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A143929 Eigentriangle by rows, termwise products of the natural numbers decrescendo and the bisected Fibonacci series. 1
1, 2, 1, 3, 2, 3, 4, 3, 6, 8, 5, 4, 9, 16, 21, 6, 5, 12, 24, 42, 55, 7, 6, 15, 32, 63, 110, 144, 8, 7, 18, 40, 84, 165, 288, 377, 9, 8, 21, 48, 105, 220, 432, 754, 987, 10, 9, 24, 56, 126, 275, 576, 1131, 1974, 2584 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Row sums = even indexed Fibonacci terms A001906.
Sum of n-th row terms = rightmost term of next row.
LINKS
FORMULA
Given A004736: (1; 2,1; 3,2,1; 4,3,2,1; ...), we apply the termwise products of the sequence A088305(n-1)}_{n>=1} starting (1, 1, 3, 8, 21, ...).
From Wolfdieter Lang, Jan 07 2021: (Start)
T(n, m) = 0 if n < m; T(n, 1) = n, for n >= 1, and T(n, m) = F(2*(m-1))*(n-m+1) for n >= m >= 2, with F = A000045 (Fibonacci).
G.f. column m: G(1, x) = x/(1-x)^2, G(m, x) = F(2*(m-1))*x^m/(1-x)^2, for m >= 2. (End)
With offset 0: g.f. of row polynomials R(n, x) := Sum_{m=0..n} t(n, m)*x^m, that is, g.f. of triangle t(n,m) = T(n+1, m+1):
G(z, x) = (1 - x*z)^2 / ((1 - z)^2*(1 - 3*x*z + (x*z)^2)). - Wolfdieter Lang, Apr 09 2021
EXAMPLE
First rows of the triangle T(n, m), n >= 1, m = 1..n:
1;
2, 1;
3, 2, 3;
4, 3, 6, 8;
5, 4, 9, 16, 21;
6, 5, 12, 24, 42, 55;
7, 6, 15, 32, 63, 110, 144;
8, 7, 18, 40, 84, 165, 288, 377;
9, 8, 21, 48, 105, 220, 432, 754, 987;
...
Example: row 4 = (4, 3, 6, 8) = termwise product of (4, 3, 2, 1) and (1, 1, 3, 8).
CROSSREFS
Sequence in context: A115872 A133926 A144337 * A153583 A346797 A029163
KEYWORD
nonn,easy,tabl
AUTHOR
Gary W. Adamson, Sep 05 2008
STATUS
approved

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Last modified March 28 13:21 EDT 2024. Contains 371254 sequences. (Running on oeis4.)