A099589 Expansion of x^3 / (1 - 4*x + 6*x^2 - 4*x^3 + 2*x^4).

0, 0, 0, 1, 4, 10, 20, 34, 48, 48, 0, -164, -560, -1352, -2704, -4616, -6528, -6528, 0, 22288, 76096, 183712, 367424, 627232, 887040, 887040, 0, -3028544, -10340096, -24963200, -49926400, -85229696, -120532992, -120532992, 0, 411525376, 1405035520, 3392055808

Offset: 0

Author

Ralf Stephan, Oct 24 2004

Keywords

• sign
• easy

Comments

{A099586, A099587, A099588, A099589} is the difference analog of the trigonometric functions {k_1(x), k_2(x), k_3(x), k_4(x)} of order 4. For the definition, see [Erdelyi] and the Shevelev link. - Vladimir Shevelev, Jul 04 2017

References

• A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
• Vladimir Shevelev, Coefficient of x^k in ((x+1)^n modulo x^N+1), seqfan, Thu Jul 20 2017.
• G. Tollisen and T. Lengyel, A congruential identity and the 2-adic order of lacunary sums of binomial coefficients, Integers 4 (2004), #A4.
• Maran van Heesch, The multiplicative complexity of symmetric functions over a field with characteristic p, Thesis, 2014.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-2).

Crossrefs

Cf. A099586, A099587, A099588.

Programs

Mathematica

Round@Table[(1/(2*Sqrt[2]))*((2-Sqrt[2])^(n/2)*(Cos[3*Pi*n/8] + Sin[3*Pi*n/8]) + (2+Sqrt[2])^(n/2)*(Sin[Pi*n/8] - Cos[Pi*n/8])), {n, 0, 40}] (* G. C. Greubel, Nov 07 2015 *)
RecurrenceTable[{a[n] == 4*a[n-1] - 6*a[n-2] + 4*a[n-3] - 2*a[n-4], a[0]==0, a[1]==0, a[2]==0, a[3]==1}, a, {n, 0, 40}] (* G. C. Greubel, Nov 10 2015 *)
Table[Sum[(-1)^k*Binomial[n, 4 k + 3], {k, 0, n}], {n, 0, 37}] (* Michael De Vlieger, Jun 30 2017 *)
a[n_] := n*(n-1)*(n-2)/6 HypergeometricPFQ[{(3-n)/4, (4-n)/4, (5-n)/4, (6-n)/4}, {5/4, 3/2, 7/4}, -1]; Array[a, 40, 0] (* Jean-François Alcover, Jul 20 2017, from Vladimir Shevelev's first formula *)

PARI

a(n) = polcoeff(((1+x)^n)%(x^4+1),3)

PARI

concat([0, 0], Vec(x^3/((1-x)^4+x^4) + O(x^50))) \\ Altug Alkan, Nov 08 2015

PARI

a(n) = sum(t=0, (n-3)\4, (-1)^t*binomial(n,4*t+3)); \\ Michel Marcus, Jun 30 2017

Formula

G.f.: x^3/((1-x)^4 + x^4), the binomial transform of x^3/(1+x^4). - Paul Barry, Apr 01 2005

Coefficient of x^3 in (1+x)^n mod (1 + x^4).

a(n) = (1/(2*sqrt(2)))*((2-sqrt(2))^(n/2)*(cos(3*Pi*n/8) + sin(3*Pi*n/8)) + (2+sqrt(2))^(n/2)*(sin(Pi*n/8) - cos(Pi*n/8))). - Paul Barry, Apr 01 2005

From Colin Barker, Nov 08 2015: (Start)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - 2*a(n-4) for n > 4.

G.f.: x^3 / (2*x^4 - 4*x^3 + 6*x^2 - 4*x + 1). (End)

From Vladimir Shevelev, Jul 04 2017: (Start)

a(n) = Sum_{t >= 0} (-1)^t*binomial(n,4*t+3).

a(n) = round((2+sqrt(2))^(n/2)*cos(Pi*(n-6)/8)/2), where round(x) is the integer nearest to x.

a(n+m) = a(n)*K_1(m) + K_3(n)*K_2(m) + K_2(n)*K_3(m) + K_1(n)*a(m), where

K_1 is A099586, K_2 is A099587, K_3 is A099588. (End)

Extensions

a(0)=0 added by N. J. A. Sloane, Jul 04 2017