A006230 Bitriangular permutations. (Formerly M4902)

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

Colin Barker, Table of n, a(n) for n = 4..1000

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

J. Riordan, Letter to N. J. A. Sloane, Dec. 1976

Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

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

Adjacent sequences: A006227 A006228 A006229 * A006231 A006232 A006233

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved