A030536 -id:A030536

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A030533

Expansion of Molien series for 4-D extraspecial group 2^{1+2*2}.

+10
5

1, 1, 5, 6, 15, 19, 35, 44, 69, 85, 121, 146, 195, 231, 295, 344, 425, 489, 589, 670, 791, 891, 1035, 1156, 1325, 1469, 1665, 1834, 2059, 2255, 2511, 2736, 3025, 3281, 3605, 3894, 4255, 4579, 4979, 5340, 5781, 6181, 6665, 7106, 7635

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

Index entries for Molien series.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

FORMULA

G.f.: (x^8-x^6+2*x^4-x^2+1)/(1+x^2)^2/(-1+x^2)^2/(1+x)^2/(-1+x)^2 (not simplified).

G.f.: (x^2-x+1)*(x^2+1) / ((x-1)^4*(x+1)^2). [Colin Barker, Jan 31 2013]

a(n) = n*(2*n^2-9*(-1)^n+13)/24. [Bruno Berselli, Jan 31 2013]

EXAMPLE

1+x^2+5*x^4+6*x^6+15*x^8+19*x^10+35*x^12+44*x^14+69*x^16+...

MATHEMATICA

CoefficientList[Series[(x^2 - x + 1) (x^2 + 1)/((x - 1)^4 (x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 19 2013 *)

LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 1, 5, 6, 15, 19}, 50] (* Harvey P. Dale, May 05 2022 *)

PROG

(PARI) select(n->n, Vec((x^8-x^6+2*x^4-x^2+1)/(1+x^2)^2/(-1+x^2)^2/(1+x)^2/(-1+x)^2+O(x^99))) \\ Charles R Greathouse IV, Sep 24 2012

(Magma) [(n+1)*(2*n^2+4*n+15+9*(-1)^n)/24: n in [0..50]]; // Vincenzo Librandi, Oct 19 2013

CROSSREFS

Cf. A014095, A030535, A030536, A030537.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

A030535

Expansion of Molien series for 16-D extraspecial group 2^{1+2*4}.

+10
4

1, 1, 51, 219, 2244, 12815, 69615, 303165, 1180395, 4052070, 12706650, 36580770, 98256600, 247786866, 592040266, 1347148374, 2936245389, 6154632399, 12456241445, 24415459445, 46484089740, 86164059465, 155843612865, 275546946795, 477079706295, 810057618396

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

The first formula intersperses the terms with zeros, the second formula does not. - Colin Barker, Apr 01 2015

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for Molien series

Index entries for linear recurrences with constant coefficients, signature (8, -20, -8, 126, -168, -196, 680, -239, -1072, 1240, 560, -1820, 560, 1240, -1072, -239, 680, -196, -168, 126, -8, -20, 8, -1).

FORMULA

G.f.: (1 -7*x^2 +63*x^4 -161*x^6 +1394*x^8 -307*x^10 +7665*x^12 +987*x^14 +13498*x^16 +987*x^18 +7665*x^20 -307*x^22 +1394*x^24 -161*x^26 +63*x^28 -7*x^30 +x^32)/( (1-x)^8*(1+x)^8*(1-x^2)^8*(1+x^2)^8 ), even terms only.

G.f.: (1 -7*x +63*x^2 -161*x^3 +1394*x^4 -307*x^5 +7665*x^6 +987*x^7 +13498*x^8 +987*x^9 +7665*x^10 -307*x^11 +1394*x^12 -161*x^13 +63*x^14 -7*x^15 +x^16)/( (1-x)^8*(1-x^2)^8 ). - Colin Barker, Apr 01 2015

EXAMPLE

1 + l^2 + 51*l^4 + 219*l^6 + 2244*l^8 + 12815*l^10 + ...

MAPLE

f(x):=(1 -7*x +63*x^2 -161*x^3 +1394*x^4 -307*x^5 +7665*x^6 +987*x^7 +13498*x^8 +987*x^9 +7665*x^10 -307*x^11 +1394*x^12 -161*x^13 +63*x^14 -7*x^15 +x^16)/( (1-x)^8*(1-x^2)^8 ); seq(coeff(series(f(x), x, n+1), x, n), n = 0..30); # G. C. Greubel, Feb 01 2020

MATHEMATICA

CoefficientList[Series[(1 -7*x +63*x^2 -161*x^3 +1394*x^4 -307*x^5 +7665*x^6 +987*x^7 +13498*x^8 +987*x^9 +7665*x^10 -307*x^11 +1394*x^12 -161*x^13 +63*x^14 -7*x^15 +x^16)/( (1-x)^8*(1-x^2)^8 ), {x, 0, 30}], x] (* G. C. Greubel, Feb 01 2020 *)

LinearRecurrence[{8, -20, -8, 126, -168, -196, 680, -239, -1072, 1240, 560, -1820, 560, 1240, -1072, -239, 680, -196, -168, 126, -8, -20, 8, -1}, {1, 1, 51, 219, 2244, 12815, 69615, 303165, 1180395, 4052070, 12706650, 36580770, 98256600, 247786866, 592040266, 1347148374, 2936245389, 6154632399, 12456241445, 24415459445, 46484089740, 86164059465, 155843612865, 275546946795}, 30] (* Harvey P. Dale, Aug 20 2022 *)

PROG

(PARI) Vec((1-7*x+63*x^2-161*x^3+1394*x^4-307*x^5+7665*x^6+987*x^7+13498*x^8 +987*x^9+7665*x^10-307*x^11+1394*x^12-161*x^13+63*x^14-7*x^15+x^16)/((1-x)^8 *(1-x^2)^8) + O(x^30)) \\ Colin Barker, Apr 01 2015

(Sage)

def f(x): return (1-7*x+63*x^2-161*x^3+1394*x^4-307*x^5+7665*x^6+987*x^7 +13498*x^8+987*x^9+7665*x^10-307*x^11+1394*x^12-161*x^13+63*x^14-7*x^15 +x^16)/((1-x)^8 *(1-x^2)^8)

[( f(x) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Feb 01 2020

CROSSREFS

Cf. A014095, A030533, A030536, A030537.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A030537

Expansion of Molien series for 64-D extraspecial group 2^{1+2*6}.

+10
3

1, 1, 715, 29359, 2649790, 151696457, 6380226905, 205074295943, 5265339073280, 111499999690365, 1997024724163755, 30863092117426803, 418329099990698130, 5040543679296888213, 54605890017229741905, 537020682260761903611

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..15.

Index entries for Molien series

CROSSREFS

Cf. A014095, A030533, A030535, A030536.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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