A090132 -id:A090132 - cp's OEIS frontend

A009116 Expansion of e.g.f. cos(x) / exp(x).

1, -1, 0, 2, -4, 4, 0, -8, 16, -16, 0, 32, -64, 64, 0, -128, 256, -256, 0, 512, -1024, 1024, 0, -2048, 4096, -4096, 0, 8192, -16384, 16384, 0, -32768, 65536, -65536, 0, 131072, -262144, 262144, 0, -524288, 1048576, -1048576, 0, 2097152, -4194304

Offset: 0

R. H. Hardin

Apart from signs, generated by 1,1 position of H_2^n = [1,1;-1,1]^n; and a(n) = 2^(n/2)*cos(Pi*n/2). - Paul Barry, Feb 18 2004
Equals binomial transform of "Period 4, repeat [1, 0, -1, 0]". - Gary W. Adamson, Mar 25 2009
Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - R. J. Mathar, Aug 10 2012

			G.f. = 1 - x + 2*x^3 - 4*x^4 + 4*x^5 - 8*x^7 + 16*x^8 - 16*x^9 + 32*x^11 - 64*x^12 + ...

N. J. A. Sloane, Table of n, a(n) for n = 0..2000
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
N. J. A. Sloane, Table of n, (I-1)^n for n=0..100
Index entries for linear recurrences with constant coefficients, signature (-2,-2).

Cf. A009545, A016116, A075553, A090132, A098158, A099087, A146559.
(With different signs) row sums of triangle A104597.
Also related to A066321 and A271472.

R:=PowerSeriesRing(Rationals(), 50); Coefficients(R!(Laplace( Exp(-x)*Cos(x) ))); // G. C. Greubel, Jul 22 2018; Apr 17 2023

A009116 := n->add((-1)^j*binomial(n,2*j),j=0..floor(n/2));

n = 50; (* n = 2 mod 4 *) (CoefficientList[ Series[ Cos[x]/Exp[x], {x, 0, n}], x]* Table[k!, {k,0,n-1}] )[[1 ;; 45]] (* Jean-François Alcover, May 18 2011 *)
Table[(1/2)*((-1-I)^n + (-1+I)^n), {n,0,50}] (* Jean-François Alcover, Jan 31 2018 *)

{a(n) = if( n<0, 0, polcoeff( (1 + x) / (1 + 2*x + 2*x^2) + x * O(x^n), n))} /* Michael Somos, Nov 17 2002 */

def A009116(n): return 2^(n/2)*chebyshev_T(n, -1/sqrt(2))
[A009116(n) for n in range(41)] # G. C. Greubel, Apr 17 2023

Real part of (-1-i)^n. See A009545 for imaginary part. - Marc LeBrun
a(n) = -2 * (a(n-1) + a(n-2)); a(0)=1, a(1)=-1. - Michael Somos, Nov 17 2002
G.f.: (1 + x) / (1 + 2*x + 2*x^2).
E.g.f.: cos(x) / exp(x).
a(n) = Sum_{k=0..n} (-1)^k*A098158(n,k). - Philippe Deléham, Dec 04 2006
a(n)*(-1)^n = A099087(n) - A099087(n-1). - R. J. Mathar, Nov 18 2007
a(n) = (-1)^n*A146559(n). - Philippe Deléham, Dec 01 2008
From Paul Curtz, Jul 22 2011: (Start)
a(n) = -4*a(n-4).
a(n) = A016116(n) * A075553(n+6). (End)
E.g.f.: cos(x)/exp(x) = 1 - x/(G(0)+1), where G(k) = 4k+1-x+(x^2)*(4k+1)/((2k+1)*(4k+3)-(x^2)+x*(2k+1)*(4k+3)/( 2k+2-x+x*(2k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
a(n) = (-1)^n*2^(n/2)*cos(n*Pi/4). - Nordine Fahssi, Dec 18 2013
a(n) = (-1)^floor((n+1)/2)*2^(n-1)*H(n, n mod 2, 1/2) for n >= 3 where H(n, a, b) = hypergeom([a - n/2, b - n/2], [1 - n], 2). - Peter Luschny, Sep 03 2019
a(n) = 2^(n/2)*ChebyshevT(n, -1/sqrt(2)). - G. C. Greubel, Apr 17 2023
a(n) = A108520(n-1)+A108520(n). - R. J. Mathar, May 09 2023

Extended with signs by Olivier Gérard, Mar 15 1997

Definition corrected by Joerg Arndt, Apr 29 2011

A108520 Expansion of 1/(1+2x+2x^2).

