A094646 -id:A094646 - cp's OEIS frontend

A049444 Generalized Stirling number triangle of first kind.

1, -2, 1, 6, -5, 1, -24, 26, -9, 1, 120, -154, 71, -14, 1, -720, 1044, -580, 155, -20, 1, 5040, -8028, 5104, -1665, 295, -27, 1, -40320, 69264, -48860, 18424, -4025, 511, -35, 1, 362880, -663696, 509004, -214676, 54649, -8624, 826, -44, 1, -3628800, 6999840, -5753736

Offset: 0

Wolfdieter Lang

T(n, k) = ^2P_n^k in the notation of the given reference with T(0, 0) := 1. The monic row polynomials s(n,x) := Sum_{m=0..n} T(n, k)*x^k which are s(n, x) = Product_{j=0..n-1} (x-(2+j)), n >= 1 and s(0, x)=1 satisfy s(n, x+y) = Sum_{k=0..n} binomial(n, k)*s(k,x)*S1(n-k, y), with the Stirling1 polynomials S1(n, x) = Sum_{m=1..n} (A008275(n, m)*x^m) 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 polynomials for (exp(2*t), exp(t)-1). This translates to the usual exponential Riordan (Sheffer) notation (1/(1+x)^2, log(1+x)).
See A143491 for the unsigned version of this array and A143494 for the inverse. - Peter Bala, Aug 25 2008
Corresponding to the generalized Stirling number triangle of second kind A137650. - Peter Luschny, Sep 18 2011
Unsigned, reversed rows (cf. A145324, A136124) are the dimensions of the cohomology of a complex manifold with a symmetric group (S_n) action. See p. 17 of the Hyde and Lagarias link. See also the Murri link for an interpretation as the Betti numbers of the moduli space M(0,n) of smooth Riemann surfaces. - Tom Copeland, Dec 09 2016
The row polynomials s(n, x) = (-1)^n*risingfactorial(2 - x, n) are related to the column sequences of the unsigned Abel triangle A137452(n, k), for k >= 2. See the formula there. - Wolfdieter Lang, Nov 21 2022

			The Triangle  begins:
n\k       0       1        2       3       4      5      6    7   8 9 ...
0:        1
1:       -2       1
2:        6      -5        1
3:      -24      26       -9       1
4:      120    -154       71     -14       1
5      -720    1044     -580     155     -20      1
6:     5040   -8028     5104   -1665     295    -27      1
7:   -40320   69264   -48860   18424   -4025    511    -35    1
8:   362880 -663696   509004 -214676   54649  -8624    826  -44
9: -3628800 6999840 -5753736 2655764 -761166 140889 -16884 1266 -54 1
...  [reformatted by _Wolfdieter Lang_, Nov 21 2022]

Y. Manin, Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, American Math. Soc. Colloquium Publications Vol. 47, 1999. [From Tom Copeland, Jun 29 2008]
S. Roman, The Umbral Calculus, Academic Press, 1984 (also Dover Publications, 2005).

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
E. Getzler, Operads and moduli spaces of genus 0 Riemann surfaces, arXiv:alg-geom/9411004, 1994, (see p. 23, g(x,t)). [From _Tom Copeland_, Dec 11 2011]
T. Hyde and J. Lagarias Polynomial splitting measures and cohomology of the pure braid group, arXiv preprint arXiv:1604.05359 [math.RT], 2016.
Y. Manin, Generating functions in algebraic geometry and sums over trees, arXiv:alg-geom/9407005, 1994, (Eqn. 0.7 and 1.7). [From _Tom Copeland_, Dec 10 2011]
D. S. Mitrinovic and M. 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.
R. Murri, Fatgraph Algorithms and the Homology of the Kontsevich Complex, arXiv:1202.1820 [math.AG], 2012, (see Table 1, p. 3). [From _Tom Copeland_, Sep 18 2012]

Unsigned column sequences are A000142(n+1), A001705-A001709. Row sums (signed triangle): n!*(-1)^n, row sums (unsigned triangle): A001710(n-2). Cf. A008275 (Stirling1 triangle).
Cf. A000035, A084938, A094645, A094646.
Cf. A143491, A143494.
Cf. A136124, A137452, A137650.

