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%I #23 Mar 12 2014 06:29:40
%S 1,-6,1,42,-13,1,-336,146,-21,1,3024,-1650,335,-30,1,-30240,19524,
%T -5000,635,-40,1,332640,-245004,74524,-11985,1075,-51,1,-3991680,
%U 3272688,-1139292,218344,-24885,1687,-63,1,51891840,-46536624,18083484,-3977764,541849,-46816,2506,-76,1
%N Generalized Stirling number triangle of first kind.
%C a(n,m)= ^6P_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-(6+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(6*t),exp(t)-1).
%D 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.
%H Reinhard Zumkeller, <a href="/A051338/b051338.txt">Rows n = 0..125 of triangle, flattened</a>
%F a(n, m)= a(n-1, m-1) - (n+5)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n<m; a(n, -1) := 0, a(0, 0)=1.
%F E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^6).
%F Triangle (signed) = [ -6, -1, -7, -2, -8, -3, -9, -4, -10, ...] DELTA A000035; triangle (unsigned) = [6, 1, 7, 2, 8, 3, 9, 4, 10, 5, 11, ...] DELTA A000035; where DELTA is Deléham's operator defined in A084938.
%F 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,6), for n=1,2,...;i=0...n. - _Milan Janjic_, Dec 21 2008
%e {1}; {-6,1}; {42,-13,1}; {-336,146,-21,1}; ... s(2,x)= 42-13*x+x^2; S1(2,x)= -x+x^2 (Stirling1).
%t t[n_, i_] = Sum[(-1)^k*Binomial[n, k]*Pochhammer[6, k]*StirlingS1[n - k, i], {k, 0, n - i}]; Flatten[Table[t[n, i], {n, 0, 8}, {i, 0, n}]][[1 ;; 45]] (* _Jean-François Alcover_, Jun 01 2011, after _Milan Janjic_ *)
%o (Haskell)
%o a051338 n k = a051338_tabl !! n !! k
%o a051338_row n = a051338_tabl !! n
%o a051338_tabl = map fst $ iterate (\(row, i) ->
%o (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 6)
%o -- _Reinhard Zumkeller_, Mar 11 2014
%Y Unsigned m=0 column sequence is: A001725. Row sums (signed triangle): A001720(n+4)*(-1)^n. Row sums (unsigned triangle): A001730(n+6).
%Y Cf. A000035 A084938.
%K sign,easy,tabl
%O 0,2
%A _Wolfdieter Lang_
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