Original entry on oeis.org

1, -2, 2, 0, -4, 8, -8, 0, 16, -32, 32, 0, -64, 128, -128, 0, 256, -512, 512, 0, -1024, 2048, -2048, 0, 4096, -8192, 8192, 0, -16384, 32768, -32768, 0, 65536, -131072, 131072, 0, -262144, 524288, -524288, 0, 1048576, -2097152, 2097152, 0, -4194304, 8388608, -8388608

Offset: 0

Michael Somos, Jun 07 2005

Yet another variation on A009545.

Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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 (-2,-2).

a(n) = (-1)^n * A099087(n). a(n) = -A084102(n) if n>0.

Cf. A009116, A009545, A016116, A075553, A084102, A099087.

[n le 2 select n*(-1)^(n-1) else -2*(Self(n-1)+Self(n-2)): n in [1..47]];  // Bruno Berselli, Apr 26 2011

A108520 := n -> `if`(n=0, 1, (-2)^n*hypergeom([1/2-n/2, -n/2], [-n], 2)):
seq(simplify(A108520(n)), n=0..46); # Peter Luschny, Dec 17 2015

CoefficientList[Series[1/(1+2x+2x^2), {x,0,50}], x] (* or *) LinearRecurrence[{-2,-2}, {1,-2}, 50] (* Harvey P. Dale, Sep 30 2012 *)
Table[-(-1-I)^(n-1) - (-1+I)^(n-1), {n, 0, 50}] (* Bruno Berselli, Nov 08 2015 *)
Im[(-1+I)^Range[51]] (* G. C. Greubel, Apr 24 2023 *)

a(n)=if(n<0, 0, polcoeff(1/(1+2*x+2*x^2)+x*O(x^n),n))

a(n)=if(n<1, n==0, -polsym(2+2*x+x^2,n-1)[n])

vector(66,n,imag((-1+I)^n)) /* Joerg Arndt, May 13 2011 */

[imag((-1+I)^(n+1)) for n in range(51)] # G. C. Greubel, Apr 24 2023

G.f.: 1/(1+2*x+2*x^2).
E.g.f.: exp(-x)*(cos(x) - sin(x)).
a(n) = -2*(a(n-1) + a(n-2)).
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(k,j)*C(k,n-j)*(-2)^(n-j). - Paul Barry, Mar 09 2006
a(n) = -4 * a(n-4). - Paul Curtz, Apr 24 2011
a(n) = A016116(n+1) * A075553(n+1). - Paul Curtz, Apr 25 2011
From Bruno Berselli, Apr 26 2011: (Start)
a(n) = -(-1-i)^(n-1) - (-1+i)^(n-1), where i=sqrt(-1).
a(n) = -2*A009116(n-1) for n > 0. (End)
Imaginary part of (-1+i)^n, negated real part is A090132. - Joerg Arndt, May 13 2011
E.g.f.: (cos(x) - sin(x))*exp(-x) = G(0); G(k) = 1 - 2*x/(4*k+1+x*(4*k+1)/(2*(2*k+1) -x -2*(x^2)*(2*k+1)/((x^2) -(2*k+2)*(4*k+3)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2011
G.f.: G(0)/(2*(1+x)), where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
a(n) = (-2)^n*hypergeom([1/2-n/2, -n/2], [-n], 2) for n >= 1. - Peter Luschny, Dec 17 2015

A026998 Triangular array T read by rows: T(n, k) = t(n, 2k), t given by A027960, 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 4, 8, 1, 1, 4, 11, 13, 1, 1, 4, 11, 26, 19, 1, 1, 4, 11, 29, 54, 26, 1, 1, 4, 11, 29, 73, 101, 34, 1, 1, 4, 11, 29, 76, 171, 174, 43, 1, 1, 4, 11, 29, 76, 196, 370, 281, 53, 1, 1, 4, 11, 29, 76, 199, 487, 743, 431, 64, 1

Offset: 0

Clark Kimberling

Right-edge columns are polynomials approximating Lucas(2n+1).

