A227430 Expansion of x^2*(1-x)^3/((1-2*x)*(1-x+x^2)*(1-3*x+3x^2)).

0, 0, 1, 3, 6, 10, 15, 21, 29, 45, 90, 220, 561, 1365, 3095, 6555, 13110, 25126, 46971, 87381, 164921, 320001, 640002, 1309528, 2707629, 5592405, 11450531, 23166783, 46333566, 91869970, 181348455, 357913941, 708653429, 1410132405, 2820264810, 5662052980

Offset: 0

Author

Paul Curtz, Jul 11 2013

Keywords

• nonn
• easy

Comments

Consider the binomial transform of 0, 0, 0, 0, 0, 1 (period 6) with its differences:

0, 0, 0, 0, 0, 1, 6, 21, 56, 126,... d(n): after 0, it is A192080.

0, 0, 0, 0, 1, 5, 15, 35, 70, 126,... e(n)

0, 0, 0, 1, 4, 10, 20, 35, 56, 85,... f(n)

0, 0, 1, 3, 6, 10, 15, 21, 29, 45,... a(n)

0, 1, 2, 3, 4, 5, 6, 8, 16, 45,... b(n)

1, 1, 1, 1, 1, 1, 2, 8, 29, 85,... c(n)

0, 0, 0, 0, 0, 1, 6, 21, 56, 126,... d(n).

a(n) + d(n) = A024495(n),

b(n) + e(n) = A131708(n),

c(n) + f(n) = A024493(n).

a(n) - d(n) = 0, 0, 1, 3, 6, 9, 9, 0,... A057083(n-2)

b(n) - e(n) = 0, 1, 2, 3, 3, 0, -9, -27,... A057682(n)

c(n) - f(n) = 1, 1, 1, 0, -3, -9, -18, -27,... A057681(n)

d(n) - a(n) = 0, 0, -1, -3, -6, -9, -9, 0,... -A057083(n-2)

e(n) - b(n) = 0, -1, -2, -3, -3, 0, 9, 27,... -A057682(n)

f(n) - c(n) = -1, -1, -1, 0, 3, 9, 18, 27,... -A057681(n).

The first column is A131531(n).

The first two trisections are multiples of 3. Is the third (1, 10, 29,...) mod 9 A029898(n)?

Examples

			a(6)=6*10-15*6+20*3-15*1+6*0=15, a(7)=90-150+120-45+6=21.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6).

Programs

Mathematica

Join[{0},LinearRecurrence[{6,-15,20,-15,6},{0,1,3,6,10},40]] (* Harvey P. Dale, Dec 17 2014 *)

PARI

{a(n) = sum(k=0, n\6, binomial(n, 6*k+2))} \\ Seiichi Manyama, Mar 23 2019

Formula

a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) for n>5, a(0)=a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(5)=10.

a(n) = A024495(n) - A192080(n-5) for n>4.

G.f.: -(x^5 - 3*x^4 + 3*x^3 - x^2)/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)). - Ralf Stephan, Jul 13 2013

a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k+2). - Seiichi Manyama, Mar 23 2019

Extensions

Definition uses the g.f. of Ralf Stephan.

More terms from Harvey P. Dale, Dec 17 2014