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 A155537 Scaled Narayana recursion: m = 0; p = 2; q = 1; a(n,k)=(m*n - m*k + 1)*a(n - 1, k - 1) + (m*k - (m - 1))*a(n - 1, k); f(n) = Product[k + 1, {k, 0, n}]; a0(n,m) = f[n]/(f[m]*f[n - m]); t(n,k)=(p^(n - m)*q^m + p^m*q^(n - m))*a0(n - 1, k - 1)*a(n, k). 0
 3, 5, 5, 9, 27, 9, 17, 102, 102, 17, 33, 330, 660, 330, 33, 65, 975, 3250, 3250, 975, 65, 129, 2709, 13545, 22575, 13545, 2709, 129, 257, 7196, 50372, 125930, 125930, 50372, 7196, 257, 513, 18468, 172368, 603288, 904932, 603288, 172368, 18468, 513, 1025 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Row sums are: {3, 10, 45, 238, 1386, 8580, 55341, 367510, 2494206, 17215900,...}. LINKS FORMULA m = 0; p = 2; q = 1; a(n,k)=(m*n - m*k + 1)*a(n - 1, k - 1) + (m*k - (m - 1))*a(n - 1, k); f(n) = Product[k + 1, {k, 0, n}]; a0(n,m) = f[n]/(f[m]*f[n - m]); t(n,k)=(p^(n - m)*q^m + p^m*q^(n - m))*a0(n - 1, k - 1)*a(n, k). EXAMPLE {3}, {5, 5}, {9, 27, 9}, {17, 102, 102, 17}, {33, 330, 660, 330, 33}, {65, 975, 3250, 3250, 975, 65}, {129, 2709, 13545, 22575, 13545, 2709, 129}, {257, 7196, 50372, 125930, 125930, 50372, 7196, 257}, {513, 18468, 172368, 603288, 904932, 603288, 172368, 18468, 513}, {1025, 46125, 553500, 2583000, 5424300, 5424300, 2583000, 553500, 46125, 1025} MATHEMATICA Clear[A, a0, b0, n, k, m]; A[n_, 1, m_] := 1; A[n_, n_, m_] := 1; A[n_, k_, m_] := (m*n - m*k + 1)*A[n - 1, k - 1, m] + (m*k - (m - 1))*A[n - 1, k, m]; f[n_] = Product[k + 1, {k, 0, n}]; a0[n_, m_] = f[n]/(f[m]*f[n - m]);; m = 0; p = 2; q = 1; Table[(p^(n - m)*q^m + p^m*q^(n - m))*a0[n - 1, k - 1]*A[n, k, m], {n, 10}, {k, n}]; Flatten[%] CROSSREFS Sequence in context: A145282 A049757 A190864 * A164663 A098971 A093572 Adjacent sequences:  A155534 A155535 A155536 * A155538 A155539 A155540 KEYWORD nonn,tabl,uned AUTHOR Roger L. Bagula, Jan 23 2009 STATUS approved

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Last modified July 17 14:44 EDT 2019. Contains 325106 sequences. (Running on oeis4.)