A051231 Generalized Stirling number triangle of the first kind.

1, -9, 1, 162, -27, 1, -4374, 891, -54, 1, 157464, -36450, 2835, -90, 1, -7085880, 1797714, -164025, 6885, -135, 1, 382637520, -104162436, 10655064, -535815, 14175, -189, 1, -24106163760, 6944870988, -775431468, 44411409, -1428840, 26082, -252, 1

Offset: 1

Author

Wolfdieter Lang

Keywords

• sign
• easy
• tabl

Comments

T(n,m) = R_n^m(a=0, b=9) in the notation of the given 1962 reference.

T(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} T(n,m)*x^m = Product_{j=0..n-1} (x - 9*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

From Petros Hadjicostas, Jun 06 2020: (Start)

For nonnegative integers n, m and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) using slightly different notation. They were further examined by Mitrinovic and Mitrinovic (1962). Special cases were tabulated in this and other related papers.

Special cases of these numbers are related to numbers introduced by Nörlund (1924).

These numbers are defined via the g.f. Product_{r=0..n-1} (x - (a + b*r)) = Sum_{m=0..n} R_n^m(a,b)*x^m for n >= 0. As a result, R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b) for n >= m >= 1 with R_1^0(a,b) = a, R_1^1(a,b) = 1, and R_n^m(a,b) = 0 for n < m. (Because an empty product is by definition 1, we may let R_0^0(a,b) = 1.)

With a = 0 and b = 1, we get the Stirling numbers of the first kind S1(n,m) = R_n^m(a=0, b=1) = A048994(n,m) which satisfy Product_{r=0}^{n-1} (x - r) = Sum_{m=0..n} S1(n,m)*x^m with S1(n,n) = 1 for n >= 0, S1(n,0) = 0 for n >= 1, and S1(n, m) = 0 for m > n. (Array A008275 is the same as array A048994 but with no zero row and no zero column.)

We have R_n^m(a,b) = Sum_{k=0}^{n-m} (-1)^k * a^k * b^(n-m-k) * binomial(m+k, k) * S1(n, m+k) for n >= m >= 0.

For the current array, T(n,m) = R_n^m(a=0, b=9) but with no zero row or column. (End)

Examples

			Triangle T(n,m) (with rows n >= 1 and columns m = 1..n) begins:
         1;
        -9,       1;
       162,     -27,       1;
     -4374,     891,     -54,    1;
    157464,  -36450,    2835,  -90,    1;
  -7085880, 1797714, -164025, 6885, -135, 1;
   ...
3rd row o.g.f.: E(3,x) = Product_{j=0..2} (x-9*j) = 162*x - 27*x^2 + x^3. [Edited by _Petros Hadjicostas_, Jun 06 2020]
		

Links

• D. S. Mitrinovic, Sur une relation de récurrence relative aux nombres de Bernoulli, Comptes rendus de l'Académie des sciences de Paris, t. 250 (1960), 4266-4267.
• D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356.
• D. S. Mitrinovic and R. S. Mitrinovic, Sur les polynômes de Stirling, Bulletin de la Société des mathématiciens et physiciens de la R. P. de Serbie, t. 10 (1958), 43-49.
• D. S. Mitrinovic and R. S. Mitrinovic, Sur les nombres de Stirling et les nombres de Bernoulli d'ordre supérieux, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 43 (1960), 1-63.
• D. S. Mitrinovic and R. S. Mitrinovic, Sur une classe de nombres se rattachant aux nombres de Stirling--Appendice: Table des nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 60 (1961), 1-15 and 17-62.
• D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77.
• D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77 [jstor stable version].
• D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling. V, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 132/142 (1965), 1-22.
• Niels Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, 1924.
• Wikipedia, Dragoslav Mitrinovic.

Crossrefs

First (m=1) column sequence is A051232(n-1).

Row sums (signed triangle): A049211(n-1)*(-1)^(n-1).

Row sums (unsigned triangle): A045756(n).

Cf. A008275 (b=1 triangle), A048994 (b=1 triangle), A051187 (b=8 triangle).

Formula

T(n, m) = T(n-1, m-1) - 9*(n-1)*T(n-1, m), n >= m >= 1; T(n, m) := 0, n < m; T(n, 0) := 0 for n >= 1; T(0, 0) = 1.

E.g.f. for the m-th column of the signed triangle: (log(1 + 9*x)/9)^m/m!.

From Petros Hadjicostas, Jun 07 2020: (Start)

T(n,m) = 9^(n-m)*Stirling1(n,m) = 9^(n-m)*A048994(n,m) = 9^(n-m)*A008275(n,m) for n >= m >= 1.

Bivariate e.g.f.-o.g.f.: Sum_{n,m >= 1} T(n,m)*x^n*y^m/n! = exp((y/9)*log(1 + 9*x)) - 1 = (1 + 9*x)^(y/9) - 1. (End)