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%I M4902 #39 Apr 12 2024 14:00:06
%S 1,13,73,301,1081,3613,11593,36301,111961,342013,1038313,3139501,
%T 9467641,28501213,85700233,257493901,773268121,2321377213,6967277353,
%U 20908123501,62736953401,188236026013,564758409673,1694375892301,5083329003481,15250389663613
%N Bitriangular permutations.
%C Prepending the term 0 and setting the offset to 0 makes this sequence row 3 of A371761. In this form it can be generated by the Akiyama-Tanigawa algorithm for powers (see the Python script). - _Peter Luschny_, Apr 12 2024
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Colin Barker, <a href="/A006230/b006230.txt">Table of n, a(n) for n = 4..1000</a>
%H 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.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H J. Riordan, <a href="/A006230/a006230.pdf">Letter to N. J. A. Sloane, Dec. 1976</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).
%F a(n) = 12*S(n-2) + 1, with S(n)=A000392(n) the Stirling numbers of second kind, 3rd column. - _Ralf Stephan_, Jul 07 2003
%F a(n+3) = Sum_{i=1..3} A008277(n,i) * A008277(3,i) * i!^2. - Brian Parsonnet, Feb 25 2011
%F From _Colin Barker_, Dec 27 2017: (Start)
%F G.f.: x^4*(1 + x)*(1 + 6*x) / ((1 - x)*(1 - 2*x)*(1 - 3*x)).
%F a(n) = 12*(3 - 3*2^(n-2) + 3^(n-2))/6 + 1.
%F a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>6
%F (End)
%p A006230:=-(z+1)*(6*z+1)/(z-1)/(3*z-1)/(2*z-1); # Conjectured by _Simon Plouffe_ in his 1992 dissertation.
%t 12*StirlingS2[n+1, 3]+1; (* Brian Parsonnet, Feb 25 2011 *)
%t Sum[ StirlingS2[n,i] * StirlingS2[ 3,i ] * i!^2, {i,3} ]; (* alternative, Brian Parsonnet, Feb 25 2011 *)
%o (PARI)
%o Vec(x^4*(1 + x)*(1 + 6*x) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^40))
%o \\ _Colin Barker_, Dec 27 2017
%o (Python) # Using the Akiyama-Tanigawa algorithm for powers from A371761.
%o print(ATPowList(3, 27)) # _Peter Luschny_, Apr 12 2024
%Y Cf. A136301 (row 4), A371761 (row 3).
%K nonn,easy,changed
%O 4,2
%A _N. J. A. Sloane_
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