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A051142 Generalized Stirling number triangle of first kind.

%I A051142 #39 Jun 10 2020 17:37:29
%S A051142 1,-4,1,32,-12,1,-384,176,-24,1,6144,-3200,560,-40,1,-122880,70144,
%T A051142 -14400,1360,-60,1,2949120,-1806336,415744,-47040,2800,-84,1,
%U A051142 -82575360,53526528,-13447168,1732864,-125440,5152,-112,1,2642411520,-1795424256,483835904
%N A051142 Generalized Stirling number triangle of first kind.
%C A051142 a(n,m) = R_n^m(a=0, b=4) in the notation of the given 1961 and 1962 references.
%C A051142 a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x - 4*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
%C A051142 This is the signed Stirling1 triangle with diagonal d >= 0 (main diagonal d = 0) scaled with 4^d.
%C A051142 Also the Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+4) (A047053) giving unsigned values and adding 1, 0, 0, 0, ... as column 0. For the definition of the Bell transform, see A264428 and for cross-references A265606. - _Peter Luschny_, Dec 31 2015
%H A051142 Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Celeste/celeste3.html">Two Approaches to Normal Order Coefficients</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
%H A051142 Wolfdieter Lang, <a href="/A051142/a051142.txt">First 10 rows</a>.
%H A051142 D. S. Mitrinovic, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k762d/f996.image.r=1961%20mitrinovic">Sur une classe de nombres reliés aux nombres de Stirling</a>, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356.
%H A051142 D. S. Mitrinovic and M. S. Mitrinovic, <a href="http://pefmath2.etf.rs/files/47/77.pdf">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77.
%H A051142 D. S. Mitrinovic and R. S. Mitrinovic, <a href="https://www.jstor.org/stable/43667130">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77 [jstor stable version].
%F A051142 a(n, m) = a(n-1, m-1) - 4*(n-1)*a(n-1, m) for n >= m >= 1; a(n, m) := 0 for n < m; a(n, 0) := 0 for n >= 1; a(0, 0) = 1.
%F A051142 E.g.f. for the m-th column of the signed triangle: (log(1 + 4*x)/4)^m/m!.
%F A051142 a(n, m) = S1(n, m)*4^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).
%e A051142 Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
%e A051142         1;
%e A051142        -4,     1;
%e A051142        32,   -12,      1;
%e A051142      -384,   176,    -24,    1;
%e A051142      6144, -3200,    560,  -40,   1,
%e A051142   -122880, 70144, -14400, 1360, -60, 1;
%e A051142   ...
%e A051142 3rd row o.g.f.: E(3,x) = 32*x - 12*x^2 + x^3.
%t A051142 Table[StirlingS1[n, m] 4^(n - m), {n, 9}, {m, n}] // Flatten (* _Michael De Vlieger_, Dec 31 2015 *)
%o A051142 (Sage) # uses[bell_transform from A264428]
%o A051142 # Unsigned values and an additional first column (1,0,0,0, ...).
%o A051142 def A051142_row(n):
%o A051142     multifact_4_4 = lambda n: prod(4*k + 4 for k in (0..n-1))
%o A051142     mfact = [multifact_4_4(k) for k in (0..n)]
%o A051142     return bell_transform(n, mfact)
%o A051142 [A051142_row(n) for n in (0..9)] # _Peter Luschny_, Dec 31 2015
%Y A051142 First (m=1) column sequence is: A047053(n-1).
%Y A051142 Row sums (signed triangle): A008545(n-1)*(-1)^(n-1).
%Y A051142 Row sums (unsigned triangle): A007696(n).
%Y A051142 Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051141 (b=3).
%Y A051142 Cf. A039692, A264428, A265606.
%K A051142 sign,easy,tabl
%O A051142 1,2
%A A051142 _Wolfdieter Lang_