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%I #23 Jun 13 2023 07:16:26
%S 1,1,2,1,6,9,1,12,48,64,1,20,150,500,625,1,30,360,2160,6480,7776,1,42,
%T 735,6860,36015,100842,117649,1,56,1344,17920,143360,688128,1835008,
%U 2097152,1,72,2268,40824,459270,3306744,14880348,38263752,43046721,1,90,3600,84000,1260000,12600000,84000000,360000000,900000000,1000000000
%N Triangle A061356 read right to left.
%C 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
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA. Second ed. 1994.
%D Peter D. Schumer (2004), Mathematical Journeys, page 168, Proposition 16.1 (c)
%H P. Bala, <a href="/A251592/a251592.pdf">Fractional iteration of a series inversion operator</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lambert_W_function">Lambert W function</a>
%F 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
%F From _Peter Bala_, Nov 16 2015: (Start)
%F 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.
%F A(x,t)^m = 1 + Sum_{n >= 1} m*(m + n*t)^(n-1)*x^n/n!.
%F 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.
%F A(x,t) = ( 1/x* Revert( x*exp(-x*t)) )^(1/t), where Revert is the series reversion operator with respect to x.
%F 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)
%e (1) times (1) = (1)
%e (1 1) * (1 2) = (1 2)
%e (1 2 1 ) * (1 3 9) = (1 6 9)
%e (1 3 3 1) * (1 4 16 64) = (1 12 48 64)
%e etc.
%p 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
%t T[n_, k_] := (n - 1)^k*Binomial[n - 2, n - k - 2];
%t Table[T[n, k], {n, 2, 11}, {k, 0, n - 2}] // Flatten (* _Jean-François Alcover_, Jun 13 2023 *)
%o (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
%Y Cf. A000272 (row sums), A061356 (row reverse), A028421, A074909, A000169 (main diagonal), A251592, A260687.
%K nonn,tabl
%O 2,3
%A _Alford Arnold_, Apr 24 2008
%E More terms from _R. J. Mathar_ and Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008
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