			  .................................... 1;
  ................................. 1, 1;
  ............................. 1,  4, 1;
  ........................ 1,   4,  8, 1;
  ................... 1,   4,  11, 13, 1;
  .............. 1,   4,  11,  26, 19, 1;
  .......... 1,  4,  11,  29,  54, 26, 1;
  ...... 1,  4, 11,  29,  73, 101, 34, 1;
  .. 1,  4, 11, 29,  76, 171, 174, 43, 1;
  1, 4, 11, 29, 76, 196, 370, 281, 53, 1;

G. C. Greubel, Table of n, a(n) for n = 0..1325

This is a bisection of the "Lucas array" A027960, see A027011 for the other bisection.
Row sums give A095121.
Signed row sums give A090132.
Diagonal sums give A027010.
Right-edge columns include A034856, A027966, A027968, A027970, A027972.
Cf. A000032.

function t(n, k) // t = A027960
      if k le n then return Lucas(k+1);
      elif k gt 2*n then return 0;
      else return t(n-1, k-2) + t(n-1, k-1);
      end if;
end function;
A026998:= func< n,k | t(n, 2*k) >;
[A026998(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2025

f[n_, k_]:= f[n, k]= Sum[Binomial[2*n-k+j,j]*LucasL[2*(k-n-j)], {j,0,k-n-1}];
A027960[n_, k_]:= LucasL[k+1] - f[n,k]*Boole[k>n];
A026998[n_, k_]:= A027960[n,2*k];
Table[A026998[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 09 2025 *)

@CachedFunction
def t(n, k): # t = A027960
    if (k>2*n): return 0
    elif (kA026998(n,k): return t(n, 2*k)
print(flatten([[A026998(n, k) for k in (0..n)] for n in (0..12)])) # G. C. Greubel, Jul 09 2025

T(n, k) = Lucas(2*n+1) = A002878(n) for 2*k <= n, otherwise the (2*n-2*k)-th coefficient of the power series for (1+2*x)/( (1-x-x^2)*(1-x)^(2*k-n) ).

Edited by Ralf Stephan, May 05 2005

A104967 Matrix inverse of triangle A104219, read by rows, where A104219(n,k) equals the number of Schroeder paths of length 2n having k peaks at height 1.

Original entry on oeis.org

1, -1, 1, -1, -2, 1, -1, -1, -3, 1, -1, 0, 0, -4, 1, -1, 1, 2, 2, -5, 1, -1, 2, 3, 4, 5, -6, 1, -1, 3, 3, 3, 5, 9, -7, 1, -1, 4, 2, 0, 0, 4, 14, -8, 1, -1, 5, 0, -4, -6, -6, 0, 20, -9, 1, -1, 6, -3, -8, -10, -12, -14, -8, 27, -10, 1, -1, 7, -7, -11, -10, -10, -14, -22, -21, 35, -11, 1, -1, 8, -12, -12, -5, 0, 0, -8, -27, -40, 44, -12, 1

Offset: 0

Paul D. Hanna, Mar 30 2005

Row sums equal A090132 with odd-indexed terms negated. Absolute row sums form A104968. Row sums of squared terms gives A104969.

Riordan array ((1-2*x)/(1-x), x(1-2*x)/(1-x)). - Philippe Deléham, Dec 05 2015

			Triangle begins:
   1;
  -1,  1;
  -1, -2,  1;
  -1, -1, -3,  1;
  -1,  0,  0, -4,  1;
  -1,  1,  2,  2, -5,  1;
  -1,  2,  3,  4,  5, -6,  1;
  -1,  3,  3,  3,  5,  9, -7,  1;
  -1,  4,  2,  0,  0,  4, 14, -8,  1;
  -1,  5,  0, -4, -6, -6,  0, 20, -9, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1080

Cf. A078011, A090132, A104968, A104969, A134824, A153881.

Cf. A347171 (rows reversed, up to signs).

A104967:= func< n,k | (&+[(-2)^j*Binomial(k+1, j)*Binomial(n-j, k): j in [0..n-k]]) >;
[A104967(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 09 2021

A104967:= (n,k)-> add( (-2)^j*binomial(k+1, j)*binomial(n-j, k), j=0..n-k);
seq(seq( A104967(n,k), k=0..n), n=0..12); # G. C. Greubel, Jun 09 2021

T[n_, k_]:= T[n, k]= Which[k==n, 1, k==0, 0, True, T[n-1, k-1] - Sum[T[n-i, k-1], {i, 2, n-k+1}]];
Table[T[n, k], {n, 13}, {k, n}]//Flatten (* Jean-François Alcover, Jun 11 2019, after Peter Luschny *)

T(n,k):=sum((-2)^i*binomial(k+1,i)*binomial(n-i,k),i,0,n-k); /* Vladimir Kruchinin, Nov 02 2011 */

{T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1-2*X)/(1-X-X*Y*(1-2*X)),n,x),k,y)}
for(n=0, 16, for(k=0, n, print1(T(n, k), ", ")); print(""))

def A104967_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-sum(prec(n-i,k-1) for i in (2..n-k+1))
    return [prec(n, k) for k in (1..n)]
for n in (1..10): print(A104967_row(n)) # Peter Luschny, Mar 16 2016

