A006230 Bitriangular permutations.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A136301 (row 4), A371761 (row 3).

Programs

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 *)

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

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)