VOOZH about

URL: https://oeis.org/A006230

⇱ A006230 - OEIS

login
A006230
Bitriangular permutations.
4
1, 13, 73, 301, 1081, 3613, 11593, 36301, 111961, 342013, 1038313, 3139501, 9467641, 28501213, 85700233, 257493901, 773268121, 2321377213, 6967277353, 20908123501, 62736953401, 188236026013, 564758409673, 1694375892301, 5083329003481, 15250389663613
OFFSET
4,2
COMMENTS
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
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
FORMULA
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
a(n+3) = Sum_{i=1..3} A008277(n,i) * A008277(3,i) * i!^2. - Brian Parsonnet, Feb 25 2011
From Colin Barker, Dec 27 2017: (Start)
G.f.: x^4*(1 + x)*(1 + 6*x) / ((1 - x)*(1 - 2*x)*(1 - 3*x)).
a(n) = 12*(3 - 3*2^(n-2) + 3^(n-2))/6 + 1.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>6. (End)
MAPLE
A006230:=-(z+1)*(6*z+1)/(z-1)/(3*z-1)/(2*z-1); # Conjectured by Simon Plouffe in his 1992 dissertation.
MATHEMATICA
12*StirlingS2[n+1, 3]+1; (* Brian Parsonnet, Feb 25 2011 *)
Sum[ StirlingS2[n, i] * StirlingS2[ 3, i ] * i!^2, {i, 3} ]; (* alternative, Brian Parsonnet, Feb 25 2011 *)
PROG
(PARI)
Vec(x^4*(1 + x)*(1 + 6*x) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^40))
\\ Colin Barker, Dec 27 2017
(Python) # Using the Akiyama-Tanigawa algorithm for powers from A371761.
print(ATPowList(3, 27)) # Peter Luschny, Apr 12 2024
CROSSREFS
Cf. A136301 (row 4), A371761 (row 3).
Sequence in context: A081586 A143008 A107963 * A066110 A020527 A146618
KEYWORD
nonn,easy
STATUS
approved