A051523 Generalized Stirling number triangle of first kind.

1, -10, 1, 110, -21, 1, -1320, 362, -33, 1, 17160, -6026, 791, -46, 1, -240240, 101524, -17100, 1435, -60, 1, 3603600, -1763100, 358024, -38625, 2335, -75, 1, -57657600, 31813200, -7491484, 976024, -75985, 3535, -91, 1, 980179200, -598482000, 159168428, -24083892, 2267769, -136080, 5082, -108, 1

Offset: 0

Author

Wolfdieter Lang

Keywords

• sign
• easy
• tabl

Comments

a(n,m)= ^10P_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-(10+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(10*t),exp(t)-1).

Examples

			{1}; {-10,1}; {110,-21,1}; {-1320,362,-331}; ... s(2,x)= 110-21*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

The first (m=0) unsigned column sequence is A049398. Row sums (signed triangle): A049389(n)*(-1)^n. Row sums (unsigned triangle): A051431(n).

See also A049444 A049458 A049459 A049460 A051338 A051339 A051379 A051380

Cf. A000035 A084938.

Programs

Haskell

a051523 n k = a051523_tabl !! n !! k
a051523_row n = a051523_tabl !! n
a051523_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 10)
-- Reinhard Zumkeller, Mar 12 2014

Mathematica

a[n_, m_] := Pochhammer[m + 1, n - m] SeriesCoefficient[Log[1 + x]^m/(1 + x)^10, {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+9)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n

E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^10).

Triangle (signed) = [ -10, -1, -11, -2, -12, -3, -13, -14, -4, ...] DELTA A000035; triangle (unsigned) = [10, 1, 11, 2, 12, 3, 13, 4, 14, 5, 15, ...] DELTA A000035; 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,10), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

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