

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
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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. AddisonWesley, 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)^(n1)*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)^(n1)*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)^(n1)*x^n/n!.
log(A(x,t)) = Sum_{n >= 1} (n*t)^(n1)*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(n2, k1)*(n1)^(nk1); end: A139526 := proc(n, k) A061356(n, nk1) ; end: for n from 2 to 14 do for k from 0 to n2 do printf("%d, ", A139526(n, k)) ; od: od: # R. J. Mathar, May 22 2008


PROG

(PARI) for(n=2, 12, forstep(k=n1, 1, 1, print1(binomial(n2, k1)*(n1)^(nk1)", "))) \\ 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



