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A122832 Exponential Riordan array (e^(x(1+x)),x). 4
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 (list; table; graph; refs; listen; history; text; internal format)
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!}

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. [Peter Bala, May 14 2012]

EXAMPLE

Triangle begins:

   1;

   1,   1;

   3,   2,  1;

   7,   9,  3,  1;

  25,  28, 18,  4, 1;

  81, 125, 70, 30, 5, 1;

  ...

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]. [Peter Bala, May 14 2012]

MATHEMATICA

(* The function RiordanArray is defined in A256893. *)

RiordanArray[E^(#(1+#))&, #&, 10, True] // Flatten (* Jean-Fran├žois 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

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

Sequence in context: A145035 A192020 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

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Last modified March 3 14:58 EST 2021. Contains 341762 sequences. (Running on oeis4.)