a049444 n k = a049444_tabl !! n !! k
a049444_row n = a049444_tabl !! n
a049444_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 2)
-- Reinhard Zumkeller, Mar 11 2014

A049444_row := proc(n) local k,i;
add(add(Stirling1(n, n-i), i=0..k)*x^(n-k-1),k=0..n-1);
seq(coeff(%,x,k),k=1..n-1) end:
seq(print(A049444_row(n)),n=1..7); # Peter Luschny, Sep 18 2011
A049444:= (n, k)-> add((-1)^(n-j)*(n-j+1)!*binomial(n, j)*Stirling1(j, k), j=0..n):
seq(print(seq(A049444(n, k), k=0..n)), n=0..11);  # Mélika Tebni, May 02 2022

t[n_, i_] = Sum[(-1)^k*Binomial[n, k]*(k+1)!*StirlingS1[n-k, i], {k, 0, n-i}]; Flatten[Table[t[n, i], {n, 0, 9}, {i, 0, n}]] [[1 ;; 48]]
(* Jean-François Alcover, Apr 29 2011, after Milan Janjic *)

T(n, k) = T(n-1, k-1) - (n+1)*T(n-1, k), n >= k >= 0; T(n, k) = 0, n < k; T(n, -1) = 0, T(0, 0) = 1.
E.g.f. for k-th column of signed triangle: ((log(1+x))^k)/(k!*(1+x)^2).
Triangle (signed) = [-2, -1, -3, -2, -4, -3, -5, -4, -6, -5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [2, 1, 3, 2, 4, 3, 5, 4, 6, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...], where DELTA is Deléham's operator defined in A084938 (unsigned version in A143491).
E.g.f.: (1 + x)^(y-2). - Vladeta Jovovic, May 17 2004 [For row polynomials s(n, y)]
With P(n, t) = Sum_{j=0..n-2} T(n-2,j) * t^j and P(1, t) = -1 and P(0, t) = 1, then G(x, t) = -1 + exp[P(.,t)*x] = [(1+x)^t - 1 - t^2 * x] / [t(t-1)], whose compositional inverse in x about 0 is given in A074060. G(x, 0) = -log(1+x) and G(x, 1) = (1+x) log(1+x) - 2x. G(x, q^2) occurs in formulas on pages 194-196 of the Manin reference. - Tom Copeland, Feb 17 2008
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then T(n,i) = f(n,i,2), for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008
T(n, k) = Sum_{j=0..n} (-1)^(n-j)*(n-j+1)!*binomial(n, j)*Stirling1(j, k). - Mélika Tebni, May 02 2022
From Wolfdieter Lang, Nov 24 2022: (Start)
Recurrence for row polynomials {s(n, x)}_{n>=0}: s(0, x) = 1, s(n, x) = (x - 2)*exp(-(d/dx)) s(n-1, x), for n >= 1. This is adapted from the general Sheffer result given by S. Roman, Corollary 3.7.2., p. 50.
Recurrence for column sequence {T(n, k)}{n>=k}: T(n, n) = 1, T(n, k) = (n!/(n-k))*Sum{j=k..n-1} (1/j!)*(a(n-1-j) + k*beta(n-1-j))*T(n-1, k), for k >= 0, where alpha = repeat(-2, 2) and beta(n) = [x^n] (d/dx)log(log(x)/x) = (-1)^(n+1)*A002208(n+1)/A002209(n+1), for n >= 0. This is the adapted Boas-Buck recurrence, also given in Rainville, Theorem 50., p. 141, For the references and a comment see A046521. (End)

Second formula corrected by Philippe Deléham, Nov 09 2008

A049458 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -3, 1, 12, -7, 1, -60, 47, -12, 1, 360, -342, 119, -18, 1, -2520, 2754, -1175, 245, -25, 1, 20160, -24552, 12154, -3135, 445, -33, 1, -181440, 241128, -133938, 40369, -7140, 742, -42, 1, 1814400, -2592720, 1580508, -537628

Offset: 0

Wolfdieter Lang

a(n,m)= ^3P_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-(3+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 polynomials for (exp(3*t),exp(t)-1).
See A143492 for the unsigned version of this array and A143495 for the inverse. - Peter Bala, Aug 25 2008

			1;
-3, 1;
12, -7, 1;
-60, 47, -12, 1;
360, -342, 119, -18, 1;
s(2,x) = 12-7*x+x^2. S1(2,x) = -x+x^2 (Stirling1 polynomial).

