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 A193561 Augmentation of the triangle A004736. See Comments. 2
 1, 2, 1, 6, 6, 3, 24, 36, 30, 15, 120, 240, 270, 210, 105, 720, 1800, 2520, 2520, 1890, 945, 5040, 15120, 25200, 30240, 28350, 20790, 10395, 40320, 141120, 272160, 378000, 415800, 374220, 270270, 135135, 362880, 1451520, 3175200, 4989600 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091. Regarding A193561, if the triangle is written as (w(n,k)), then w(n,n)=A001147(n), "double factorial numbers"; w(n,n-1)=A097801(n), (2n)!/(n!*2^(n-1)) col 1: A000142, n! col 2: A001286, Lah numbers, (n-1)*n!/2 LINKS EXAMPLE First 5 rows of A193560: 1 2.....1 6.....6....3 24....36...30...15 120...240..270..210..105 MATHEMATICA p[n_, k_] := n + 1 - k Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A004736 *) m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}] TableForm[m[4]] w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1]; v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]}; v[n_] := v[n - 1].m[n] TableForm[Table[v[n], {n, 0, 6}]] (* A193561 *) Flatten[Table[v[n], {n, 0, 8}]] CROSSREFS Cf. A193091. Sequence in context: A075181 A052121 A193895 * A328349 A117965 A111646 Adjacent sequences: A193558 A193559 A193560 * A193562 A193563 A193564 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Jul 30 2011 STATUS approved

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Last modified January 27 13:42 EST 2023. Contains 359844 sequences. (Running on oeis4.)