A143929 Eigentriangle by rows, termwise products of the natural numbers decrescendo and the bisected Fibonacci series.

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

Offset

1,2

Comments

Row sums = even-indexed Fibonacci terms A001906.

Sum of n-th row terms = rightmost term of next row.

Links

Table of n, a(n) for n=1..55.

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

Cf. A000045, A001906, A004736, A088305.

Sequence in context: A115872 A133926 A144337 * A153583 A346797 A029163

Adjacent sequences: A143926 A143927 A143928 * A143930 A143931 A143932

Keyword

nonn,easy,tabl

Author

Gary W. Adamson, Sep 05 2008

Status

approved