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A193229 A double factorial triangle.

%I A193229 #26 Apr 07 2019 09:06:07
%S A193229 1,1,1,3,3,2,15,15,12,6,105,105,90,60,24,945,945,840,630,360,120,
%T A193229 10395,10395,9450,7560,5040,2520,720,135135,135135,124740,103950,
%U A193229 75600,45360,20160,5040,2027025,2027025,1891890,1621620,1247400,831600,453600,181440,40320
%N A193229 A double factorial triangle.
%C A193229 The double factorial triangle coefficients are T(n,k), n >= 0 and 0 <= k <= n.
%C A193229 The T(n,0) equal the double factorial numbers A001147(n) = (2*n-1)!!.
%C A193229 The T(n,n) equal the factorial numbers A000142(n) = n!.
%C A193229 The row sums equal the double factorial numbers A000165(n) = (2*n)!!.
%C A193229 The Kn21(n) sums, see A180662 for the definition of these and other triangle sums, equal A130905(n) while the Kn2p(n) sums equal A130905(n+2*p-2) - (n+2*p-2)!*A010844(p-2)/A000165(p-2), p >= 2. - _Johannes W. Meijer_, Jul 21 2011
%H A193229 Seiichi Manyama, <a href="/A193229/b193229.txt">Rows n = 0..139, flattened</a>
%F A193229 T(n,k) = the (k+1)-th term in the top row of M^n, where M is an infinite square production matrix; M[i,j] = i, i >= 1 and 1 <= j <= i+1, and M[i,j] = 0, i >= 1 and j >= i+2, see the examples.
%F A193229 It appears that T(n,k) = (2*n-k)!/(2^(n-k)*(n-k)!) with conjectural e.g.f. 1/(x*(1-2*z) + (1-x)*sqrt(1-2*z)) = 1 + (1+x)*z + (3+3*x+2*x^2)*z^2/2! + .... Cf. A102625. - _Peter Bala_, Jul 09 2012
%e A193229 The first few rows of matrix M[i,j] are:
%e A193229   1, 1, 0, 0, 0, 0, ...
%e A193229   2, 2, 2, 0, 0, 0, ...
%e A193229   3, 3, 3, 3, 0, 0, ...
%e A193229   4, 4, 4, 4, 4, 0, ...
%e A193229   5, 5, 5, 5, 5, 5, ...
%e A193229 The first few rows of triangle T(n,k) are:
%e A193229        1;
%e A193229        1,      1;
%e A193229        3,      3,      2;
%e A193229       15,     15,     12,      6;
%e A193229      105,    105,     90,     60,    24;
%e A193229      945,    945,    840,    630,   360,   120;
%e A193229    10395,  10395,   9450,   7560,  5040,  2520,   720;
%e A193229   135135, 135135, 124740, 103950, 75600, 45360, 20160, 5040;
%p A193229 nmax:=7: M := Matrix(1..nmax+1,1..nmax+1): for i from 1 to nmax do for j from 1 to i+1 do M[i,j] := i od: od: for n from 0 to nmax do B := M^n: for k from 0 to n do T(n,k) := B[1,k+1] od: od: for n from 0 to nmax do seq(T(n,k),k=0..n) od: seq(seq(T(n,k),k=0..n),n=0..nmax); # _Johannes W. Meijer_, Jul 21 2011
%o A193229 (PARI) row(n)=(matrix(n,n,i,j,(i>j-2)*i)^(n-1))[1,]  \\ _M. F. Hasler_, Jul 24 2011
%Y A193229 Cf. A000142, A000165, A001147, A102625.
%Y A193229 Cf. A180662, A130905, A010844.
%Y A193229 T(2*n,n) gives A166334.
%K A193229 nonn,tabl
%O A193229 0,4
%A A193229 _Gary W. Adamson_, Jul 18 2011
%E A193229 Corrected, edited and extended by _Johannes W. Meijer_, Jul 21 2011
%E A193229 More terms from _Seiichi Manyama_, Apr 06 2019