This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A297195 #19 Oct 18 2024 20:18:11
%S A297195 1,0,1,2,7,28,133,726,4483,30896,235105,1957930,17712799,172980804,
%T A297195 1813760317,20323234814,242353047355,3064550705752,40958281206169,
%U A297195 576917769130578,8541793624670551,132623408805525740,2154730841214003061,36560670776303600422,646697046042017004787
%N A297195 Number of bitriangular permutations (row sums of A272644 if that triangle is prefixed with two rows for n=0,1).
%C A297195 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
%H A297195 A.N.W. Hone, N. Joshi and A.V. Kitaev, <a href="https://doi.org/10.1112/S0024610702003423">An Entire Function Defined by a Nonlinear Recurrence Relation</a>, J. of the London Math. Soc., Oct. 2002, vol. 66, iss. 2, pp. 377-387.
%H A297195 Irving Kaplansky and John Riordan, <a href="http://projecteuclid.org/euclid.dmj/1077473616">The problem of the rooks and its applications</a>, Duke Mathematical Journal 13.2 (1946): 259-268. The array is on page 267.
%H A297195 Irving Kaplansky and John Riordan, <a href="/A274105/a274105.pdf">The problem of the rooks and its applications</a>, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]
%e A297195 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
%p A297195 A297195 := proc(n)
%p A297195 add(A272644(n, m), m=0..n) ;
%p A297195 end proc:
%p A297195 seq(A297195(n), n=0..30) ; # _R. J. Mathar_, Mar 04 2018
%t A297195 A272644[n_, m_] := Sum[StirlingS2[m+1, i+1] (-1)^(m-i) i^(n-m) i!, {i, 0, m}];
%t A297195 a[n_] := If[n == 1, 1, Sum[A272644[n, m], {m, 1, n-1}]];
%t A297195 Array[a, 24] (* _Jean-François Alcover_, Apr 03 2020 *)
%o A297195 (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 */
%Y A297195 Cf. A272644.
%K A297195 nonn,easy
%O A297195 0,4
%A A297195 _N. J. A. Sloane_, Jan 10 2018
%E A297195 Some terms corrected by _Alois P. Heinz_, Oct 17 2024