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A139526
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Triangle A061356 read right to left.
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4
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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, 3600, 84000, 1260000, 12600000, 84000000, 360000000, 900000000, 1000000000
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OFFSET
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2,3
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COMMENTS
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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
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REFERENCES
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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)
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LINKS
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FORMULA
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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
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)
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EXAMPLE
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(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.
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MAPLE
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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
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MATHEMATICA
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T[n_, k_] := (n - 1)^k*Binomial[n - 2, n - k - 2];
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from R. J. Mathar and Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008
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STATUS
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approved
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