G.f.: A(x, y) = (1-2*x)/(1-x - x*y*(1-2*x)).
Sum_{k=0..n} T(n, k) = (-1)^n*A090132(n).
Sum_{k=0..n} abs(T(n, k)) = A104968(n).
Sum_{k=0..n} T(n, k)^2 = A104969(n).
T(n,k) = Sum_{i=0..n-k} (-2)^i*binomial(k+1,i)*binomial(n-i,k). - Vladimir Kruchinin, Nov 02 2011
Sum_{k=0..floor(n/2)} T(n-k, k) = A078011(n+2). - G. C. Greubel, Jun 09 2021

A137470 Inverse binomial transform of 1, 2, 2, 4, 10, 20, ... = A100088.

Original entry on oeis.org

1, 1, -1, 3, -1, -1, 7, -9, 7, 7, -25, 39, -25, -25, 103, -153, 103, 103, -409, 615, -409, -409, 1639, -2457, 1639, 1639, -6553, 9831, -6553, -6553, 26215, -39321, 26215, 26215, -104857, 157287, -104857, -104857, 419431

Offset: 0

Paul Curtz, Apr 20 2008

Index entries for linear recurrences with constant coefficients, signature (-1,0,2)

a(n)=[3+(1-2i)(i-1)^n+(1+2i)(-1-i)^n]/5 where i=sqrt(-1). - R. J. Mathar, Apr 25 2008
O.g.f.: -(1+2x)/((1+2x+2x^2)(-1+x)). - R. J. Mathar, Apr 25 2008
a(n+1)-a(n)=A090132(n+1). - R. J. Mathar, Apr 25 2008
G.f.: Q(0)*(1+2*x)/(2- 2*x), where Q(k) = 1 + 1/(1 - x*(4*k+2 +2*x)/(x*(4*k+4 +2*x) - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 01 2014

More terms from R. J. Mathar, Apr 25 2008

A108086 Triangle, read by rows, where T(0,0) = 1, T(n,k) = (-1)^(n+k)*T(n-1,k) + T(n-1,k-1); a signed version of Pascal's triangle.

Original entry on oeis.org

1, -1, 1, -1, -2, 1, 1, -3, -3, 1, 1, 4, -6, -4, 1, -1, 5, 10, -10, -5, 1, -1, -6, 15, 20, -15, -6, 1, 1, -7, -21, 35, 35, -21, -7, 1, 1, 8, -28, -56, 70, 56, -28, -8, 1, -1, 9, 36, -84, -126, 126, 84, -36, -9, 1, -1, -10, 45, 120, -210, -252, 210, 120, -45, -10, 1, 1, -11, -55, 165, 330, -462, -462, 330, 165, -55, -11, 1

Offset: 0

Gerald McGarvey, Jun 05 2005

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000984, A001700, A002054, A007318, A009116.

Cf. A090132, A108520, A260192, A333378.

A108086:= func< n,k | (-1)^Floor((n-k+1)/2)*Binomial(n,k) >;
[A108086(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 02 2022

A108086[n_, k_]:= (-1)^(Floor[(n-k+1)/2])*Binomial[n, k];
Table[A108086[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 02 2022 *)

def A108086(n,k): return (-1)^int((n-k+1)/2)*binomial(n,k)
flatten([[A108086(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 02 2022

T(n,k) = (-1)^(n+k)*T(n-1,k) + T(n-1,k-1), with T(0, 0) = 1.
T(n, k) = (-1)^floor((n-k+1)/2) * A007318(n, k).
From G. C. Greubel, Dec 02 2022: (Start)
T(2*n, n) = (-1)^binomial(n+1,2) * A000984(n).
T(2*n, n+1) = (-1)^binomial(n,2) * A001791(n), n >= 1.
T(2*n, n-1) = (-1)^binomial(n+2,2) * A001791(n).
T(2*n+1, n-1) = (-1)^binomial(n-1,2) * A002054(n).
T(2*n+1, n+1) = (-1)^binomial(n+1,2) * A001700(n+1).
Sum_{k=0..n} T(n, k) = (-1)^n * A090132(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A108520(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n * A260192(n-1).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A333378(n+1). (End)

A229502 Expansion of q * f(-q) * f(-q^4) * f(-q^16) * f(q, -q^3) in powers of q where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 0, -2, -2, 0, 2, 2, 4, -1, -2, 0, 0, -2, -4, 2, -4, -2, 2, 6, 4, 0, 2, -6, -4, 3, 2, 0, 4, 6, 0, -8, 0, -2, 0, -4, -2, 0, -6, 2, -4, 0, 4, 0, -4, 2, 12, 8, 8, 3, -6, 4, 0, 0, -8, 2, 0, -6, -6, 0, -4, -18, 0, 2, 8, 2, 0, -10, 4, 0, 4, 10, 0, 4, 6, 0, 0, 4

Offset: 1

Michael Somos, Oct 02 2013

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q - 2*q^3 - 2*q^4 + 2*q^6 + 2*q^7 + 4*q^8 - q^9 - 2*q^10 - 2*q^13 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500 (terms 1..77 from Michael Somos)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A090132, A229893.

