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%I #27 Jun 19 2018 12:31:08
%S 1,-4,1,20,-9,1,-120,74,-15,1,840,-638,179,-22,1,-6720,5944,-2070,355,
%T -30,1,60480,-60216,24574,-5265,625,-39,1,-604800,662640,-305956,
%U 77224,-11515,1015,-49,1,6652800,-7893840,4028156,-1155420,203889
%N Generalized Stirling number triangle of first kind.
%C a(n,m)= ^4P_n^m in the notation of the given reference with a(0,0) := 1.
%C The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(4+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.
%C In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(4*t),exp(t)-1).
%C See A143493 for the unsigned version of this array and A143496 for the inverse. - _Peter Bala_, Aug 25 2008
%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="/A049459/b049459.txt">Rows n = 0..125 of triangle, flattened</a>
%F a(n, m)= a(n-1, m-1) - (n+3)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n<m; a(n, -1) := 0, a(0, 0)=1. E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^4).
%F Triangle (signed) = [ -4, -1, -5, -2, -6, -3, -7, -4, -8, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [4, 1, 5, 2, 6, 3, 7, 4, 8, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938 (unsigned version in A143493).
%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,4), for n=1,2,...;i=0...n. - _Milan Janjic_, Dec 21 2008
%e 1;
%e -4, 1;
%e 20, -9, 1;
%e -120, 74, -15, 1;
%e 840, -638, 179, -22, 1;
%p A049459_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x+4, n)), x, k), k=0..n): seq(print(A049459_row(n)),n=0..8); # _Peter Luschny_, May 16 2013
%t a[n_, m_] /; 0 <= m <= n := a[n, m] = a[n-1, m-1] - (n+3)*a[n-1, m];
%t a[n_, m_] /; n < m = 0;
%t a[_, -1] = 0; a[0, 0] = 1;
%t Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 19 2018 *)
%o (Haskell)
%o a049459 n k = a049459_tabl !! n !! k
%o a049459_row n = a049459_tabl !! n
%o a049459_tabl = map fst $ iterate (\(row, i) ->
%o (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 4)
%o -- _Reinhard Zumkeller_, Mar 11 2014
%Y Unsigned column sequences are: A001715-A001719. Cf. A008275 (Stirling1 triangle, A049458, A049460. Row sums (signed triangle): A001710(n+2)*(-1)^n. Row sums (unsigned triangle): A001720(n+4).
%Y Cf. A000035 A084938.
%Y A143493, A143496. - _Peter Bala_, Aug 25 2008
%K sign,easy,tabl
%O 0,2
%A _Wolfdieter Lang_
%E Second formula corrected by _Philippe Deléham_, Nov 09 2008
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