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 A129652 Exponential Riordan array [e^(x/(1-x)),x]. 4
 1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 73, 52, 18, 4, 1, 501, 365, 130, 30, 5, 1, 4051, 3006, 1095, 260, 45, 6, 1, 37633, 28357, 10521, 2555, 455, 63, 7, 1, 394353, 301064, 113428, 28056, 5110, 728, 84, 8, 1, 4596553, 3549177, 1354788, 340284, 63126, 9198, 1092, 108, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Satisfies the equation e^[x/(1-x),x] = e*[e^(x/(1-x)),x]. Row sums are A052844. Diagonal sums are A129653. LINKS T.-X. He, A symbolic operator approach to power series transformation-expansion formulas, JIS 11 (2008) 08.2.7. FORMULA Number triangle T(n,k)=(n!/k!)*sum{i=0..n-k, C(n-k-1,i)/(n-k-i)!} From Peter Bala, May 14 2012 : (Start) Array is exp(S*(I-S)^(-1)) where S is A132440 the infinitesimal generator for Pascal's triangle. Column 0 is A000262. T(n,k) = binomial(n,k)*A000262(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. (End) T(n,k) = (-1)^(k-n+1)*C(n,k)*KummerU(k-n+1, 2, -1). - Peter Luschny, Sep 17 2014 From Tom Copeland, Mar 11 2016: (Start) The row polynomials P_n(x) form an Appell sequence with e.g.f. e^(t*P.(x)) = e^[t / (1-t)] e^(x*t), so the lowering and raising operators are L = d/dx = D and the R = x + 1 / (1-D)^2 = x + 1 + 2 D + 3 D^2 + ..., satisfying L P_n(x) = n * P_(n-1)(x) and R P_n(x) = P_(n+1)(x). (P.(x) + y)^n = Sum_{k=0..n} binomial(n,k) P_k(x) * y^(n-k) = P_n(x+y). The Appell polynomial umbral compositional inverse sequence has the e.g.f. e^(t*Q.(x)) = e^[-t / (1-t)] e^(x*t) (see A111884 and A133314), so Q_n(P.(x)) = P_n(Q.(x)) = x^n. The lower triangular matrices for the coefficients of these two Appell sequences are a multiplicative inverse pair. (End) EXAMPLE Triangle begins 1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 73, 52, 18, 4, 1, 501, 365, 130, 30, 5, 1, 4051, 3006, 1095, 260, 45, 6, 1 MAPLE A129652 := (n, k) -> (-1)^(k-n+1)*binomial(n, k)*KummerU(k-n+1, 2, -1); seq(seq(round(evalf(A129652(n, k), 99)), k=0..n), n=0..9); # Peter Luschny, Sep 17 2014 MATHEMATICA T[n_, k_] := If[k==n, 1, n!/k! Sum[Binomial[n-k-1, j]/(j+1)!, {j, 0, n-k-1}]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 14 2019 *) CROSSREFS Cf. A000262 (column 0), A052844 (row sums). Cf. A111884, A133314. Sequence in context: A316566 A134090 A132845 * A154921 A127126 A161133 Adjacent sequences:  A129649 A129650 A129651 * A129653 A129654 A129655 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, Apr 26 2007 STATUS approved

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Last modified September 24 16:23 EDT 2021. Contains 347645 sequences. (Running on oeis4.)