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%I #29 Oct 29 2023 18:15:35
%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 From _Peter Bala_, May 14 2012: (Start)
%F Array is exp(S + S^2) where S is A132440 the infinitesimal generator for Pascal's triangle.
%F T(n,k) = binomial(n,k)*A047974(n-k).
%F 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)
%F From _Peter Bala_, Oct 24 2023: (Start)
%F 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.
%F 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)
%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 From _Peter Bala_, May 14 2012: (Start)
%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]. (End)
%t (* The function RiordanArray is defined in A256893. *)
%t RiordanArray[E^(#(1+#))&, #&, 10, True] // Flatten (* _Jean-François 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 A000898 (row sums), A047974 (column 0), A291632 (column 1), A122833 (inverse array).
%Y Cf. A059344, A111062.
%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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