OFFSET
0,2
COMMENTS
T(n, m) = ^4P_n^m 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, m)*x^m which are s(n, x) = Product_{k=0..n-1} x-(4+k), 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 for (exp(4*t), exp(t)-1).
See A143493 for the unsigned version of this array and A143496 for the inverse. - Peter Bala, Aug 25 2008
LINKS
Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Dragoslav S. Mitrinović and Ružica S. Mitrinović, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962); alternative link.
FORMULA
T(n, m)= a(n-1, m-1) - (n+3)*T(n-1, m), n >= m >= 0; T(n, m) = 0, n < m; T(n, -1) = 0, T(0, 0)=1. E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^4).
Triangle (signed) = [ -4, -1, -5, -2, -6, -3, -7, -4, -8, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [4, 1, 5, 2, 6, 3, 7, 4, 8, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938 (unsigned version in A143493).
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, 4), for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008
EXAMPLE
Triangle begins:
1;
-4, 1;
20, -9, 1;
-120, 74, -15, 1;
840, -638, 179, -22, 1;
...
MAPLE
A049459_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x+4, n)), x, k), k=0..n): seq(print(A049459_row(n)), n=0..8); # Peter Luschny, May 16 2013
MATHEMATICA
a[n_, m_] /; 0 <= m <= n := a[n, m] = a[n-1, m-1] - (n+3)*a[n-1, m];
a[n_, m_] /; n < m = 0;
a[_, -1] = 0; a[0, 0] = 1;
Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)
PROG
(Haskell)
a049459 n k = a049459_tabl !! n !! k
a049459_row n = a049459_tabl !! n
a049459_tabl = map fst $ iterate (\(row, i) ->
(zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 4)
-- Reinhard Zumkeller, Mar 11 2014
CROSSREFS
Unsigned column sequences are: A001715-A001719. Cf. A008275 (Stirling1 triangle), A049458, A049460. Row sums (signed triangle): A001710(n+2)*(-1)^n. Row sums (unsigned triangle): A001720(n+4).
KEYWORD
AUTHOR
EXTENSIONS
Second formula corrected by Philippe Deléham, Nov 09 2008
Name changed by Thomas Scheuerle, Feb 04 2026
STATUS
approved
