

A196776


Triangle T(n,k) gives the number of ordered partitions of an n set into k oddsized blocks.


5



1, 0, 2, 1, 0, 6, 0, 8, 0, 24, 1, 0, 60, 0, 120, 0, 32, 0, 480, 0, 720, 1, 0, 546, 0, 4200, 0, 5040, 0, 128, 0, 8064, 0, 40320, 0, 40320, 1, 0, 4920, 0, 115920, 0, 423360, 0, 362880, 0, 512, 0, 130560, 0, 1693440, 0, 4838400, 0, 3628800
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OFFSET

1,3


COMMENTS

See A136630 for the case of unordered partitions into oddsized blocks. See A193474 for this triangle in row reverse form (but with an offset of 0).


LINKS

Table of n, a(n) for n=1..55.


FORMULA

T(n,k) = 1/(2^k)*sum {j = 0..k}(1)^(kj)*binomial(k,j)*(2*jk)^n.
Recurrence: T(n+2,k) = k^2*T(n,k) + k*(k1)*T(n,k2).
E.g.f.: x*sinh(t)/(1x*sinh(t)) = x*t + 2*x^2*t^2/2! + (x+6*x^3)*t^3/3! + (8*x^2+24*x^4)*t^4/4! + (x+60*x^3+120*x^5)*t^5/5! + ....
O.g.f. for column 2*k: (2*k)!*x^(2*k)/Product {j = 0..k} (1  (2*j)^2*x^2).
O.g.f. for column 2*k+1: (2*k+1)!*x^(2*k+1)/Product {j = 0..k} (1  (2*j+1)^2*x^2).
Let P denote Pascal's triangle A070318 and put M = 1/2*(PP^1). M is A162590 (see also A131047). Then the first column of (It*M)^1 (apart from the initial 1) lists the row polynomials for the present triangle.
nth row sum = A006154(n).
Row generating polynomials equal D^n(1/(1x*t)) evaluated at x = 0, where D is the operator sqrt(1+x^2)*d/dx. Cf. A136630.  Peter Bala, Dec 06 2011


EXAMPLE

Triangle begins
.n\k...1....2....3....4.....5....6.....7
= = = = = = = = = = = = = = = = = = = = =
..1....1
..2....0....2
..3....1....0....6
..4....0....8....0...24
..5....1....0...60....0...120
..6....0...32....0..480.....0..720
..7....1....0..546....0..4200....0..5040
...
T(4,2) = 8: The 8 ordered partitions of the set {1,2,3,4} into 2 oddsized blocks are {1}{2,3,4}, {2,3,4}{1}, {2}{1,3,4}, {1,3,4}{2}, {3}{1,2,4}, {1,2,4}{3}, {4}{1,2,3} and {1,2,3}{4}.
Example of recurrence relation: T(7,3) = 3^2*T(5,3) + 3*(31)*T(5,1) = 9*60 + 6*1 = 546.


CROSSREFS

Cf. A006154 (row sums), A136630, A162590, A193474 (row reverse).
Sequence in context: A299198 A137477 A181297 * A336345 A157982 A119275
Adjacent sequences: A196773 A196774 A196775 * A196777 A196778 A196779


KEYWORD

nonn,easy,tabl


AUTHOR

Peter Bala, Oct 06 2011


STATUS

approved



