A122832 Exponential Riordan array (e^(x(1+x)),x).

1, 1, 1, 3, 2, 1, 7, 9, 3, 1, 25, 28, 18, 4, 1, 81, 125, 70, 30, 5, 1, 331, 486, 375, 140, 45, 6, 1, 1303, 2317, 1701, 875, 245, 63, 7, 1, 5937, 10424, 9268, 4536, 1750, 392, 84, 8, 1, 26785, 53433, 46908, 27804, 10206, 3150, 588, 108, 9, 1

Offset

0,4

Comments

Row sums are A000898. Inverse is A122833. Product of A007318 and A067147.

Links

Michel Marcus, Rows n=0..50 of triangle, flattened

Formula

Number triangle T(n,k) = (n!/k!)*Sum_{i = 0..n-k} C(i,n-k-i)/i!.

From Peter Bala, May 14 2012: (Start)

Array is exp(S + S^2) where S is A132440 the infinitesimal generator for Pascal's triangle.

T(n,k) = binomial(n,k)*A047974(n-k).

So T(n,k) gives the number of ways to choose a subset of {1,2,...,n} of size k and then arrange the remaining n-k elements into a set of lists of length 1 or 2. (End)

From Peter Bala, Oct 24 2023: (Start)

n-th row polynomial: R(n,x) = exp(D + D^2) (x^n) = exp(D^2) (1 + x)^n, where D denotes the derivative operator d/dx. Cf. A111062.

The sequence of polynomials defined by R(n,x-1) = exp(D^2) (x^n) begins [1, 1, 2 + x^2, 6*x + x^3, 12 + 12*x^2 + x^4, ...] and is related to the Hermite polynomials. See A059344. (End)

Example

Triangle begins:
   1;
   1,   1;
   3,   2,  1;
   7,   9,  3,  1;
  25,  28, 18,  4, 1;
  81, 125, 70, 30, 5, 1;
  ...
From Peter Bala, May 14 2012: (Start)
T(3,1) = 9. The 9 ways to select a subset of {1,2,3} of size 1 and arrange the remaining elements into a set of lists (denoted by square brackets) of length 1 or 2 are:
{1}[2,3], {1}[3,2], {1}[2][3],
{2}[1,3], {2}[3,1], {2}[1][3],
{3}[1,2], {3}[2,1], {3}[1][2]. (End)

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[E^(#(1+#))&, #&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Prog

(PARI) T(n, k) = (n!/k!)*sum(i=0, n-k, binomial(i, n-k-i)/i!); \\ Michel Marcus, Aug 28 2017

Crossrefs

A000898 (row sums), A047974 (column 0), A291632 (column 1), A122833 (inverse array).

Cf. A059344, A111062.

Sequence in context: A192020 A367564 A171128 * A056151 A134436 A306226

Adjacent sequences: A122829 A122830 A122831 * A122833 A122834 A122835

Keyword

easy,nonn,tabl

Author

Paul Barry, Sep 12 2006

Extensions

More terms from Michel Marcus, Aug 28 2017

Status

approved