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Unsigned column sequences are: A001710-A001714. Row sums (signed triangle): (n+1)!*(-1)^n. Row sums (unsigned triangle): A001715(n+3).
Cf. A000035 A084938.
Cf. A094645, A094646.
A143492, A143495. - Peter Bala, Aug 25 2008

a049458 n k = a049458_tabl !! n !! k
a049458_row n = a049458_tabl !! n
a049458_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 3)
-- Reinhard Zumkeller, Mar 11 2014

A049458_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x+3, n)), x, k), k=0..n): seq(print(A049458_row(n)),n=0..8); # Peter Luschny, May 16 2013

t[n_, k_] := (-1)^(n - k)*Coefficient[ Pochhammer[x + 3, n], x, k]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 17 2013, after Peter Luschny *)

a(n, m)= a(n-1, m-1) - (n+2)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n

Triangle (signed) = [ -3, -1, -4, -2, -5, -3, -6, -4, -7, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [3, 1, 4, 2, 5, 3, 6, 4, 7, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938 (unsigned version in A143492).

E.g.f.: (1+y)^(x-3). - Vladeta Jovovic, May 17 2004

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,3), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

Second formula corrected by Philippe Deléham, Nov 09 2008

A094645 Triangle of generalized Stirling numbers of the first kind.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 0, -1, 0, 1, 0, -2, -1, 2, 1, 0, -6, -5, 5, 5, 1, 0, -24, -26, 15, 25, 9, 1, 0, -120, -154, 49, 140, 70, 14, 1, 0, -720, -1044, 140, 889, 560, 154, 20, 1, 0, -5040, -8028, -64, 6363, 4809, 1638, 294, 27, 1, 0, -40320, -69264, -8540, 50840, 44835, 17913, 3990, 510, 35, 1

Offset: 0

Vladeta Jovovic, May 17 2004

From Wolfdieter Lang, Jun 20 2011: (Start)
The row polynomials s(n,x) := Sum_{j=0..n} T(n,k)*x^k satisfy risefac(x-1,n) = s(n,x), with the rising factorials risefac(x-1,n) := Product_{j=0..n-1} (x-1+j), n >= 1, risefac(x-1,0) = 1. Compare with the formula risefac(x,n) = s1(n,x), with the row polynomials s1(n,x) of A132393 (unsigned Stirling1).
This is the lower triangular Sheffer array with e.g.f.
  T(x,z) = (1-z)*exp(-x*log(1-z)) (the rewritten e.g.f. from the formula section). See the W. Lang link under A006232 for Sheffer matrices and the Roman reference. In the notation which indicates the column e.g.f.s this is Sheffer (1-z,-log(1-z)). In the umbral notation (cf. Roman) this is called Sheffer for (exp(t),1-exp(-t)).
The row polynomials satisfy s(n,x) = (x+n-1)*s(n-1,x), s(0,x)=1, and s(n,x) = (x-1)*s1(n-1,x), n >= 1, s1(0,x) = 1, with the unsigned Stirling1 row polynomials s1(n,x).
The row polynomials also satisfy
  s(n,x) - s(n,x-1) = n*s(n-1,x), n > 1, s(0,x) = 1
  (from the Meixner identity, see the Meixner reference given at A060338).
The row polynomials satisfy as well (from corollary 3.7.2. p. 50 of the Roman reference)
  s(n,x) = (x-1)*s(n-1,x+1), n >= 1, s(0,n) = 1.
The exponential convolution identity is
  s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,y)*s1(n-k,x),
  n >= 0, with symmetry x <-> y.
The row sums are 1 for n=0 and 0 otherwise, and the alternating row sums are 1,-2,2, followed by zeros, with e.g.f. (1-x)^2.
The Sheffer a-sequence Sha(n) = A164555(n)/A027642(n) with e.g.f. x/(1-exp(-x)), and the z-sequence is Shz(n) = -1 with e.g.f. -exp(x).
The inverse Sheffer matrix is ((-1)^(n-k))*A105794(n,k) with e.g.f. exp(z)*exp(x*(1-exp(-z))). (End)
Triangle T(n,k), read by rows, given by (-1, 1, 0, 2, 1, 3, 2, 4, 3, 5, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 16 2012
Also coefficients of t in t*(t-1)*Sum[(-1)^(n+m) t^(m-1) StirlingS1[n,m], {m,n}] in which setting t^k equal to k gives n!, from this follows that the dot product of row n with [0,...,n] equals (n-1)!. - Wouter Meeussen, May 15 2012

