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

1,3

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.

Links

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

Example

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.

Prog

(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

Crossrefs

Row sums are A171844.

Diagonals include A001906, A003945, A259968.

Cf. A004736.

Sequence in context: A139123 A133580 A019603 * A132476 A328807 A103279

Adjacent sequences: A171840 A171841 A171842 * A171844 A171845 A171846

Keyword

nonn,tabl

Author

Gary W. Adamson, Dec 19 2009

Extensions

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

Status

approved