A002109 - OEIS (original) (raw)

COMMENTS

A054374 gives the discriminants of the Hermite polynomials in the conventional (physicists') normalization, and A002109 (this sequence) gives the discriminants of the Hermite polynomials in the (in my opinion more natural) probabilists' normalization. See refs Wikipedia and Szego, eq. (6.71.7). - Alan Sokal, Mar 02 2012

a(n) = (-1)^n/det(M_n) where M_n is the n X n matrix m(i,j) = (-1)^i/i^j. - Benoit Cloitre, May 28 2002

a(n) = determinant of the n X n matrix M(n) where m(i,j) = B(n,i,j) and B(n,i,x) denote the Bernstein polynomial: B(n,i,x) = binomial(n,i)*(1-x)^(n-i)*x^i. - Benoit Cloitre, Feb 02 2003

Number of trailing zeros (A246839) increases every 5 terms since the exponent of the factor 5 increases every 5 terms and the exponent of the factor 2 increases every 2 terms. - Chai Wah Wu, Sep 03 2014

Also the number of minimum distinguishing labelings in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017

Also shows up in a term in the solution to the generalized version of Raabe's integral. - Jibran Iqbal Shah, Apr 24 2021

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 477.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. Szego, Orthogonal Polynomials, American Mathematical Society, 1981 edition, 432 Pages.

LINKS

Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.

Eric Weisstein's World of Mathematics, K-Function.

FORMULA

Determinant of n X n matrix m(i, j) = binomial(i*j, i). - Benoit Cloitre, Aug 27 2003

a(n) = exp(zeta'(-1, n + 1) - zeta'(-1)) where zeta(s, z) is the Hurwitz zeta function. - Peter Luschny, Jun 23 2012

G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k^k*x). - Paul D. Hanna, Oct 02 2013

a(n) ~ A * n^(n*(n+1)/2 + 1/12) / exp(n^2/4), where A = 1.2824271291006226368753425... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Feb 20 2015

a(n) = Product_{k=1..n} ff(n,k) where ff denotes the falling factorial. - Peter Luschny, Nov 29 2015

log a(n) = (1/2) n^2 log n - (1/4) n^2 + (1/2) n log n + (1/12) log n + log(A) + o(1), where log(A) = A225746 is the logarithm of Glaisher's constant. - Charles R Greathouse IV, Mar 27 2020

Sum_{n>=1} (-1)^(n+1)/a(n) = A347352. (End)

a(n) = e^(Integral_{x=1..n+1} (x - 1/2 - log(sqrt(2*Pi)) + (n+1-x)*Psi(x)) dx), where Psi(x) is the digamma function.

a(n) = e^(Integral_{x=1..n} (x + 1/2 - log(sqrt(2*Pi)) + log(Gamma(x+1))) dx). (End)

MAPLE

f := proc(n) local k; mul(k^k, k=1..n); end;

A002109 := n -> exp(Zeta(1, -1, n+1)-Zeta(1, -1));

MATHEMATICA

Table[Hyperfactorial[n], {n, 0, 11}] (* Zerinvary Lajos, Jul 10 2009 *)

Join[{1}, FoldList[Times, #^#&/@Range[15]]] (* Harvey P. Dale, Nov 02 2023 *)

PROG

(PARI) a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1, k+1, (1+j^j*x+x*O(x^n)) )), n) \\ Paul D. Hanna, Oct 02 2013

(Haskell)

a002109 n = a002109_list !! n

a002109_list = scanl1 (*) a000312_list -- Reinhard Zumkeller, Jul 07 2012

(Python)

for n in range(1, 10):

(SageMath)

a = lambda n: prod(falling_factorial(n, k) for k in (1..n))

CROSSREFS

Cf. A000178, A000142, A000312, A001358, A002981, A002982, A100015, A005234, A014545, A018239, A006794, A057704, A057705, A054374.

Cf. A074962 [Glaisher-Kinkelin constant, also gives an asymptotic approximation for the hyperfactorials].

Cf. A261175 (number of digits).