			Triangle begins
   1;
  -1,   1;
   0,  -1,   1;
   0,  -1,   0,   1;
   0,  -2,  -1,   2,   1;
   0,  -6,  -5,   5,   5,   1;
   0, -24, -26,  15,  25,   9,   1;
   ...
Recurrence:
  -2 = T(4,1) = T(3,0) + (4-2)*T(3,1) = 0 + 2*(-1).
Row polynomials:
  s(3,x) = -x+x^3 = (x-1)*s1(2,x) = (x-1)*(x+x^2).
  s(3,x) = (x-1)*s(2,x+1) = (x-1)*(-(x+1)+(x+1)^2).
  s(3,x) - s(3,x-1) = -x+x^3 -(-(x-1)+(x-1)^3) = 3*(-x+x^2) = 3*s(2,x).

S. Roman, The Umbral Calculus, Academic Press, New York, 1984.

M. W. Coffey and M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.
Igor Victorovich Statsenko, On the ordinal numbers of triangles of generalized special numbers, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.

Cf. A049444, A049458, A094646, A132393, A105794.

A094645_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x-n+2, n)), x, k), k=0..n): seq(print(A094645_row(n)), n=0..6); # Peter Luschny, May 16 2013

t[n_, k_] /; n >= k >= 0 := t[n, k] = t[n-1, k-1] + (n-2)*t[n-1, k]; t[n_, k_] /; n < k = 0; t[, -1] = 0; t[0, 0] = 1; Flatten[ Table[ t[n, k], {n, 0, 10}, {k, 0, n}] ] (* _Jean-François Alcover, Sep 29 2011, after recurrence *);
Table[CoefficientList[t*(t-1)*Sum[(-1)^(n+m)*t^(m-1)*StirlingS1[n,m],{m,n}],t],{n,1,7}] (* Wouter Meeussen, May 15 2012 *)

E.g.f.: (1-y)^(1-x).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000142(n), A000142(n+1), A001710(n+2), A001715(n+3), A001720(n+4), A001725(n+5), A001730(n+6), A049388(n), A049389(n), A049398(n), A051431(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively. - Philippe Deléham, Nov 13 2007
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then |T(n,i)| = |f(n,i,-1)|, for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008
From Wolfdieter Lang, Jun 20 2011: (Start)
T(n,k) = |S1(n-1,k-1)| - |S1(n-1,k)|, n >= 1, k >= 1, with |S1(n,k)| = A132393(n,k) (unsigned Stirling1).
Recurrence: T(n,k) = T(n-1,k-1) + (n-2)*T(n-1,k) if n >= k >= 0; T(n,k) = 0 if n < k; T(n,-1) = 0; T(0,0) = 1.
E.g.f. column k: (1-x)*((-log(1-x))^k)/k!. (End)
T(n,k) = Sum_{i=0..n} binomial(n,i)*(n-i)!*Stirling1(i,k)*TC(m,n,i) where TC(m,n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+m+1,i+1)*(-1)^i, m = 1 for n >= 0. See A130534, A370518 for m=0 and m=2. - Igor Victorovich Statsenko, Feb 27 2024

A192001 Triangle with sums of nonnegative integer powers of positive first n integers in the columns.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 5, 1, 5, 10, 14, 9, 1, 6, 15, 30, 36, 17, 1, 7, 21, 55, 100, 98, 33, 1, 8, 28, 91, 225, 354, 276, 65, 1, 9, 36, 140, 441, 979, 1300, 794, 129, 1, 10, 45, 204, 784, 2275, 4425, 4890, 2316, 257, 1

