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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

%I #15 Mar 24 2017 00:47:53

%S 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,

%T 5040,0,128,0,8064,0,40320,0,40320,1,0,4920,0,115920,0,423360,0,

%U 362880,0,512,0,130560,0,1693440,0,4838400,0,3628800

%N Triangle T(n,k) gives the number of ordered partitions of an n set into k odd-sized blocks.

%C 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).

%F T(n,k) = 1/(2^k)*sum {j = 0..k}(-1)^(k-j)*binomial(k,j)*(2*j-k)^n.

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

%F 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! + ....

%F O.g.f. for column 2*k: (2*k)!*x^(2*k)/Product {j = 0..k} (1 - (2*j)^2*x^2).

%F 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).

%F 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.

%F n-th row sum = A006154(n).

%F 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

%e Triangle begins

%e .n\k.|..1....2....3....4.....5....6.....7

%e = = = = = = = = = = = = = = = = = = = = =

%e ..1..|..1

%e ..2..|..0....2

%e ..3..|..1....0....6

%e ..4..|..0....8....0...24

%e ..5..|..1....0...60....0...120

%e ..6..|..0...32....0..480.....0..720

%e ..7..|..1....0..546....0..4200....0..5040

%e ...

%e 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}.

%e Example of recurrence relation: T(7,3) = 3^2*T(5,3) + 3*(3-1)*T(5,1) = 9*60 + 6*1 = 546.

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

%K nonn,easy,tabl

%O 1,3

%A _Peter Bala_, Oct 06 2011

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Last modified March 28 12:26 EDT 2024. Contains 371254 sequences. (Running on oeis4.)