A210682 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.

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

Offset

1,3

Comments

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

Links

Alois P. Heinz, Rows n = 1..40, flattened

Keisuke Uchimura, An identity for the divisor generating function arising from sorting theory, J. Combin. Theory Ser. A 31 (1981), no. 2, 131--135. MR0629588 (82k:05015)

Example

Triangle begins:
1
1 2 -1
1 2  2 -1 -2  1
1 2  2  3 -3 -1 -2 1 2 -1
...

Maple

U:= proc(n) U(n):= `if`(n=1, x, expand (n*x^n + (1-x^n)*U(n-1))) end:
T:= (n, k)-> coeff (U(n), x, k):
seq(seq(T(n, k), k=1..n*(n+1)/2), n=1..10); # Alois P. Heinz, May 30 2012

Mathematica

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 *)

Crossrefs

Cf. A000005.

Sequence in context: A122497 A350330 A154402 * A293433 A177025 A379552

Adjacent sequences: A210679 A210680 A210681 * A210683 A210684 A210685

Keyword

sign,tabf

Author

N. J. A. Sloane, May 09 2012

Status

approved