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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.

%I A133289 #19 Sep 10 2015 04:04:00
%S A133289 1,1,1,5,2,1,37,15,3,1,363,148,30,4,1,4441,1815,370,50,5,1,65133,
%T A133289 26646,5445,740,75,6,1,1114009,455931,93261,12705,1295,105,7,1,
%U A133289 21771851,8912072,1823724,248696,25410,2072,140,8,1
%N 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.
%C A133289 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.
%C A133289 T and TI are also reciprocals under the list partition transform described in A133314.
%H A133289 Vincenzo Librandi, <a href="/A133289/b133289.txt">Rows n = 0..100, flattened</a>
%H A133289 T.-X. He, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/He/he51.html">A symbolic operator approach to power series transformation-expansion formulas</a>, JIS 11 (2008) 08.2.7
%F A133289 T(n,k) = binomial(n,k) * A084358(n-k).
%F A133289 E.g.f.: exp(xt) / { 2 - exp[x/(1-x)] }.
%e A133289 Triangle starts:
%e A133289 1,
%e A133289 1, 1,
%e A133289 5, 2, 1,
%e A133289 37, 15, 3, 1,
%e A133289 363, 148, 30, 4, 1,
%e A133289 4441, 1815, 370, 50, 5, 1,
%e A133289 ...
%t A133289 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 *)
%Y A133289 Cf. A131202.
%K A133289 easy,nonn,tabl
%O A133289 0,4
%A A133289 _Tom Copeland_, Oct 16 2007, Nov 30 2007