Search: id:A049403
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%I A049403
%S A049403 1,1,1,0,3,1,0,3,6,1,0,0,15,10,1,0,0,15,45,15,1,0,0,0,105,105,21,1,0,0,
%T A049403 0,105,420,210,28,1,0,0,0,0,945,1260,378,36,1,0,0,0,0,945,4725,3150,
%U A049403 630,45,1,0,0,0,0,0,10395,17325,6930,990,55,1,0,0,0,0,0,10395,62370
%N A049403 A triangle of numbers related to triangle A030528.
%C A049403 a(n,1)= A019590(n)= A008279(1,n). a(n,m)=: S1(-1; n,m), a member of a 
               sequence of lower triangular Jabotinsky matrices, including S1(1; 
               n,m)= A008275 (signed Stirling first kind), S1(2; n,m)= A008297(n,
               m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix 
               ((-1)^(n-m))*A001497(n-1,m-1) (signed Bessel triangle). The monic 
               row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are 
               exponential convolution polynomials (see A039692 for the definition 
               and a Knuth reference).
%C A049403 Exponential Riordan array [1+x,x(1+x/2)]. T(n,k)=A001498(k+1,n-k). [From 
               Paul Barry (pbarry(AT)wit.ie), Jan 15 2009]
%H A049403 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               On generalizations of Stirling number triangles</a>, J. Integer Seqs., 
               Vol. 3 (2000), #00.2.4.
%H A049403 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A049403.text">
               First 10 rows of the array and more.</a> [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.d\
               e), Oct 17 2008]
%F A049403 a(n, m) = n!*A030528(n, m)/(m!*2^(n-m)); a(n, m) = (2*m-n+1)*a(n-1, m) 
               + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n<m; a(n, 0) := 0; a(1, 1)=1. 
               E.g.f. for m-th column: ((x*(1+x/2))^m)/m!.
%e A049403 {1}; {1,1}; {0,3,1}; {0,3,6,1}; ... E.g. row polynomial E(3,x)= 3*x^2+x^3.
%Y A049403 Cf. A000085 (row sums).
%Y A049403 Sequence in context: A099546 A036870 A036874 this_sequence A104556 A116089 
               A122016
%Y A049403 Adjacent sequences: A049400 A049401 A049402 this_sequence A049404 A049405 
               A049406
%K A049403 easy,nonn,tabl
%O A049403 1,5
%A A049403 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

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