Basis( CuspForms( Gamma1(16), 2), 78)[1];

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^3, -q^4] QPochhammer[ -q, -q^4] QPochhammer[ -q^4] QPochhammer[ q] QPochhammer[ q^4] QPochhammer[ q^16], {q, 0, n}];

{a(n) = my(A, m); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^16 + A) * sum( k=0, n, if( issquare( 8*k + 1, &m), (-1)^((m\2 + 2) \ 4) * x^k, 0), A), n))};

Sage
```
CuspForms( Gamma1(16), 2, prec=78).0;
```

Euler transform of period 16 sequence [0, -2, -2, -1, -2, -1, 0, -4, 0, -1, -2, -1, -2, -2, 0, -4, ...].

a(2^n) = A090132(n). a(16*n + 5) = a(16*n + 11) = 0. 2 * a(n) = A229893(8*n). a(2*n) = -2 * A229893(n).

A138232 First differences of A138231.

Original entry on oeis.org

1, 0, 0, 1, -2, 2, -4, 2, -4, 0, 0, -4, 8, -8, 16, -8, 16, 0, 0, 16, -32, 32, -64, 32, -64, 0, 0, -64, 128, -128, 256, -128, 256, 0, 0, 256, -512, 512, -1024, 512, -1024, 0, 0, -1024, 2048, -2048, 4096, -2048, 4096, 0, 0, 4096, -8192, 8192, -16384, 8192, -16384, 0, 0, -16384, 32768

Offset: 0

Paul Curtz, May 05 2008

sign

The sequence contains 2 copies of 1 and 3 copies of the higher powers 2^j (up to sign).

Index entries for linear recurrences with constant coefficients, signature (0, 2, 0, -2).

Cf. A000079, A131577.

LinearRecurrence[{0,2,0,-2},{1,0,0,1},70] (* Harvey P. Dale, Apr 23 2022 *)

a(n) = 2a(n-2)-2a(n-4). a(n) = -4a(n-8).
a(2n) = (-1)^(n+1)*A090132(n). a(2n+1) = A009545(n).
O.g.f.: (x-1)(x^2-x-1)/(1-2x^2+2x^4). - R. J. Mathar, Jul 08 2008

Edited by R. J. Mathar, Jul 08 2008

A226075 Expansion of (eta(q) * eta(q^11))^2 + 2 * (eta(q^2) * eta(q^22))^2 in powers of q.

Original entry on oeis.org

1, 0, -1, -2, 1, 0, -2, 4, -2, 0, 1, 2, 4, 0, -1, -4, -2, 0, 0, -2, 2, 0, -1, -4, -4, 0, 5, 4, 0, 0, 7, 0, -1, 0, -2, 4, 3, 0, -4, 4, -8, 0, -6, -2, -2, 0, 8, 4, -3, 0, 2, -8, -6, 0, 1, -8, 0, 0, 5, 2, 12, 0, 4, 8, 4, 0, -7, 4, 1, 0, -3, -8, 4, 0, 4, 0, -2, 0

Offset: 1

Michael Somos, May 25 2013

			G.f. = q - q^3 - 2*q^4 + q^5 - 2*q^7 + 4*q^8 - 2*q^9 + q^11 + 2*q^12 + 4*q^13 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500

Cf. A006571, A090132, A214733.

Basis( CuspForms( Gamma0(22), 2), 79)[1];

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^11])^2 + 2 q^2 ( QPochhammer[ q^2] QPochhammer[ q^22])^2, {q, 0, n}]; (* Michael Somos, Apr 25 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^2 + 2 * x * (eta(x^2 + A) * eta(x^22 + A))^2, n))};

Sage
```
CuspForms( Gamma0(22), 2, prec=79).0;
```

a(n) is multiplicative with a(11^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) if p != 11.
G.f. is a period 1 Fourier series which satisfies f(-1 / (22 t)) = 22 (t/i)^2 f(t) where q = exp(2 Pi i t).
a(4*n + 2) = 0. a(4*n) = -2 * A006571(n). a(2^n) = A090132(n). a(3^n) = A214733(n+1).

cp's OEIS Frontend

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