A297195 Number of bitriangular permutations (row sums of A272644 if that triangle is prefixed with two rows for n=0,1).

1, 0, 1, 2, 7, 28, 133, 726, 4483, 30896, 235105, 1957930, 17712799, 172980804, 1813760317, 20323234814, 242353047355, 3064550705752, 40958281206169, 576917769130578, 8541793624670551, 132623408805525740, 2154730841214003061, 36560670776303600422, 646697046042017004787

Offset: 0

Author

N. J. A. Sloane, Jan 10 2018

Keywords

• nonn
• easy

Comments

Define c(n) = Sum_{m=2..n-1} C(n-1, m-1)^(-2). Define b(1) = x and b(n+1) = b(n) + (Sum_{m=2..n-1} b(m)*b(n+1-m)*C(n-1, m-1)^(-2))/n^2 for n>0. Then b(n) is a polynomial in x and so is (b(n+1)-b(n))/x^2 whose constant term is c(n)/n^2. The Hone et.al.[2002] link denotes x with alpha_2 and alpha_k = (k-1)!^2*b(k). Conjecture: Asymptotic expansion of c(n) = 2*Sum_{i>1} a(i)/n^i. - Michael Somos, Oct 17 2024

Examples

			G.f. = 1 + x^2 + 2*x^3 + 7*x^4 + 28*x^5 + 133*x^6 + 726*x^7 + ... - _Michael Somos_, Oct 17 2024
		

Links

• A.N.W. Hone, N. Joshi and A.V. Kitaev, An Entire Function Defined by a Nonlinear Recurrence Relation, J. of the London Math. Soc., Oct. 2002, vol. 66, iss. 2, pp. 377-387.
• Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. The array is on page 267.
• Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]

Crossrefs

Cf. A272644.

Programs

Maple

A297195 := proc(n)
    add(A272644(n, m), m=0..n) ;
end proc:
seq(A297195(n), n=0..30) ; # R. J. Mathar, Mar 04 2018

Mathematica

A272644[n_, m_] := Sum[StirlingS2[m+1, i+1] (-1)^(m-i) i^(n-m) i!, {i, 0, m}];
a[n_] := If[n == 1, 1, Sum[A272644[n, m], {m, 1, n-1}]];
Array[a, 24] (* Jean-François Alcover, Apr 03 2020 *)

PARI

{a(n) = if(n<2, n==0, sum(m=1, n-1, sum(i=0, m, (-1)^(m-i)*i^(n-m)*i!*stirling(m+1, i+1, 2))))}; /* Michael Somos, Oct 17 2024 */

Extensions

Some terms corrected by Alois P. Heinz, Oct 17 2024