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%I #14 Apr 14 2021 05:23:28
%S 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,
%T 1,3,6,10,15,30,96,987,1,3,6,10,15,23,53,192,2584,1,3,6,10,15,21,37,
%U 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
%N Triangle read by rows = truncated columns of an array formed by variants of the natural number decrescendo triangle, A004736.
%C Rows tend to the triangular series, A000217.
%C 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.
%H Andrew Howroyd, <a href="/A171843/b171843.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%e First few rows of the array are:
%e .
%e 1, 3, 8, 21, 55, 144, 377, 987, ...
%e 1, 1, 3, 6, 12, 24, 48, 96, ...
%e 1, 1, 1, 3, 6, 10, 17, 30, ...
%e 1, 1, 1, 1, 3, 6, 10, 15, ...
%e 1, 1, 1, 1, 1, 3, 6, 10, ...
%e ...
%e First few rows of the triangle =
%e 1;
%e 1, 3;
%e 1, 3, 8;
%e 1, 3, 6, 21;
%e 1, 3, 6, 12, 55;
%e 1, 3, 6, 10, 24, 144;
%e 1, 3, 6, 10, 17, 48, 377;
%e 1, 3, 6, 10, 15, 30, 96, 987;
%e 1, 3, 6, 10, 15, 23, 53, 192, 2584;
%e 1, 3, 6, 10, 15, 21, 37, 93, 384, 6765;
%e 1, 3, 6, 10, 15, 21, 30, 61, 163, 768, 17711;
%e 1, 3, 6, 10, 15, 21, 28, 45, 100, 286, 1536, 46368;
%e ...
%e Example: Row 2 of the array is generated from a variant of A004736, the leftmost column with two prepended 1's, = T(2):
%e 1;
%e 1;
%e 1;
%e 2, 1;
%e 3, 2, 1;
%e ...
%e 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.
%o (PARI)
%o 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)))]}
%o { my(A=T(10)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Apr 13 2021
%Y Row sums are A171844.
%Y Diagonals include A001906, A003945, A259968.
%Y Cf. A004736.
%K nonn,tabl
%O 1,3
%A _Gary W. Adamson_, Dec 19 2009
%E a(52) corrected and terms a(56) and beyond from _Andrew Howroyd_, Apr 13 2021
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