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A196776 Triangle T(n,k) gives the number of ordered partitions of an n set into k odd-sized 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

See A136630 for the case of unordered partitions into odd-sized 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)^(k-j)*binomial(k,j)*(2*j-k)^n.

Recurrence: T(n+2,k) = k^2*T(n,k) + k*(k-1)*T(n,k-2).

E.g.f.: x*sinh(t)/(1-x*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*(P-P^-1). M is A162590 (see also A131047). Then the first column of (I-t*M)^-1 (apart from the initial 1) lists the row polynomials for the present triangle.

n-th row sum = A006154(n).

Row generating polynomials equal D^n(1/(1-x*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 odd-sized 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*(3-1)*T(5,1) = 9*60 + 6*1 = 546.

CROSSREFS

Cf. A006154 (row sums), A136630, A162590, A193474 (row reverse).

Sequence in context: A266904 A137477 A181297 * A157982 A119275 A129462

Adjacent sequences:  A196773 A196774 A196775 * A196777 A196778 A196779

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Bala, Oct 06 2011

STATUS

approved

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Last modified February 25 16:14 EST 2018. Contains 299653 sequences. (Running on oeis4.)