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 A133289 Riordan matrix T from A084358 (lists of sets of lists) inverse to the Riordan matrix TI = 2I-A129652 formed from A000262 (number of sets of lists) and reciprocal under a partition transform. 3
 1, 1, 1, 5, 2, 1, 37, 15, 3, 1, 363, 148, 30, 4, 1, 4441, 1815, 370, 50, 5, 1, 65133, 26646, 5445, 740, 75, 6, 1, 1114009, 455931, 93261, 12705, 1295, 105, 7, 1, 21771851, 8912072, 1823724, 248696, 25410, 2072, 140, 8, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS T(n,k) is simply constructed from Pascal's triangle PT and A084358 through multiplication along the diagonals. Taking the matrix inverse gives TI = 2I-A129652 = PT times diagonal multiplication by -A000262 with the sign of the first term flipped to positive. T and TI are also reciprocals under the list partition transform described in A133314. LINKS Vincenzo Librandi, Rows n = 0..100, flattened T.-X. He, A symbolic operator approach to power series transformation-expansion formulas, JIS 11 (2008) 08.2.7 FORMULA T(n,k) = binomial(n,k) * A084358(n-k). E.g.f.: exp(xt) / { 2 - exp[x/(1-x)] }. EXAMPLE Triangle starts: 1, 1, 1, 5, 2, 1, 37, 15, 3, 1, 363, 148, 30, 4, 1, 4441, 1815, 370, 50, 5, 1, ... MATHEMATICA max = 7; s = Series[Exp[x*t]/(2-Exp[x/(1-x)]), {x, 0, max}, {t, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, k}]*n!; t[0, 0] = 1; Table[t[n, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 23 2014 *) CROSSREFS Cf. A131202. Sequence in context: A281890 A111544 A109281 * A107719 A229959 A174485 Adjacent sequences:  A133286 A133287 A133288 * A133290 A133291 A133292 KEYWORD easy,nonn,tabl AUTHOR Tom Copeland, Oct 16 2007, Nov 30 2007 STATUS approved

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Last modified April 5 16:53 EDT 2020. Contains 333245 sequences. (Running on oeis4.)