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

0,4

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)?

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).

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Sequence in context: A175724 A335184 A101551 * A051166 A373934 A188278

Adjacent sequences: A227427 A227428 A227429 * A227431 A227432 A227433

Keyword

nonn,easy

Author

Paul Curtz, Jul 11 2013

Extensions

Definition uses the g.f. of Ralf Stephan.

More terms from Harvey P. Dale, Dec 17 2014

Status

approved