OFFSET
0,2
COMMENTS
This is the member N = 2 of a family of signed triangles with row sums n! = A000142(n): T(N; n, k) = (-1)^(n-k)*binomial(n, k)*(k + N + 1)^n, for integer N, n >= 0 and k = 0, 1, ..., n. The row polynomials PS(N; n, z) = Sum_{k=0..n} T(N; n, k)*z^k = ((-1)^n/z^N)*D_{n,N+1,n}(z) in [Sidi 1980].
For N = -1, 0 and 1 see A258773(n, k), A075513(n+1, k) and (-1)^(n-k) * A154715(n, k), respectively.
The column sequences, for k = 0, 1, ..., 6 and n >= k, are A141413(n+2), (-1)^(n+1)*A018215(n) = 4*(-1)^(n+1)*A002697(n), 5^2*(-1)^n*A081135(n), (-1)^(n+1)*A128964(n-1) = 6^3*(-1)^(n+1)*A081144(n), 7^4*(-1)^n*A139641(n-4), 2^15*(-1)^(n+1)*A173155(n-5), 3^12*(-1)^n*A173191(n-6), respectively.
The e.g.f. of the triangle (see below) needs the exponential convolution (LambertW(-z)/(-z))^2 = Sum_{n>=0} c(2; n)*z^n/n!, where c(2; n) = Sum_{m=0..n} |A137352(n+1, m)|*2^m = A007334(n+2).
The row sums give n! = A000142(n).
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..5049 (rows 0..100 of the triangle, flattened)
Wolfdieter Lang, On a Certain Family of Sidi Polynomials, May 2023.
Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874.
FORMULA
T(n, k) = (-1)^(n-k)*binomial(n, k)*(k + 3)^n, for n >= 0, k = 0, 1, ..., n.
O.g.f. of column k: (x*(k + 3))^k/(1 - (k + 3)*x)^(k+1), for k >= 0.
E.g.f. of column k: exp(-(k + 3)*x)*((k + 3)*x)^k/k!, for k >= 0.
E.g.f. of the triangle, that is, the e.g.f. of its row polynomials {PS(2;n,y)}_{n>=0}): ES(2;y,x) = exp(-3*x)*(1/3)*(d/dz)(W(-z)/(-z))^2, after replacing z by x*y*exp(-x), where W is the Lambert W-function for the principal branch. This becomes ES(2;y,x) = exp(-3*x)*exp(3*(-W(-z)))/(1 - (-W(-z)), with z = x*y*exp(-x).
EXAMPLE
The triangle T begins:
n\k 0 1 2 3 4 5 6 7
0: 1
1: -3 4
2: 9 -32 25
3: -27 192 -375 216
4: 81 -1024 3750 -5184 2401
5: -243 5120 -31250 77760 -84035 32768
6: 729 -24576 234375 -933120 1764735 -1572864 531441
7: -2187 114688 -1640625 9797760 -28824005 44040192 -33480783 10000000
...
n = 8: 6561 -524288 10937500 -94058496 403536070 -939524096 1205308188 -800000000 2143588,
n = 9: -19683 2359296 -70312500 846526464 -5084554482 16911433728 -32543321076 36000000000 -21221529219 5159780352.
MATHEMATICA
A362353row[n_]:=Table[(-1)^(n-k)Binomial[n, k](k+3)^n, {k, 0, n}]; Array[A362353row, 10, 0] (* Paolo Xausa, Jul 30 2023 *)
CROSSREFS
KEYWORD
AUTHOR
Harlan J. Brothers and Wolfdieter Lang, Apr 27 2023
EXTENSIONS
a(41)-a(44) from Paolo Xausa, Jul 31 2023
STATUS
approved