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 A107267 A square array of Motzkin related transforms, read by antidiagonals. 10
 1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 6, 3, 1, 0, 9, 20, 12, 4, 1, 0, 21, 72, 54, 20, 5, 1, 0, 51, 272, 261, 112, 30, 6, 1, 0, 127, 1064, 1323, 672, 200, 42, 7, 1, 0, 323, 4272, 6939, 4224, 1425, 324, 56, 8, 1, 0, 835, 17504, 37341, 27456, 10625, 2664, 490, 72, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Rows are transforms of k^n, k>=0, under the matrix A107131. As a number triangle, with T(n,k)=if(k<=n,sum{j=0..n-k, (1/(j+1))C(j+1,n-k-j+1)C(n-k,j)k^j},0), row sums are A107268 and diagonal sums are A107269. Rows are series reversions of x/(1+kx+kx^2), k>=0. Conjecture: rows count weighted Motzkin paths. Row k counts colored Motzkin paths, where H(1,0) and U(1,1) each have k colors and D(1,-1) one color. - Paul Barry, May 16 2005 LINKS Seiichi Manyama, Antidiagonals n = 0..139, flattened FORMULA Number array T(n,k) = Sum_{j=0..k} n^j * binomial(k,j) * binomial(j+1,k-j+1)/(j+1). G.f. of row k: 1/(1 - k*x - k*x^2/(1 - k*x - k*x^2/(1 - k*x - k*x^2/(1 - k*x - k*x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017 From Seiichi Manyama, May 05 2019: (Start) T(n,k) = Sum_{j=0..floor(k/2)} n^(k-j) * binomial(k,2*j) * binomial(2*j,j)/(j+1) = Sum_{j=0..floor(k/2)} n^(k-j) * binomial(k,2*j) * A000108(j). (k+2) * T(n,k) = n * (2*k+1) * T(n,k-1) - n * (n-4) * (k-1) * T(n,k-2). (End) EXAMPLE Array begins   1, 0,  0,   0,    0,     0,      0, ...   1, 1,  2,   4,    9,    21,     51, ...   1, 2,  6,  20,   72,   272,   1064, ...   1, 3, 12,  54,  261,  1323,   6939, ...   1, 4, 20, 112,  672,  4224,  27456, ...   1, 5, 30, 200, 1425, 10625,  81875, ...   1, 6, 42, 324, 2664, 22896, 203256, ... CROSSREFS Rows n=0..6 give A000007, A001006, A071356, A107264, A003645, A107265, A107266. Main diagonal gives A292716. Cf. A000108. Sequence in context: A191348 A198792 A196182 * A320530 A347986 A191239 Adjacent sequences:  A107264 A107265 A107266 * A107268 A107269 A107270 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, May 15 2005 STATUS approved

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Last modified May 17 13:24 EDT 2022. Contains 353746 sequences. (Running on oeis4.)