A182541 Coefficients in g.f. for certain marked mesh patterns.

1, 4, 19, 107, 702, 5274, 44712, 422568, 4407120, 50292720, 623471040, 8344624320, 119938250880, 1842662908800, 30136443724800, 522780938265600, 9587900602828800, 185371298306611200, 3768248516336640000, 80349669847157760000, 1793238207723325440000, 41806479141525288960000

Offset: 3

Author

N. J. A. Sloane, May 04 2012

Keywords

• nonn

Comments

See Kitaev and Remmel for precise definition.

The listed terms a(3)-a(10) of this sequence can be produced by the formula (n-1)!*(H(n-1)-1/2)/2, where H(n) = A001008(n)/A002805(n) is the n-th harmonic number. - Gary Detlefs, May 28 2012

a(n) is also the number of nonzero elements left in the matrix where all the rows consist of permutations of 11...n after we delete for each element with the value of 'k' k elements of this type, and repeat this operation until no more elements with the value of k can be deleted. The whole operation should be done for all the values of k from 1 to n. - Anton Zakharov, Jun 28 2016

Examples

			a(1) corresponds to the 1 X 2 matrix 11 -> 1 element is left and there are no more ones to delete => n(1) = 1. a(2) corresponds to the 3 X 3 matrix 112 121 211 -> 102 120 210 -> 102 100 010 only 4 nonzero elements are left and a(2) = 4 = 3 + 3/3. a(3) = 12 + 12/3 + 12/4 = 19 = 19 nonzero elements left in the 4 X 12 matrix after the deletion for each element with the value of 1 one element with the value of 1, for every element with the value of 2 - two elements with the value of 2 and for each element with the value of 3 - three elements with the value of 3). - _Anton Zakharov_, Jun 28 2016
		

Links

• Joerg Arndt, Table of n, a(n) for n = 3..101
• Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, arXiv preprint arXiv:1201.1323 [math.CO], 2012.
• Anton Zakharov, Matrix-related sequences

Crossrefs

Cf. A001710, A307642.

Programs

Mathematica

Table[Numerator[(n+1)!/2] *(1 + Sum[1/(k+1), {k, 2, n}]),{n, 1, 22}] (* Indranil Ghosh, Mar 12 2017 *)

PARI

for(n=1, 22, print1(numerator((n + 1)!/2) * (1 + sum(k=2, n, 1/(k+1))),", ")) \\ Indranil Ghosh, Mar 12 2017

Formula

a(n) = A001710(n+1) * (1 + Sum_{k=2..n} 1/(k+1) ). - Anton Zakharov, Jun 28 2016

a(n) ~ sqrt(Pi/2)*exp(-n)*n^(n-1/2)*log(n). - Ilya Gutkovskiy, Jul 12 2016

From Pedro Caceres, Apr 19 2019: (Start)

a(n) = (n-3)! * Sum_{i=1..n-2} (Sum_{j=1..i} (i/j)).

a(n) = (1/4) * (n-1)! * (2*harmonic(n-1)-1). (End)

a(n) = (-(n-1)! + 2 * |Stirling1(n,2)|)/4. - Seiichi Manyama, Sep 05 2024

Extensions

More terms from Anton Zakharov, Jun 28 2016