A210682 - OEIS

%I #32 Mar 07 2014 10:14:08

%S 1,1,2,-1,1,2,2,-1,-2,1,1,2,2,3,-3,-1,-2,1,2,-1,1,2,2,3,2,-2,-4,-1,-1,

%T 2,1,2,-1,-2,1,1,2,2,3,2,4,-5,-3,-3,-1,-1,4,3,-1,2,-2,-1,-2,1,2,-1,1,

%U 2,2,3,2,4,2,-4,-5,-3,-4,2,-1,4,5,1,0,-1,-3,-1,0

%N Triangle read by rows: T(n,k) = coefficient of x^k in polynomial U_n(x) defined by U_1 = x, U_n = n*x^n + (1-x^n)*U_(n-1), n >= 1, 1 <= k <= n(n+1)/2.

%C T(n,m) = d(m) for m <= n (cf. A000005).

%H Alois P. Heinz, <a href="/A210682/b210682.txt">Rows n = 1..40, flattened</a>

%H Keisuke Uchimura, <a href="http://dx.doi.org/10.1016/0097-3165(81)90009-1">An identity for the divisor generating function arising from sorting theory</a>, J. Combin. Theory Ser. A 31 (1981), no. 2, 131--135. MR0629588 (82k:05015)

%e Triangle begins:

%e 1

%e 1 2 -1

%e 1 2  2 -1 -2  1

%e 1 2  2  3 -3 -1 -2 1 2 -1

%e ...

%p U:= proc(n) U(n):= `if`(n=1, x, expand (n*x^n + (1-x^n)*U(n-1))) end:

%p T:= (n, k)-> coeff (U(n), x, k):

%p seq(seq(T(n, k), k=1..n*(n+1)/2), n=1..10); # _Alois P. Heinz_, May 30 2012

%t U[1] = x; U[n_] := U[n] = n*x^n + (1-x^n)*U[n-1]; T[n_, k_] := Coefficient[U[n], x, k]; Table[T[n, k], {n, 1, 10}, {k, 1, n*(n+1)/2}] // Flatten (* _Jean-François Alcover_, Mar 07 2014 *)

%Y Cf. A000005.

%K sign,tabf

%O 1,3

%A _N. J. A. Sloane_, May 09 2012