A051186 Generalized Stirling number triangle of first kind.

1, -7, 1, 98, -21, 1, -2058, 539, -42, 1, 57624, -17150, 1715, -70, 1, -2016840, 657874, -77175, 4165, -105, 1, 84707280, -29647548, 3899224, -252105, 8575, -147, 1, -4150656720, 1537437132, -220709524, 16252369, -672280, 15778, -196, 1

Offset: 1

Author

Wolfdieter Lang

Keywords

• sign
• easy
• tabl

Comments

T(n,m) = R_n^m(a=0, b=7) 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-7*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 07 2020: (Start)

For integers n, m >= 0 and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) and further examined by Mitrinovic and Mitrinovic (1962).

They are defined via 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, R_n^m(a,b) = 0 for n < m, and 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).

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=7) but with no zero row or column. (End)

Examples

			Triangle T(n,m) (with rows n >= 1 and columns m = 1..n) begins:
         1;
        -7,      1;
        98,    -21,      1;
     -2058,    539,    -42,    1;
     57624, -17150,   1715,  -70,    1;
  -2016840, 657874, -77175, 4165, -105, 1;
  ...
3rd row o.g.f.: E(3,x) = Product_{j=0..2} (x - 7*j) = 98*x - 21*x^2 + x^3.
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened
• Wolfdieter Lang, First ten rows.
• 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 nombres de Stirling et les nombres de Bernoulli d'ordre supérieur, 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].

Crossrefs

Cf. A000142, A045754 (unsigned row sums), A049209 (row sums), A051188.

The b=1..6 triangles are: A008275 (Stirling1 triangle), A039683, A051141, A051142, A051150, A051151.

Programs

Magma

[7^(n-k)*StirlingFirst(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 22 2022

Mathematica

Table[7^(n-k)*StirlingS1[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Feb 22 2022 *)

Sage

flatten([[(-7)^(n-k)*stirling_number1(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 22 2022

Formula

T(n, m) = T(n-1, m-1) - 7*(n-1)*T(n-1, m) for n >= m >= 1, T(n, m) = 0 for n < m, T(n, 0) = 0 for n >= 1, and T(0, 0) = 1.

T(n, 1) = A051188(n-1).

Sum_{k=0..n} T(n, k) = (-1)^(n-1)*A049209(n-1).

Sum_{k=0..n} (-1)^(n-k)*T(n, k) = A045754(n).

E.g.f. for m-th column of signed triangle: (log(1 + 7*x)/7)^m/m!.

T(n,m) = 7^(n-m)*S1(n,m) with the (signed) Stirling1 triangle S1(n,m) = A008275(n,m).

Bivariate e.g.f.-o.g.f.: Sum_{n,m >= 1} T(n,m)*x^n*y^m/n! = exp((y/7)*log(1 + 7*x)) - 1 = (1 + 7*x)^(y/7) - 1. - Petros Hadjicostas, Jun 07 2020

T(n, 0) = (-7)^(n-1)*A000142(n-1). - G. C. Greubel, Feb 22 2022