A099588 Coefficient of x^2 in (1+x)^n mod 1+x^4.

0, 0, 1, 3, 6, 10, 14, 14, 0, -48, -164, -396, -792, -1352, -1912, -1912, 0, 6528, 22288, 53808, 107616, 183712, 259808, 259808, 0, -887040, -3028544, -7311552, -14623104, -24963200, -35303296, -35303296, 0, 120532992, 411525376, 993510144, 1987020288, 3392055808, 4797091328, 4797091328

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.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-2).

Crossrefs

Cf. A099586, A099587, A099589.

Programs

Maple

f:= rectoproc({rec, a(0)=0,a(1)=0,a(2)=1,a(3)=3},a(n),remember):
map(f, [$0..100]); # Robert Israel, Jun 30 2017

Mathematica

RecurrenceTable[{a[n]==4*a[n-1] - 6*a[n-2] + 4*a[n-3] - 2*a[n-4], a[1]=0, a[2]=1, a[3]=3, a[4]=6}, a, {n, 1, 200}] (* G. C. Greubel, Nov 10 2015 *)
Table[Sum[(-1)^k*Binomial[n, 4 k + 2], {k, 0, n}], {n, 0, 36}] (* Michael De Vlieger, Jun 30 2017 *)
a[n_] := n*(n-1)/2 HypergeometricPFQ[{(2-n)/4, (3-n)/4, (4-n)/4, (5-n)/4}, {3/4, 5/4, 3/2}, -1]; Array[a, 40, 0] (* Jean-François Alcover, Jul 20 2017, from Vladimir Shevelev's first formula *)

PARI

x='x+O('x^55); concat([0, 0], Vec(-x^2*(x-1)/(2*x^4-4*x^3+6*x^2-4*x+1))) \\ Altug Alkan, Nov 11 2015

PARI

a(n) = sum(t=0, (n-2)\4, (-1)^t*binomial(n,4*t+2)); \\ Michel Marcus, Jun 30 2017

PARI

a(n)=polcoeff(lift(Mod(1+x,1+x^4)^n),2); \\ Joerg Arndt, Feb 22 2018

Formula

G.f.: -x^2*(x-1) / (2*x^4-4*x^3+6*x^2-4*x+1). - Colin Barker, Jul 15 2013

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - 2*a(n-4). - G. C. Greubel, Nov 10 2015

a(n) = (1/2)*((2+sqrt(2))^(n/2) * sin(n*Pi/8) - (2-sqrt(2))^(n/2)*sin(3*n*Pi/8)). - G. C. Greubel, Nov 10 2015

From Vladimir Shevelev, Jul 04 2017 (Start)

a(n) = Sum_{k>=0}(-1)^k*binomial(n,4*k+2);

a(n) = round((2+sqrt(2))^(n/2)*cos(Pi*(n-4)/8)/2), where round(x) is the integer nearest to x;

a(n+m) = a(n)*K_1(m) + K_2(n)*K_2(m) + K_1(n)*a(m) - K_4(n)*K_4(m), where K_1 is A099586, K_2 is A099587, K_4 is A099589. (End)

Extensions

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

a(673) in b-file corrected by Andrew Howroyd, Feb 21 2018