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A139526 Triangle A061356 read right to left. 4
1, 1, 2, 1, 6, 9, 1, 12, 48, 64, 1, 20, 150, 500, 625, 1, 30, 360, 2160, 6480, 7776, 1, 42, 735, 6860, 36015, 100842, 117649, 1, 56, 1344, 17920, 143360, 688128, 1835008, 2097152, 1, 72, 2268, 40824, 459270, 3306744, 14880348, 38263752, 43046721, 1, 90 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,3

COMMENTS

Related to the two Appell sequences the Bernoulli polynomials B(n,x) and their umbral compositional inverses (cf. A074909) Up(n,x) = [(x+1)^(n+1)-x^(n+1)] / (n+1). With offset 0, the row polynomials of this entry P(n,x) = (Up(n,0))^(-n) * [x + Up(n,0)]^n = (n+1)^n * [x + 1/(n+1)]^n. Compare to the Abel polynomials of A061356, which are also an Appell sequence. - Tom Copeland, Nov 14 2014

REFERENCES

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA. Second ed. 1994.

Peter D. Schumer (2004), Mathematical Journeys, page 168, Proposition 16.1 (c)

LINKS

Table of n, a(n) for n=2..48.

P. Bala, Fractional iteration of a series inversion operator

Wikipedia, Lambert W function

FORMULA

E.g.f. (with offset 1) Sum_{n >= 1} (1 + n*t)^(n-1)*x^n/n! = x + (1 + 2*t)*x^2/2! + (1 + 6*t + 9*t^2)*x^3/3! + .... For properties of this function see Graham et al., equations 5.60, 5.61 and 7.71. The e.g.f. is the series reversion with respect to x of the function log(1 + x)/(1 + x)^t, which is the e.g.f. for a signed version of A028421. - Peter Bala, Jul 18 2013

From Peter Bala, Nov 16 2015: (Start)

E.g.f. with offset 0 and constant term 1: A(x,t) = ( Sum_{n >= 0} (n + 1)^(n-1)*t^n*x^n/n! )^(1/t). This is the generalized exponential series E_t(x) in the terminology of Graham et al., Section 5.4.

A(x,t)^m = 1 + Sum_{n >= 1} m*(m + n*t)^(n-1)*x^n/n!.

log(A(x,t)) = Sum_{n >= 1} (n*t)^(n-1)*x^n/n! = 1/t*T(t*x), where T(z) is Euler's tree function. See A000169.

A(x,t) = ( 1/x* Revert( x*exp(-x*t)) )^(1/t), where Revert is the series reversion operator with respect to x.

In the notation of the Bala link the e.g.f. is I^t(e^x), where I^t is a fractional series inversion operator. Cf. A251592, which has o.g.f. I^t(1 + x), and A260687, which has o.g.f. I^t(1/(1 - x)). (End)

EXAMPLE

(1) times (1) = (1)

(1 1) * (1 2) = (1 2)

(1 2 1 ) * (1 3 9) = (1 6 9)

(1 3 3 1) * (1 4 16 64) = (1 12 48 64)

etc.

MAPLE

A061356 := proc(n, k) binomial(n-2, k-1)*(n-1)^(n-k-1); end: A139526 := proc(n, k) A061356(n, n-k-1) ; end: for n from 2 to 14 do for k from 0 to n-2 do printf("%d, ", A139526(n, k)) ; od: od: # R. J. Mathar, May 22 2008

PROG

(PARI) for(n=2, 12, forstep(k=n-1, 1, -1, print1(binomial(n-2, k-1)*(n-1)^(n-k-1)", "))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 10 2008

CROSSREFS

Cf. A000272 (row sums), A061356 (row reverse), A028421, A074909, A000169 (main diagonal), A251592, A260687.

Sequence in context: A156034 A160581 A021465 * A248927 A176013 A263255

Adjacent sequences:  A139523 A139524 A139525 * A139527 A139528 A139529

KEYWORD

nonn,tabl

AUTHOR

Alford Arnold, Apr 24 2008

EXTENSIONS

More terms from R. J. Mathar and Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008

STATUS

approved

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Last modified October 20 10:18 EDT 2019. Contains 328257 sequences. (Running on oeis4.)