A082137 - OEIS

%I #54 Feb 15 2021 02:00:54

%S 1,1,1,1,2,2,1,3,6,4,1,4,12,16,8,1,5,20,40,40,16,1,6,30,80,120,96,32,

%T 1,7,42,140,280,336,224,64,1,8,56,224,560,896,896,512,128,1,9,72,336,

%U 1008,2016,2688,2304,1152,256,1,10,90,480,1680,4032,6720,7680,5760,2560,512

%N Square array of transforms of binomial coefficients, read by antidiagonals.

%C Rows are associated with the expansions of (x^k/k!)exp(x)cosh(x) (leading zeros dropped). Rows include A011782, A057711, A080929, A082138, A080951, A082139, A082140, A082141. Columns are of the form 2^(k-1)C(n+k, k). Diagonals include A069723, A082143, A082144, A082145, A069720.

%C T(n, k) is also the number of idempotent order-preserving and order-decreasing partial transformations (of an n-chain) of width k (width(alpha)= |Dom(alpha)|). - _Abdullahi Umar_, Oct 02 2008

%C Read as a triangle this is A119468 with rows reversed. A119468 has e.g.f. exp(z*x)/(1-tanh(x)). - _Peter Luschny_, Aug 01 2012

%C Read as a triangle this is a subtriangle of A198793. - _Philippe Deléham_, Nov 10 2013

%H G. C. Greubel, <a href="/A082137/b082137.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Laradji, A. and Umar, <a href="http://dx.doi.org/10.1016/j.jalgebra.2003.10.023">A. Combinatorial results for semigroups of order-preserving partial transformations</a>, Journal of Algebra 278, (2004), 342-359.

%H Laradji, A. and Umar, A. <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Umar/um.html">Combinatorial results for semigroups of order-decreasing partial transformations</a>, J. Integer Seq. 7 (2004), 04.3.8.

%F Square array defined by T(n, k)=(2^(n-1)+0^n/2)C(n + k, n)= Sum{k=0..n, C(n+k, k+j)C(k+j, k)(1+(-1)^j)/2 }.

%F As an infinite lower triangular matrix, equals A007318 * A134309. - _Gary W. Adamson_, Oct 19 2007

%F O.g.f. for array read as a triangle: (1-x*(1+t))/((1-x)*(1-x*(1+2*t))) = 1 + x*(1+t) + x^2*(1+2*t+2*t^2) + x^3*(1+3*t+6*t^2+4*t^3) + .... - _Peter Bala_, Apr 26 2012

%F For array read as a triangle: T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) -2*T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Nov 10 2013

%e Rows begin

%e   1 1  2   4   8 ...

%e   1 2  6  16  40 ...

%e   1 3 12  40 120 ...

%e   1 4 20  80 280 ...

%e   1 5 30 140 560 ...

%e Read as a triangle, this begins:

%e   1

%e   1, 1

%e   1, 2,  2

%e   1, 3,  6,  4

%e   1, 4, 12, 16,   8

%e   1, 5, 20, 40,  40, 16

%e   1, 6, 30, 80, 120, 96, 32

%e   ... - _Philippe Deléham_, Nov 10 2013

%p # As a triangular array:

%p T := (n,k) -> 2^(k+0^k-1)*binomial(n,k):

%p for n from 0 to 9 do seq(T(n,k), k=0..n) od; # _Peter Luschny_, Nov 10 2017

%t rows = 11; t[n_, k_] := 2^(n-1)*(n+k)!/(n!*k!); t[0, _] = 1; tkn = Table[ t[n, k], {k, 0, rows}, {n, 0, rows}]; Flatten[ Table[ tkn[[ n-k+1, k ]], {n, 1, rows}, {k, 1, n}]] (* _Jean-François Alcover_, Jan 20 2012 *)

%o (Sage)

%o def A082137_row(n) : # as a triangular array

%o     var('z')

%o     s = (exp(z*x)/(1-tanh(x))).series(x,n+2)

%o     t = factorial(n)*s.coefficient(x,n)

%o     return [t.coefficient(z,n-k) for k in (0..n)]

%o for n in (0..7) : print(A082137_row(n))  # _Peter Luschny_, Aug 01 2012

%Y Cf. A119468, A007318, A134309.

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Apr 06 2003