A349815 -id:A349815 - 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

A349816 Irregular triangle read by rows: the right-hand side of the triangle in A349815.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 4, 4, 3, 1, 8, 13, 12, 8, 4, 1, 13, 33, 41, 37, 25, 13, 5, 1, 74, 124, 136, 116, 80, 44, 19, 6, 1, 124, 334, 450, 456, 376, 259, 149, 70, 26, 7, 1, 784, 1364, 1616, 1541, 1240, 854, 504, 252, 104, 34, 8, 1, 1364, 3764, 5305, 5761, 5251, 4139, 2850, 1714, 894, 398, 147, 43, 9, 1, 9069, 16194, 20081, 20456, 18001, 13954, 9597, 5856, 3153, 1482, 597, 200, 53, 10, 1

Offset: 0

N. J. A. Sloane, Dec 23 2021

It seems more symmetrical to omit the central column of zeros in A349815.

			Triangle begins:
    1;
    1,    1;
    1,    2,    1;
    2,    4,    4,    3,    1;
    8,   13,   12,    8,    4,   1;
   13,   33,   41,   37,   25,  13,   5,   1;
   74,  124,  136,  116,   80,  44,  19,   6,   1;
  124,  334,  450,  456,  376, 259, 149,  70,  26,  7, 1;
  784, 1364, 1616, 1541, 1240, 854, 504, 252, 104, 34, 8, 1;
  ...

Cf. A348815, A349818.

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

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

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A349816 Irregular triangle read by rows: the right-hand side of the triangle in A349815.

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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.

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Maple

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A349816 Irregular triangle read by rows: the right-hand side of the triangle in A349815.

Views

Author

Keywords

Comments

Examples

Crossrefs

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.