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.

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

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: A341723 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