A091575 -id:A091575 - cp's OEIS frontend

A091574 Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type D_4.

5, 8, 15, 16, 25, 24, 35, 32, 45, 40, 55, 48, 65, 56, 75, 64, 85, 72, 95, 80, 105, 88, 115, 96, 125, 104, 135, 112, 145, 120, 155, 128, 165, 136, 175, 144, 185, 152, 195, 160, 205, 168, 215, 176, 225, 184, 235, 192, 245, 200, 255, 208

Offset: 0

Paul Boddington, Jan 22 2004

a(n) is also the number of orbits of length n for T^2, if T is a map with n orbits of length n. - Thomas Ward, Apr 08 2009

			a(2) = (1/2)*mu(2)*sigma_2(2)+(1/2)*mu(1)*sigma_2(4) = 8. - _Thomas Ward_, Apr 08 2009

I. Reiten, Dynkin diagrams and the representation theory of algebras, Notices of the AMS, May 1997, Vol. 44, Number 5.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Apisit Pakapongpun and Thomas Ward, Functorial orbit counting, Journal of Integer Sequences, 12 (2009) Article 09.2.4.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1)

Cf. A091571, A091572, A091573, A091575, A091576, A091577.

CoefficientList[ Series[ (5 + 8x + 5x^2) / (1 - 2x^2 + x^4), {x, 0, 51}], x] (* Jean-François Alcover, Dec 02 2011 *)
With[{nn=40},Riffle[10*Range[nn]-5,8*Range[nn]]] (* or *) LinearRecurrence[ {0,2,0,-1},{5,8,15,16},80] (* Harvey P. Dale, Oct 30 2013 *)

(1/n)*sumdiv(n,d,moebius(n/d)*sumdiv(2*d,e,e^2)) \\ Thomas Ward, Apr 08 2009

a(n) = 5*(2*n+1) if n even, 4*(n+1) if n odd.
G.f.: (5+8*x+5*x^2)/(1-x^2)^2.
a(n) = (1/n)*Sum_{d|n} mobius(n/d)*sigma_2(2*d). - Thomas Ward, Apr 08 2009

A091573 Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type E_6.

Original entry on oeis.org

7, 12, 17, 24, 31, 36, 41, 48, 55, 60, 65, 72, 79, 84, 89, 96, 103, 108, 113, 120, 127, 132, 137, 144, 151, 156, 161, 168, 175, 180, 185, 192, 199, 204, 209, 216, 223, 228, 233, 240, 247, 252, 257, 264, 271, 276, 281, 288, 295, 300, 305, 312, 319, 324, 329

Offset: 0

Paul Boddington, Jan 22 2004

I. Reiten, Dynkin diagrams and the representation theory of algebras, Notices of the AMS, Vol. 44, Number 5.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A004525, A091571, A091572, A091574, A091575, A091576, A091577.

CoefficientList[ Series[ (7 - 2x + 7x^2) / (1 - 2x + 2x^2 - 2x^3 + x^4), {x, 0, 49}], x] (* Jean-François Alcover, Dec 02 2011 *)

a(n) = (12+(-I)^n+I^n+12*n)/2 \\ Colin Barker, Oct 18 2015

Vec((7-2*x+7*x^2)/((1+x^2)*(1-x)^2) + O(x^100)) \\ Colin Barker, Oct 18 2015

a(n) = if(n%2 == 1, 6*n+6, if(n%4 == 0, 6*n+7, 6*n+5));
vector(100, n, a(n-1)) \\ Altug Alkan, Oct 18 2015

a(n) = 6*n+6 (n odd), 6*n+7 (n==0 (mod 4)), 6*n+5 (n==2 (mod 4)).
G.f.: (7-2*x+7*x^2) / ((1+x^2)*(1-x)^2).
From Colin Barker, Oct 18 2015: (Start)
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>3.
a(n) = (12+(-i)^n+i^n+12*n)/2 where i = sqrt(-1).
(End)

G.f. corrected by Colin Barker, Oct 18 2015

A091571 Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type E_8.

