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A172094
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The Riordan square of the little Schröder numbers A001003.
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3
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1, 1, 1, 3, 4, 1, 11, 17, 7, 1, 45, 76, 40, 10, 1, 197, 353, 216, 72, 13, 1, 903, 1688, 1145, 458, 113, 16, 1, 4279, 8257, 6039, 2745, 829, 163, 19, 1, 20793, 41128, 31864, 15932, 5558, 1356, 222, 22, 1, 103049, 207905, 168584, 90776, 35318, 10070, 2066, 290, 25, 1
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OFFSET
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0,4
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COMMENTS
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The Riordan square is defined in A321620.
Previous name was: Triangle, read by rows, given by [1,2,1,2,1,2,1,2,1,2,1,2,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
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LINKS
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FORMULA
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T(0, 0) = 1, T(n, k) = 0 if k>n, T(n, 0) = T(n-1, 0) + 2*T(n-1, 1), T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k+1) for k>0.
Sum_{0<=k<=n} T(n, k) = A109980(n).
Sum_{k>=0} T(m, k)*T(n, k)*2^k = T(m+n, 0) = A001003(m+n).
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EXAMPLE
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Triangle begins:
1
1, 1
3, 4, 1
11, 17, 7, 1
45, 76, 40, 10, 1
197, 353, 216, 72, 13, 1
903, 1688, 1345, 458, 113, 16, 1
4279, 8257, 6039, 2745, 829, 163, 19, 1
20793, 41128, 31864, 15932, 5558, 1356, 222, 22, 1
103049, 207905, 168584, 90776, 35318, 10070, 2066, 290, 25, 1
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Production matrix begins:
1, 1
2, 3, 1
0, 2, 3, 1
0, 0, 2, 3, 1
0, 0, 0, 2, 3, 1
0, 0, 0, 0, 2, 3, 1
0, 0, 0, 0, 0, 2, 3, 1
0, 0, 0, 0, 0, 0, 2, 3, 1
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MATHEMATICA
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DELTA[r_, s_, m_] := Module[{p, q, t, x, y}, q[k_] := x r[[k+1]] + y s[[k+1]]; p[0, _] = 1; p[_, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k-1] + q[k] p[n-1, k+1] // Expand; t[n_, k_] := Coefficient[p[n, 0], x^(n-k)*y^k]; t[0, 0] = p[0, 0]; Table[t[n, k], {n, 0, m}, {k, 0, n}]];
nmax = 9;
DELTA[Table[{1, 2}, (nmax+1)/2] // Flatten, Prepend[Table[0, {nmax}], 1], nmax] // Flatten (* Jean-François Alcover, Aug 07 2018 *)
(* Function RiordanSquare defined in A321620. *)
RiordanSquare[(1 + x - Sqrt[1 - 6x + x^2])/(4x), 11] // Flatten (* Peter Luschny, Nov 27 2018 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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