cp's OEIS Frontend

A082137 Square array of transforms of binomial coefficients, read by antidiagonals.

%I A082137 #54 Feb 15 2021 02:00:54
%S A082137 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 A082137 1,7,42,140,280,336,224,64,1,8,56,224,560,896,896,512,128,1,9,72,336,
%U A082137 1008,2016,2688,2304,1152,256,1,10,90,480,1680,4032,6720,7680,5760,2560,512
%N A082137 Square array of transforms of binomial coefficients, read by antidiagonals.
%C A082137 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 A082137 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 A082137 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 A082137 Read as a triangle this is a subtriangle of A198793. - _Philippe Deléham_, Nov 10 2013
%H A082137 G. C. Greubel, <a href="/A082137/b082137.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A082137 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 A082137 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 A082137 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 A082137 As an infinite lower triangular matrix, equals A007318 * A134309. - _Gary W. Adamson_, Oct 19 2007
%F A082137 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 A082137 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 A082137 Rows begin
%e A082137   1 1  2   4   8 ...
%e A082137   1 2  6  16  40 ...
%e A082137   1 3 12  40 120 ...
%e A082137   1 4 20  80 280 ...
%e A082137   1 5 30 140 560 ...
%e A082137 Read as a triangle, this begins:
%e A082137   1
%e A082137   1, 1
%e A082137   1, 2,  2
%e A082137   1, 3,  6,  4
%e A082137   1, 4, 12, 16,   8
%e A082137   1, 5, 20, 40,  40, 16
%e A082137   1, 6, 30, 80, 120, 96, 32
%e A082137   ... - _Philippe Deléham_, Nov 10 2013
%p A082137 # As a triangular array:
%p A082137 T := (n,k) -> 2^(k+0^k-1)*binomial(n,k):
%p A082137 for n from 0 to 9 do seq(T(n,k), k=0..n) od; # _Peter Luschny_, Nov 10 2017
%t A082137 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 A082137 (Sage)
%o A082137 def A082137_row(n) : # as a triangular array
%o A082137     var('z')
%o A082137     s = (exp(z*x)/(1-tanh(x))).series(x,n+2)
%o A082137     t = factorial(n)*s.coefficient(x,n)
%o A082137     return [t.coefficient(z,n-k) for k in (0..n)]
%o A082137 for n in (0..7) : print(A082137_row(n))  # _Peter Luschny_, Aug 01 2012
%Y A082137 Cf. A119468, A007318, A134309.
%K A082137 easy,nonn,tabl
%O A082137 0,5
%A A082137 _Paul Barry_, Apr 06 2003