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 A096811 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). 5
 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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,19 COMMENTS 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. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..5150 FORMULA 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. EXAMPLE T(11,5) = 6 = 5th term of convolution of row (11-5) with row (5-2) = T(6,1)*T(3,3) + T(6,2)*T(3,2) + T(6,3)*T(3,1) + T(6,4)*T(3,0). Rows begin with n=0: 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, 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, 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; 1, 1, 1, 2, 4, 8, 15, 24, 31, 38, 40, 38, 35, 25, 16, 8, 4, 2, 1, 1; 1, 1, 1, 2, 4, 8, 16, 26, 36, 45, 48, 52, 46, 40, 28, 17, 8, 4, 2, 1, 1; 1, 1, 1, 2, 4, 8, 16, 28, 40, 53, 59, 66, 64, 55, 45, 30, 17, 8, 4, 2, 1, 1; 1, 1, 1, 2, 4, 8, 16, 30, 44, 60, 71, 83, 84, 78, 66, 51, 32, 17, 8, 4, 2, 1, 1; 1, 1, 1, 2, 4, 8, 16, 31, 48, 68, 83, 102, 108, 106, 95, 76, 55, 33, 18, 8, 4, 2, 1, 1; ... Forwards row convergent forms A096812: [1,1,1,2,4,8,16,34,72,156,336,746,1652,3696,...]. Backwards row convergent forms A096813: [0,1,1,2,4,8,18,40,92,210,490,1178,2834,6908,...]. PROG (PARI) {T(n, k) = if(n

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Last modified November 29 11:36 EST 2020. Contains 338765 sequences. (Running on oeis4.)