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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-Fran├žois 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 October 17 17:08 EDT 2018. Contains 316290 sequences. (Running on oeis4.)