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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

Table of n, a(n) for n=0..54.

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

CROSSREFS

Cf. A000262 (column 0), A052844 (row sums).

Cf. A111884, A133314.

Sequence in context: A104980 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 February 17 17:27 EST 2018. Contains 299296 sequences. (Running on oeis4.)