A129334 Triangle T(n,k) read by rows: inverse of the matrix PE = exp(P)/exp(1) given in A011971.

1, -1, 1, 0, -2, 1, 1, 0, -3, 1, 1, 4, 0, -4, 1, -2, 5, 10, 0, -5, 1, -9, -12, 15, 20, 0, -6, 1, -9, -63, -42, 35, 35, 0, -7, 1, 50, -72, -252, -112, 70, 56, 0, -8, 1, 267, 450, -324, -756, -252, 126, 84, 0, -9, 1, 413, 2670, 2250, -1080, -1890, -504, 210, 120, 0, -10, 1

Offset: 0

Author

Gottfried Helms, Apr 08 2007

Keywords

• easy
• tabl
• sign

Comments

The structure of the triangle is A(r,c) = A000587(1+(r-c))*binomial(r-1,c-1) where row index r and column-index c start at 1.

Row polynomials defined recursively: P(0,x) = 1, P(n+1,x) = x*P(n,x) - P(n,x+1). The polynomials appear to be irreducible. Polynomials evaluated at x = c give sequences with e.g.f. exp(1 - cx - exp(-x)).

Examples

			Triangle starts:
[0]   1;
[1]  -1,   1;
[2]   0,  -2,    1;
[3]   1,   0,   -3,    1;
[4]   1,   4,    0,   -4,    1;
[5]  -2,   5,   10,    0,   -5,   1;
[6]  -9, -12,   15,   20,    0,  -6,  1;
[7]  -9, -63,  -42,   35,   35,   0, -7,  1;
[8]  50, -72, -252, -112,   70,  56,  0, -8,  1;
[9] 267, 450, -324, -756, -252, 126, 84,  0, -9, 1;
		

Links

• S. de Wannemacker, T. Laffey and R. Osburn, On a conjecture of Wilf, arXiv:math/0608085 [math.NT], 2006-2007.

Crossrefs

First column is A000587 (Uppuluri Carpenter numbers) which is also the negative of the row sums (=P(n, 1)). Polynomials evaluated at 2 are A074051, at -1 A109747.

Cf. A094816.

Programs

Maple

P := proc(n,x) option remember; if n=0 then 1 else
x*P(n-1, x) - P(n-1, x+1) fi end:
aRow := n -> seq(coeff(P(n, x), x, k), k = 0..n):
seq(aRow(n), n = 0..10); # Peter Luschny, Apr 15 2022

Formula

Let P be the lower-triangular Pascal-matrix, PE = exp(P-I) a matrix-exponential in exact integer arithmetic (or PE = lim exp(P)/exp(1) as limit of the exponential) then A = PE^-1 and a(n) = A(n, read sequentially). - Gottfried Helms, Apr 08 2007

T(n, k) = Sum_{j=0..n} (-1)^(j-k)*A094816(j, k)*Stirling2(n, j). - Mélika Tebni, Apr 15 2022

Extensions

Edited by Ralf Stephan, May 12 2007