,T(n-k,j)*T(k-2,k-j-1))))}

%o for(n=0,20, for(k=0,n, print1(T(n,k),", "));print(""))

%Y Cf. A096813, A096814, A091499.

%K nonn,tabl

%O 0,19

%A _Paul D. Hanna_, Jul 20 2004