A349816 -id:A349816 - cp's OEIS frontend

A349818 Central column (ignoring the zeros) of A349815, or leading entries in rows of A349816.

1, 1, 1, 2, 8, 13, 74, 124, 784, 1364, 9069, 16194, 111144, 202070, 1418196, 2612376, 18642208, 34682348, 250706548, 470066728, 3433030048, 6477333149, 47703926354, 90472092748, 670963514192, 1278016171132, 9534032792470, 18226719157820, 136658371126320

Offset: 0

N. J. A. Sloane, Dec 24 2021

nonn

Note that the analogous sequence for A349812 is the Motzkin numbers A001006.

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A001006, A349812, A349815, A349816.

a(n) = if(n==0, 1, polcoef((-1 - x + x^2 + x^3)*(1 + x + x^2 + x^3)^(n-1), 3*n\2+1)) \\ Andrew Howroyd, Feb 28 2023

a(n) = [x^floor(3*n/2+1)](-1 - x + x^2 + 3*x^3)*(1 + x + x^2 + x^3)^(n-1) for n > 0. - Andrew Howroyd, Feb 28 2023

Offset corrected and terms a(21) and beyond from Andrew Howroyd, Feb 28 2023

A349815 Triangle read by rows: row 1 is [1]; for n >= 1, row n gives coefficients of expansion of (-1 - x + x^2 + x^3)*(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.

Original entry on oeis.org

1, -1, -1, 1, 1, -1, -2, -1, 0, 1, 2, 1, -1, -3, -4, -4, -2, 2, 4, 4, 3, 1, -1, -4, -8, -12, -13, -8, 0, 8, 13, 12, 8, 4, 1, -1, -5, -13, -25, -37, -41, -33, -13, 13, 33, 41, 37, 25, 13, 5, 1, -1, -6, -19, -44, -80, -116, -136, -124, -74, 0, 74, 124, 136, 116, 80, 44, 19, 6, 1, -1, -7, -26, -70, -149, -259, -376, -456, -450, -334, -124, 124, 334, 450, 456, 376, 259, 149, 70, 26, 7, 1

Offset: 0

N. J. A. Sloane, Dec 23 2021

The row polynomials can be further factorized, since -3 - x + x^2 + 3*x^3 = -(1-x)*(1+x)^2 and 1 + x + x^2 + x^3 = (1+x)*(1+x^2).
The rule for constructing this triangle (ignoring row 0) is the same as that for A008287: each number is the sum of the four numbers immediately above it in the previous row. Here row 1 is [-1, -1, 1, 3] instead of [1, 1, 1, 1].
This is a companion to A008287 and A349813.

			Triangle begins:
   1;
  -1, -1,   1,   1;
  -1, -2,  -1,   0,   1,   2,   1;
  -1, -3,  -4,  -4,  -2,   2,   4,   4,  3,  1;
  -1, -4,  -8, -12, -13,  -8,   0,   8, 13, 12,  8,  4,  1;
  -1, -5, -13, -25, -37, -41, -33, -13, 13, 33, 41, 37, 25, 13, 5, 1;
  ...

Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965 [Mentions application to design of antenna arrays. Annotated scan.]

Cf. A007318, A008287, A349812, A349813.

The right half of the triangle gives A349816. For the central nonzero entries see A349818.

t1:=-1-x+x^2+x^3;
m:=1+x+x^2+x^3;
lprint([3]);
for n from 1 to 12 do
w1:=expand(t1*m^(n-1));
w4:=series(w1,x,3*n+1);
w5:=seriestolist(w4);
lprint(w5);
od:

cp's OEIS Frontend

A349818 Central column (ignoring the zeros) of A349815, or leading entries in rows of A349816.

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