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

Author

Tom Copeland, Oct 16 2007, Nov 30 2007

Keywords

• easy
• nonn
• tabl

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.

Examples

			Triangle starts:
1,
1, 1,
5, 2, 1,
37, 15, 3, 1,
363, 148, 30, 4, 1,
4441, 1815, 370, 50, 5, 1,
...
		

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

Crossrefs

Cf. A131202.

Programs

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 *)

Formula

T(n,k) = binomial(n,k) * A084358(n-k).

E.g.f.: exp(xt) / { 2 - exp[x/(1-x)] }.