

A317499


Coefficients in expansion of 1/(1 + 2*x  3*x^3).


2



1, 2, 4, 5, 4, 4, 23, 58, 104, 139, 104, 104, 625, 1562, 2812, 3749, 2812, 2812, 16871, 42178, 75920, 101227, 75920, 75920, 455521, 1138802, 2049844, 2733125, 2049844, 2049844, 12299063, 30747658, 55345784, 73794379, 55345784, 55345784
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

The coefficients in the expansion of 1/(1 + 2*x  3*x^3) are given by the sequence generated by the row sums in triangle A317503.
Coefficients in expansion of 1/(1 + 2*x  3*x^3) are given by the sum of numbers along second Layer skew diagonals pointing topleft in triangle A303901 ((32x)^n) and by the sum of numbers along second Layer skew diagonals pointing topright in triangle A317498 ((2+3x)^n), see links.


REFERENCES

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 9781999591403, pp. 396, 397.


LINKS

Colin Barker, Table of n, a(n) for n = 0..1000
Zagros Lalo, Second layer skew diagonals in centerjustified triangle of coefficients in expansion of (3  2x)^n
Zagros Lalo, Second layer skew diagonals in centerjustified triangle of coefficients in expansion of (2 + 3x)^n
Index entries for linear recurrences with constant coefficients, signature (2,0,3).


FORMULA

a(0)=1, a(n) = 2*a(n1) + 3*a(n3) for n = 0,1...; a(n)=0 for n < 0.
a(n) = (2^(n)*(2^n + (3i*sqrt(3))^n*(32*i*sqrt(3)) + (3+i*sqrt(3))^n*(3+2*i*sqrt(3)))) / 7 where i=sqrt(1).  Colin Barker, Aug 02 2018


MAPLE

seq(coeff(series(1/(1+2*x3*x^3), x, n+1), x, n), n=0..40); # Muniru A Asiru, Aug 01 2018


MATHEMATICA

CoefficientList[Series[1/(1 + 2 x  3 x^3), {x, 0, 40}], x].
a[0] = 1; a[n_] := a[n] = If[n < 0, 0, 2 * a[n  1] + 3 * a[n  3]]; Table[a[n], {n, 0, 40}] // Flatten.
LinearRecurrence[{2, 0, 3}, {1, 2, 4}, 41].


PROG

(GAP) a:=[1, 2, 4];; for n in [4..40] do a[n]:=2*a[n1]+3*a[n3]; od; a; # Muniru A Asiru, Aug 01 2018
(PARI) Vec(1 / ((1  x)*(1 + 3*x + 3*x^2)) + O(x^40)) \\ Colin Barker, Aug 02 2018


CROSSREFS

Cf. A317502, A317503.
Cf. A303901, A317498.
Sequence in context: A123546 A339069 A334422 * A004581 A212790 A317558
Adjacent sequences: A317496 A317497 A317498 * A317500 A317501 A317502


KEYWORD

sign,easy


AUTHOR

Zagros Lalo, Jul 31 2018


STATUS

approved



