A171843 Triangle read by rows = truncated columns of an array formed by variants of the natural number decrescendo triangle, A004736.

1, 1, 3, 1, 3, 8, 1, 3, 6, 21, 1, 3, 6, 12, 55, 1, 3, 6, 10, 24, 144, 1, 3, 6, 10, 17, 48, 377, 1, 3, 6, 10, 15, 30, 96, 987, 1, 3, 6, 10, 15, 23, 53, 192, 2584, 1, 3, 6, 10, 15, 21, 37, 93, 384, 6765, 1, 3, 6, 10, 15, 21, 30, 61, 163, 768, 17711, 1, 3, 6, 10, 15, 21, 28, 45, 100, 286, 1536, 46368

Offset: 1

Author

Gary W. Adamson, Dec 19 2009

Keywords

• nonn
• tabl

Comments

Rows tend to the triangular series, A000217.

Let T(n) be the variants of the natural number decrescendo triangle, A004736; such that T(n) = A004736, prepending n ones to the leftmost column. Then take Lim_{k=1..inf} ((T(n))^k, left-shifted vectors considered as sequences = rows of the array, deleting the first 1. The rows of this triangle sequence are the truncated columns of the array with one "1" per row.

Examples

			First few rows of the array are:
.
  1, 3, 8, 21, 55, 144, 377, 987, ...
  1, 1, 3,  6, 12,  24,  48,  96, ...
  1, 1, 1,  3,  6,  10,  17,  30, ...
  1, 1, 1,  1,  3,   6,  10,  15, ...
  1, 1, 1,  1,  1,   3,   6,  10, ...
  ...
First few rows of the triangle =
  1;
  1, 3;
  1, 3, 8;
  1, 3, 6, 21;
  1, 3, 6, 12, 55;
  1, 3, 6, 10, 24, 144;
  1, 3, 6, 10, 17, 48, 377;
  1, 3, 6, 10, 15, 30, 96, 987;
  1, 3, 6, 10, 15, 23, 53, 192, 2584;
  1, 3, 6, 10, 15, 21, 37, 93, 384, 6765;
  1, 3, 6, 10, 15, 21, 30, 61, 163, 768, 17711;
  1, 3, 6, 10, 15, 21, 28, 45, 100, 286, 1536, 46368;
  ...
Example: Row 2 of the array is generated from a variant of A004736, the leftmost column with two prepended 1's, = T(2):
  1;
  1;
  1;
  2, 1;
  3, 2, 1;
  ...
Take lim_{k->inf.} (P(2))^k, obtaining a left-shifted vector considered as a sequence; then delete the first 1, getting row 2 of the array.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Row sums are A171844.

Diagonals include A001906, A003945, A259968.

Cf. A004736.

Programs

PARI

T(n)={[Vec(p) | p<-Vec(sum(k=1, n, x^k*y^(k-1)*(1 - x^k)/((1 - x)*(1 - 2*x + x^2 - x^k)) + O(x*x^n)))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Apr 13 2021

Extensions

a(52) corrected and terms a(56) and beyond from Andrew Howroyd, Apr 13 2021