A049460 Generalized Stirling number triangle of first kind.
1, -5, 1, 30, -11, 1, -210, 107, -18, 1, 1680, -1066, 251, -26, 1, -15120, 11274, -3325, 485, -35, 1, 151200, -127860, 44524, -8175, 835, -45, 1, -1663200, 1557660, -617624, 134449, -17360, 1330, -56, 1, 19958400, -20355120, 8969148, -2231012, 342769, -33320, 2002, -68, 1
Offset: 0
Examples
{1}; {-5,1}; {30,-11,1}; {-210,107,-18,1}; ... s(2,x)= 30-11*x+x^2; S1(2,x)= -x+x^2 (Stirling1).
Links
- Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
- D. S. Mitrinovic, M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).
Crossrefs
Programs
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Haskell
a049460 n k = a049460_tabl !! n !! k a049460_row n = a049460_tabl !! n a049460_tabl = map fst $ iterate (\(row, i) -> (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 5) -- Reinhard Zumkeller, Mar 11 2014
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Mathematica
a[n_, m_] := Pochhammer[m+1, n-m] SeriesCoefficient[Log[1+x]^m/(1+x)^5, {x, 0, n}]; Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)
Formula
a(n, m)= a(n-1, m-1) - (n+4)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n
Triangle (signed) = [ -5, -1, -6, -2, -7, -3, -8, -4, -9, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, ...]; triangle (unsigned) = [5, 1, 6, 2, 7, 3, 8, 4, 9, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938.
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,5), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008
Extensions
Second formula corrected by Philippe Deléham, Nov 10 2008
A325137 Triangle T(n, k) = [x^n] (n + k + x)!/(k + x)! for 0 <= k <= n, read by rows.
1, 1, 1, 2, 5, 1, 6, 26, 12, 1, 24, 154, 119, 22, 1, 120, 1044, 1175, 355, 35, 1, 720, 8028, 12154, 5265, 835, 51, 1, 5040, 69264, 133938, 77224, 17360, 1687, 70, 1, 40320, 663696, 1580508, 1155420, 342769, 46816, 3066, 92, 1
Offset: 0
Comments
Sister triangle of A307419.
Examples
Triangle starts: [0] 1 [1] 1, 1 [2] 2, 5, 1 [3] 6, 26, 12, 1 [4] 24, 154, 119, 22, 1 [5] 120, 1044, 1175, 355, 35, 1 [6] 720, 8028, 12154, 5265, 835, 51, 1 [7] 5040, 69264, 133938, 77224, 17360, 1687, 70, 1 [8] 40320, 663696, 1580508, 1155420, 342769, 46816, 3066, 92, 1 [9] 362880, 6999840, 19978308, 17893196, 6687009, 1197273, 109494, 5154, 117, 1 A000142, A001705, A001712, A001718, A001724, ...
Programs
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Maple
T := (n, k) -> add(binomial(j+k, k)*(k+1)^j*abs(Stirling1(n, j+k)), j=0..n-k); seq(seq(T(n,k), k=0..n), n=0..8); # Note that for n > 16 Maple fails (at least in some versions) to compute the # terms properly. Inserting 'simplify' or numerical evaluation might help. A325137Row := proc(n) local ogf, ser; ogf := (n, k) -> (n+k+x)!/(k+x)!; ser := (n, k) -> series(ogf(n,k),x,k+2); seq(coeff(ser(n,k),x,k), k=0..n) end: seq(A325137Row(n), n=0..8);
Formula
T(n, k) = Sum_{j=0..n-k} binomial(j+k, k)*|Stirling1(n, j+k)|*(k+1)^j.
Comments