Offset: 1

Wolfdieter Lang, Jun 25 2011

This is the Abramowitz-Stegun table on p. 813, call it s(m,n), with an extra column n=0 with values n added, and read by antidiagonals. a(n,m) = s(n-m,m), n+1 >= m >= 0.
O.g.f. for column no. m >= 0: (x^(m+1)/(1-x)^(m+2))*E(m;x) with the row polynomials E(m;x) = Sum_{p=0..m} A173018(m,p)*x^p of the Eulerian number triangle (proof via the Worpitzky identity). See the Graham et al. reference p. 253-8 for Eulerian numbers, and the Worpitzky identity (6.37) on p. 255.
E.g.f. for diagonals (starting with k=0 for the main diagonal): g(k,x) = exp(x)*(exp((k+1)*x)-1)/(1-exp(x)).
  Compare with (7.77) on p. 353 of the Graham et al. reference.
O.g.f. for diagonals (starting with k=0 for the main diagonal): G(k,z) =(Psi(1/z+1)-Psi(1/z-k-1))/z - 1.
  with the digamma function Psi(z):=(log(Gamma(z)))'.
  Compare with Graham et al., p. 352, eq.(7.76), where H_z=Psi(z+1)+gamma, with the Euler-Mascheroni constant gamma.
The diagonal sequences are, for k=0..9: A000012, A000051, A001550-A001557.
The negative k-diagonal, -a(k+m+1,m), yields the Sheffer z-sequence Shz(k+1;m) for the Sheffer arrays |S1|(k+1) defined in a comment to A094646.
See also A196837 with a W. Lang link, where the o.g.f.s for the diagonals, numbered with k >= 1, are given as G(k,x) = Sum_{m=0..k} (k-m)*S1(k+1,k+1-m)*x^m / Product_{j=1..k} (1-j*x), with S1 the Stirling numbers of the first kind, A048994. - Wolfdieter Lang, Nov 01 2011

			The triangle a(n,m) begins:
n\m    0   1   2    3    4     5     6     7    8   9  10 ...
n=1:   1
n=2:   2   1
n=3:   3   3   1
n=4:   4   6   5    1
n=5:   5  10  14    9    1
n=6:   6  15  30   36   17     1
n=7:   7  21  55  100   98    33     1
n=8:   8  28  91  225  354   276    65     1
n=9:   9  36 140  441  979  1300   794   129    1
n=10: 10  45 204  784 2275  4425  4890  2316  257   1
n=11: 11  55 285 1296 4676 12201 20515 18700 6818 513   1
...  Reformatted and extended by _Wolfdieter Lang_, Jan 12 2013
a(4,2)= 5 = s(2,2) = 1^2 + 2^2.
Recurrence: 55 = a(7,2) = (7-2)*a(6,1) - (a(2,1) + a(3,1) + a(4,1) + a(5,1)) = 5*15 - (1 + 3 + 6 + 10) = 55. - _Wolfdieter Lang_, Jan 12 2013
The first column, m=0 holds the integers 1,2,3,..., equal to the sums of 0th powers of the n first integers. The second column is 1, 1+2, 1+2+3, ... = A000217. The third column are the sums of squares, 1^2, 1^2+2^2, 1^2+2^2+3^3, ... = A000330, etc. - _M. F. Hasler_, Jan 13 2013

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available, see the link), p. 813.
Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1991.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [scanned copy], p. 813.

Cf. A103438 (omitting the first column of zeros, reversed).

Flatten[ Table[ HarmonicNumber[-m + n, -m], {n, 1, 10}, {m, 0, n - 1}]] (* Jean-François Alcover, Sep 26 2011 *)

A192001(n,m) = sum(k=1,n-m,k^m) \\ - M. F. Hasler, Jan 13 2013

a(n,m) = s(n-m,m), n-1 >= m >= 0, n >= 1, else 0, with s(n,m) := Sum_{k=1..n} k^m.
O.g.f. column no. m: see a comment above.
O.g.f.s and e.g.f.s for diagonals k >= 0: see a comment above.
Recurrence known by Ibn al-Haytham (see a comment and link under A000537, and comments by Bruno Berselli under the A-numbers of the first column sequences):
  a(n,m) = (n-m)*a(n-1,m-1) - Sum_{j=m..n-2} a(j,m-1), n >= 1, n-1 >= m >= 1. a(n,0) = n. - Wolfdieter Lang, Jan 12 2013

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A049444 Generalized Stirling number triangle of first kind.

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A049458 Generalized Stirling number triangle of first kind.

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A094645 Triangle of generalized Stirling numbers of the first kind.

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A192001 Triangle with sums of nonnegative integer powers of positive first n integers in the columns.

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