Original entry on oeis.org

9, 16, 23, 30, 37, 44, 53, 60, 67, 74, 83, 90, 99, 106, 113, 118, 127, 134, 143, 150, 159, 164, 173, 180, 189, 194, 203, 208, 217, 224, 235, 240, 249, 254, 263, 268, 279, 284, 293, 298, 309, 314, 325, 330, 339, 342, 353, 358, 369, 374, 385, 388, 399, 404, 415

Offset: 0

Paul Boddington, Jan 22 2004

I. Reiten, Dynkin diagrams and the representation theory of algebras, Notices of the AMS, Vol. 44, No 5 (1997) 546-556. [MR1444112]

Cf. A087866, A091572, A091573, A091574, A091575, A091576, A091577.

CoefficientList[ Series[ (9-2x+9x^3-2x^4+9x^5-2x^7+9x^8) / (1-2x+x^2+x^3-2x^4+2x^5-2x^6+x^7+x^8-2x^9+x^10), {x, 0, 54}], x] (* Jean-François Alcover, Dec 02 2011 *)

a(n) = 7*(n+1) + 2*A087866(n).

G.f.: (7/(1-x)^2) + 2*(1-x^15)/((1-x^6)*(1-x^10)*(1-x)).

A091572 Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type E_7.

Original entry on oeis.org

8, 14, 20, 26, 34, 40, 48, 54, 62, 66, 74, 80, 90, 94, 102, 106, 116, 120, 130, 134, 144, 146, 156, 160, 172, 174, 184, 186, 198, 200, 212, 214, 226, 226, 238, 240, 254, 254, 266, 266, 280, 280, 294, 294, 308, 306, 320, 320, 336, 334, 348, 346, 362, 360, 376

Offset: 0

Paul Boddington, Jan 22 2004

I, Reiten, Dynkin diagrams and the representation theory of algebras, Notices of the AMS, Vol. 44, No 5.

Cf. A087504, A091571, A091573, A091574, A091575, A091576, A091577.

CoefficientList[ Series[ (8 + 6x + 6x^2 + 14x^3 + 6x^4 + 6x^5 + 8x^6) / (1 - x + x^3 - 2x^4 + x^5 - x^7 + x^8),{x,0,54}],x] (* Jean-François Alcover, Dec 02 2011 *)

a(n) = 6*(n+1) + 2*A087504(n).

G.f.: (6/(1-x)^2)+2*(1-x^9)/((1-x^4)*(1-x^6)*(1-x)).

A091576 Poincaré series [or Poincare series] of the preprojective algebra of a Dynkin diagram of type E_7.

Original entry on oeis.org

7, 12, 17, 22, 27, 30, 33, 34, 35, 34, 33, 30, 27, 22, 17, 12, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul Boddington, Jan 22 2004

The series is a polynomial because the algebra is finite dimensional. See the entry for A091575 for a general formula.

Cf. A091571, A091572, A091573, A091574, A091575, A091577.

A091577 Poincaré series [or Poincare series] of the preprojective algebra of a Dynkin diagram of type E_6.

Original entry on oeis.org

6, 10, 14, 18, 20, 20, 20, 18, 14, 10, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul Boddington, Jan 22 2004

The series is a polynomial because the algebra is finite dimensional. See the entry for A091575 for a general formula.

I. Reiten, Dynkin diagrams and the representation theory of algebras, Notices of the AMS, Vol. 44, Number 5.

Cf. A091571, A091572, A091573, A091574, A091575, A091576.

cp's OEIS Frontend

A091574 Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type D_4.

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A091573 Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type E_6.

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A091571 Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type E_8.

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A091572 Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type E_7.

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A091576 Poincaré series [or Poincare series] of the preprojective algebra of a Dynkin diagram of type E_7.

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A091577 Poincaré series [or Poincare series] of the preprojective algebra of a Dynkin diagram of type E_6.

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