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

1,2

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)

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].

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

Example

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.

Mathematica

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

Prog

(Magma) [7^(n-k)*StirlingFirst(n, k): k in [1..n], n in [1..12]]; // 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

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.

Sequence in context: A027517 A092082 A013559 * A012034 A138324 A052122

Adjacent sequences: A051183 A051184 A051185 * A051187 A051188 A051189

Keyword

sign,easy,tabl

Author

Wolfdieter Lang

Status

approved