Search: id:A051339
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%I A051339
%S A051339 1,7,1,56,15,1,504,191,24,1,5040,2414,431,34,1,55440,31594,
%T A051339 7155,805,45,1,665280,434568,117454,16815,1345,57,1,8648640,
%U A051339 6314664,1961470,336049,34300,2086,70,1,121080960,97053936,33775244,6666156,
               816249,63504,3066,84,1
%V A051339 1,-7,1,56,-15,1,-504,191,-24,1,5040,-2414,431,-34,1,-55440,31594,
%W A051339 -7155,805,-45,1,665280,-434568,117454,-16815,1345,-57,1,-8648640,
%X A051339 6314664,-1961470,336049,-34300,2086,-70,1,121080960,-97053936,33775244,
               -6666156,816249,-63504,3066,-84,1
%N A051339 Generalized Stirling number triangle of first kind.
%C A051339 a(n,m)= ^7P_n^m in the notation of the given reference with a(0,0) := 
               1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which 
               are s(n,x)= product(x-(7+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy 
               s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 
               polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1. 
               In the umbral calculus (see the S. Roman reference given in A048854) 
               the s(n,x) polynomials are called Sheffer for (exp(7*t),exp(t)-1).
%D A051339 Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres 
               relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. 
               Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
%F A051339 a(n, m)= a(n-1, m-1) - (n+6)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n<m; 
               a(n, -1) := 0, a(0, 0)=1.
%F A051339 E.g.f. for m-th column of signed triangle: ((ln(1+x))^m)/(m!*(1+x)^7).
%F A051339 Triangle (signed) = [ -7, -1, -8, -2, -9, -3, -10, -4, -11, -5, ...] 
               DELTA A000035; triangle (unsigned) = [7, 1, 8, 2, 9, 3, 10, 4, ...] 
               DELTA A000035; where DELTA is Deleham's operator defined in A084938.
%F A051339 If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,
               j=0..k-1),k=0..n-i), then T(n,i) = f(n,i,7), for n=1,2,...;i=0...n. 
               [From Milan R. Janjic (agnus(AT)blic.net), Dec 21 2008]
%e A051339 {1}; {-7,1}; {56,-15,1}; {-504,191,-24,1}; ... s(2,x)= 56-15*x+x^2; S1(2,
               x)= -x+x^2 (Stirling1).
%Y A051339 The first (m=0) column sequence is A001730. Row sums (signed triangle): 
               A001725(n+5)*(-1)^n. Row sums (unsigned triangle): A049388(n).
%Y A051339 Cf. A000035 A084938.
%Y A051339 Sequence in context: A075502 A052104 A144450 this_sequence A134141 A110788 
               A100254
%Y A051339 Adjacent sequences: A051336 A051337 A051338 this_sequence A051340 A051341 
               A051342
%K A051339 sign,easy,tabl
%O A051339 0,2
%A A051339 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

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