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
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.
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
MATHEMATICA
T[n_, k_] := (n - 1)^k*Binomial[n - 2, n - k - 2];
Table[T[n, k], {n, 2, 11}, {k, 0, n - 2}] // Flatten (* Jean-François Alcover, Jun 13 2023 *)
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
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