A076112 Triangle (read by rows) in which the n-th row contains first n terms of n geometric progression with first term 1 and common ratio (n-1).

1, 1, 2, 1, 3, 9, 1, 4, 16, 64, 1, 5, 25, 125, 625, 1, 6, 36, 216, 1296, 7776, 1, 7, 49, 343, 2401, 16807, 117649, 1, 8, 64, 512, 4096, 32768, 262144, 2097152, 1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 1, 10, 100, 1000, 10000, 100000, 1000000

Offset

1,3

Comments

Table T(n,k) = (n+k-1)^(n-1) n,k > 0 read by antidiagonals. - Boris Putievskiy, Dec 17 2012

T(n,k) = n^(k-1) is the number of rooted trees on [n] containing a rooted forest on [n] having k components (see Bona at p. 261). - Stefano Spezia, Apr 02 2026

References

Miklos Bona, Introduction to Enumerative and Analytic Combinatorics, CRC Press, 2025, p. 261.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..5000

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Formula

As a linear array, the sequence is a(n) = A002024(n)^A002260(n) or a(n) = (t+1)^(n-t*(t+1)/2-1), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012

Example

Triangle begins:
  1;
  1,2;
  1,3,9;
  1,4,16,64;
  1,5,25,125,625;
  1,6,36,216,1296,7776;
  ...

Mathematica

Table[n^Range[0, n-1], {n, 10}]//Flatten (* Harvey P. Dale, Jan 27 2020 *)

Prog

(Python)
from math import isqrt, comb
def A076112(n): return (isqrt(n<<3)+1>>1)**(n-comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)), 2)-1) # Chai Wah Wu, Jun 09 2025

Crossrefs

Cf. A000169, A076113.

Sequence in context: A236438 A101486 A086606 * A208744 A122454 A189650

Adjacent sequences: A076109 A076110 A076111 * A076113 A076114 A076115

Keyword

easy,nonn,tabl

Author

Amarnath Murthy, Oct 09 2002

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

Status

approved