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A122832 Exponential Riordan array (e^(x(1+x)),x). 4

%I

%S 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,

%T 6,1,1303,2317,1701,875,245,63,7,1,5937,10424,9268,4536,1750,392,84,8,

%U 1,26785,53433,46908,27804,10206,3150,588,108,9,1

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

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

%H Michel Marcus, <a href="/A122832/b122832.txt">Rows n=0..50 of triangle, flattened</a>

%F Number triangle T(n,k)=(n!/k!)sum{i=0..n-k, C(i,n-k-i)/i!}

%F 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]

%e Triangle begins:

%e 1;

%e 1, 1;

%e 3, 2, 1;

%e 7, 9, 3, 1;

%e 25, 28, 18, 4, 1;

%e 81, 125, 70, 30, 5, 1;

%e ...

%e 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:

%e {1}[2,3], {1}[3,2], {1}[2][3],

%e {2}[1,3], {2}[3,1], {2}[1][3],

%e {3}[1,2], {3}[2,1], {3}[1][2]. [_Peter Bala_, May 14 2012]

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

%t RiordanArray[E^(#(1+#))&, #&, 10, True] // Flatten (* _Jean-Fran├žois Alcover_, Jul 19 2019 *)

%o (PARI) T(n,k) = (n!/k!)*sum(i=0, n-k, binomial(i,n-k-i)/i!); \\ _Michel Marcus_, Aug 28 2017

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

%K easy,nonn,tabl

%O 0,4

%A _Paul Barry_, Sep 12 2006

%E More terms from _Michel Marcus_, Aug 28 2017

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Last modified April 13 12:29 EDT 2021. Contains 342936 sequences. (Running on oeis4.)