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
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)
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
Comments