A349814 -id:A349814 - cp's OEIS frontend

A349819 Central column (ignoring the zeros) of A349813, or leading entries in rows of A349814.

3, 1, 3, 4, 20, 31, 182, 304, 1932, 3364, 22407, 40044, 275132, 500522, 3515564, 6478784, 46260604, 86094668, 622636764, 1167752848, 8531618640, 16100882359, 118615471982, 225001039696, 1669094344820, 3179713880524, 23725950404610, 45364387834120, 340192536782760

Offset: 0

N. J. A. Sloane, Dec 24 2021

nonn

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

Cf. A349813, A349814.

a(n) = if(n==0, 3, polcoef((-3 - x + x^2 + 3*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)](-3 - 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

A349813 Triangle read by rows: row 1 is [3]; for n >= 1, row n gives coefficients of expansion of (-3 - x + x^2 + 3x^3)(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.

Original entry on oeis.org

3, -3, -1, 1, 3, -3, -4, -3, 0, 3, 4, 3, -3, -7, -10, -10, -4, 4, 10, 10, 7, 3, -3, -10, -20, -30, -31, -20, 0, 20, 31, 30, 20, 10, 3, -3, -13, -33, -63, -91, -101, -81, -31, 31, 81, 101, 91, 63, 33, 13, 3, -3, -16, -49, -112, -200, -288, -336, -304, -182, 0, 182, 304, 336, 288, 200, 112, 49, 16, 3

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)*(3 + 4*x + 3*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 [-3, -1, 1, 3] instead of [1, 1, 1, 1].

			Triangle begins:
   3;
  -3,  -1,   1,   3;
  -3,  -4,  -3,   0,   3,    4,   3;
  -3,  -7, -10, -10,  -4,    4,  10,  10,  7,  3;
  -3, -10, -20, -30, -31,  -20,   0,  20, 31, 30,  20, 10,  3;
  -3, -13, -33, -63, -91, -101, -81, -31, 31, 81, 101, 91, 63, 33, 13, 3;
  ...

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, A349815, A349819.

The right half of the triangle gives A349814.

t1:=-3-x+x^2+3*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

A349819 Central column (ignoring the zeros) of A349813, or leading entries in rows of A349814.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions

A349813 Triangle read by rows: row 1 is [3]; for n >= 1, row n gives coefficients of expansion of (-3 - x + x^2 + 3x^3)(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

cp's OEIS Frontend

A349819 Central column (ignoring the zeros) of A349813, or leading entries in rows of A349814.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions

A349813 Triangle read by rows: row 1 is [3]; for n >= 1, row n gives coefficients of expansion of (-3 - x + x^2 + 3*x^3)*(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A349813 Triangle read by rows: row 1 is [3]; for n >= 1, row n gives coefficients of expansion of (-3 - x + x^2 + 3x^3)(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.