A078789 -id:A078789 - cp's OEIS frontend

A049016 Expansion of 1/((1-x)^5 - x^5).

1, 5, 15, 35, 70, 127, 220, 385, 715, 1430, 3004, 6385, 13380, 27370, 54740, 107883, 211585, 416405, 826045, 1652090, 3321891, 6690150, 13455325, 26985675, 53971350, 107746282, 214978335, 429124630, 857417220, 1714834440, 3431847189

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,2).

Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), A000749 (m=4), this sequence (m=5), A192080 (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).

Cf. A000750, A078789.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/((1-x)^5-x^5) )); // G. C. Greubel, Apr 11 2023

CoefficientList[Series[1/((1-x)^5-x^5),{x,0,30}],x] (* or *) LinearRecurrence[ {5,-10,10,-5,2},{1,5,15,35,70},40] (* Harvey P. Dale, Jan 20 2014 *)

def A049016_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)^5-x^5) ).list()
A049016_list(30) # G. C. Greubel, Apr 11 2023

G.f.: 1/((1-x)^5-x^5) = 1/( (1-2*x)*(1-3*x+4*x^2-2*x^3+x^4) ).
a(10*n+3) = A078789(5*n+3).
a(10*n+5) = A078789(5*n+4).
a(n) = (-1)^n * A000750(n).
Binomial transform of expansion of (1+x)^4/(1-x^5), or (1, 4, 6, 4, 1, 1, 4, 6, 4, 1, ...). - Paul Barry, Mar 19 2004
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 2*a(n-5). - Paul Curtz, May 24 2008
G.f.: -1/( x^5 - 1 + 5*x/Q(0) ) where Q(k) = 1 + k*(x+1) + 5*x - x*(k+1)*(k+6)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 15 2013

A378254 Cogrowth sequence of the 20-element group C10 X C2 = .

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 67, 1002, 8009, 43759, 184758, 646878, 1971883, 5541966, 16231216, 60090032, 290305577, 1523150157, 7564006759, 34099637859, 139541849878, 526321168143, 1878476551128, 6603812572941, 24052434515891, 94278969044262, 396750746960712

Offset: 0

Sean A. Irvine, Nov 20 2024

Gives the even terms, all the odd terms are 0.

Cf. A377840 (C8 X C2), A377627 (C6 X C2), A078789 (D10).

G.f.: (10*x^9+75*x^8+162*x^7-48*x^6+127*x^5-126*x^4+84*x^3-36*x^2+9*x-1) / ((4*x-1) * (25*x^4+10*x^2-5*x+1) * (x^4+4*x^3+6*x^2-x+1)).

A378276 Cogrowth sequence of the 20-element dicyclic group Dic20 = .

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 6, 7, 19, 9, 61, 88, 226, 377, 945, 1473, 3427, 6154, 13758, 25327, 53967, 102865, 213345, 411700, 849706, 1661885, 3382821, 6668577, 13493719, 26725414, 53861446, 107074403, 215261971, 428704177, 860266725, 1715950208, 3439229842, 6866462849

Offset: 0

Sean A. Irvine, Nov 21 2024

Cf. A377656 (Dic12), A078789 (D10), A378254 (C10 X C2), A378278 (Frob20).

G.f.: (4*x^9-6*x^8-3*x^7-5*x^6-2*x^5-2*x^4-2*x^3-x^2+1) / ((2*x-1) * (2*x^2+2*x+1) * (x^2-x-1) * (4*x^4-2*x^3+3*x^2-x+1)).

A378278 Cogrowth sequence of the 20-element Frobenius group Frob20 = .

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 3, 7, 15, 27, 51, 99, 199, 403, 819, 1651, 3303, 6579, 13107, 26163, 52327, 104755, 209715, 419635, 839271, 1678131, 3355443, 6710067, 13420135, 26841907, 53687091, 107377459, 214754919, 429503283, 858993459, 1717973811, 3435947623

Offset: 0

Sean A. Irvine, Nov 21 2024

Cf. A078789 (D10), A378254 (C10 X C2), A378276 (Dic20).

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

cp's OEIS Frontend

A049016 Expansion of 1/((1-x)^5 - x^5).

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A378254 Cogrowth sequence of the 20-element group C10 X C2 = .

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A378276 Cogrowth sequence of the 20-element dicyclic group Dic20 = .

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A378278 Cogrowth sequence of the 20-element Frobenius group Frob20 = .

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