A126673 Third diagonal of A126671.

0, 2, 26, 274, 2844, 30708, 351504, 4292496, 55988640, 779171040, 11545476480, 181705299840, 3029581820160, 53376951801600, 991337037465600, 19363464423475200, 396915849843609600, 8520964324004966400, 191220598650009600000, 4477883953203763200000, 109242544826541772800000

Offset

2,2

Comments

It appears that a(n) = sum of invc(p) over all permutations p of {1,2,...,n}, where invc(p) is defined (by Carlitz) in the following way: express p in standard cycle form (i.e., cycles ordered by increasing smallest elements with each cycle written with its smallest element in the first position), then remove the parentheses and count the inversions in the obtained word. a(3)=2 because the six permutations 123,132,312,213,231 and 321 of {1,2,3} yield the words 123,123,132,123,123 and 132, respectively, having a total of 0+0+1+0+0+1 = 2 inversions. a(n) = Sum_{k>=0} k*A129178(n,k). - Emeric Deutsch, Oct 10 2007

References

L. Carlitz, Generalized Stirling numbers, Combinatorial Analysis Notes, Duke University, 1968, 1-7.

Links

G. C. Greubel, Table of n, a(n) for n = 2..445

M. Shattuck, Parity theorems for statistics on permutations and Catalan words, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 5, Paper A07, 2005.

N. J. A. Sloane, Notes on Carlo Wood's Polynomials

Formula

a(n) = n! * (n*(n-5)/4 + 1 + 1/2 + ... + 1/n). - Emeric Deutsch, Oct 10 2007

E.g.f.: (2*x - 3*x^2 + 2*(1-x)^2 * log(1-x)) / (2*(-1+x)^3). - G. C. Greubel, May 05 2019

Maple

seq(factorial(n)*(sum(1/k, k = 1 .. n)+(1/4)*n*(n-5)), n = 2 .. 21) # Emeric Deutsch, Oct 10 2007

Mathematica

Table[n!*(n*(n-5)/4 + HarmonicNumber[n]), {n, 2, 25}] (* G. C. Greubel, May 05 2019 *)

Prog

(PARI) my(x='x+O('x^30)); concat([0], Vec(serlaplace( (2*x - 3*x^2 + 2*(1-x)^2*log(1-x))/(2*(-1+x)^3) ))) \\ G. C. Greubel, May 05 2019
(Magma) [Factorial(n)*(n*(n-5)/4 + HarmonicNumber(n)): n in [2..25]]; // G. C. Greubel, May 05 2019
(Sage) [factorial(n)*(n*(n-5)/4 + harmonic_number(n)) for n in (2..25)] # G. C. Greubel, May 05 2019

Crossrefs

Cf. A129178.

Sequence in context: A187252 A096233 A363985 * A057351 A350436 A359924

Adjacent sequences: A126670 A126671 A126672 * A126674 A126675 A126676

Keyword

nonn

Author

N. J. A. Sloane and Carlo Wood (carlo(AT)alinoe.com), Feb 13 2007

Status

approved