A099530 -id:A099530 - cp's OEIS frontend

A099531 Expansion of (1+x)^3/((1+x)^3+x^4).

1, 0, 0, 0, -1, 3, -6, 10, -14, 15, -7, -20, 80, -188, 351, -549, 702, -622, -42, 1839, -5471, 11560, -20064, 29144, -33329, 21059, 27730, -142182, 355626, -689121, 1114937, -1490892, 1461360, -337220, -2996465, 10030587, -22226506, 39921442, -60118930, 72788383, -55703295, -31057776

Offset: 0

Paul Barry, Oct 20 2004

Binomial transform is A099530.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3,-3,-1,-1).

Cf. A099529.

CoefficientList[Series[(1+x)^3/((1+x)^3+x^4),{x,0,50}],x] (* or *) LinearRecurrence[{-3,-3,-1,-1},{1,0,0,0},50] (* Harvey P. Dale, Feb 21 2016 *)

a(n)=-3a(n-1)-3a(n-2)-a(n-3)-a(n-4); a(n)=sum{j=0..n, sum{k=0..floor(j/4), C(n, j)(-1)^(n-j)C(j-3k, k)(-1)^k}}.

A290989 Expansion of x^6(1 + x^3)/(1 - 4x + 5x^2 - x^3 - 2x^4 + x^6 + x^7 - 2*x^8 + x^9).

Original entry on oeis.org

1, 4, 11, 26, 55, 109, 208, 389, 722, 1339, 2488, 4634, 8646, 16146, 30160, 56333, 105198, 196413, 366672, 684475, 1277701, 2385080, 4452277, 8311254, 15515091, 28963012, 54067156, 100930660, 188413624, 351723304, 656583197

Offset: 6

R. J. Mathar, Aug 16 2017

This corresponds to S(213,1,x) of Langley if one uses Theorem 8. Note that all three expressions for S(213;t,x), S(213;1,x) and the series on page 22 are mutually incompatible, so we show the sequence one would most likely see in other publications.

Vincenzo Librandi, Table of n, a(n) for n = 6..1000
T. Langley, J. Liese, and J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2.
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,0,-1,-1,2,-1).

Cf. A059633, A099530, A290986, A290987.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x^6*(1+x^3)/((1-x)*(1-2*x+x^3-x^4)*(1-x+x^4)) )); // G. C. Greubel, Apr 12 2023

DeleteCases[#, 0] &@ CoefficientList[Series[x^6*(1+x^3)/(1 -4x +5x^2 -x^3 -2x^4 +x^6 +x^7 -2x^8 +x^9), {x, 0, 36}], x] (* Michael De Vlieger, Aug 16 2017 *)
LinearRecurrence[{4,-5,1,2,0,-1,-1,2,-1}, {1,4,11,26,55,109,208,389,722}, 80] (* Vincenzo Librandi, Aug 17 2017 *)

def A290989_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^6*(1+x^3)/((1-x)*(1-x+x^4)*(1-2*x+x^3-x^4)) ).list()
a=A290989_list(50); a[6:] # G. C. Greubel, Apr 12 2023

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

a(n) = -2 + (1/19)*( 9*A099530(n+1) + 15*A099530(n) + 2*A099530(n-1) - A099530(n- 2) + 10*A059633(n+4) - 6*A059633(n+3) - 16*A059633(n+2) - A059633(n+1) ). - G. C. Greubel, Apr 12 2023

A193884 Expansion of o.g.f. (1-x^2)/(1-x+x^4).

Original entry on oeis.org

1, 1, 0, 0, -1, -2, -2, -2, -1, 1, 3, 5, 6, 5, 2, -3, -9, -14, -16, -13, -4, 10, 26, 39, 43, 33, 7, -32, -75, -108, -115, -83, -8, 100, 215, 298, 306, 206, -9, -307, -613, -819, -810, -503, 110, 929, 1739, 2242, 2132, 1203, -536, -2778, -4910, -6113, -5577

Offset: 0

Johannes W. Meijer, Aug 11 2011

The Kn11 sums, see A180662, of triangle A108299 equal the terms of this sequence.

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

Cf. A099530, A108299, A180662.

A193884 := proc(n) option remember: if n=0 then 1 elif n=1 then 1 elif n=2 then 0 elif n=3 then 0 elif n>=4 then procname(n-1)-procname(n-4) fi: end: seq(A193884(n), n=0..54);

CoefficientList[Series[(1-x^2)/(1-x+x^4),{x,0,80}],x] (* or *) LinearRecurrence[{1,0,0,-1},{1,1,0,0},80] (* Harvey P. Dale, Jul 15 2020 *)

G.f.: (1+x)*(1-x)/(1-x+x^4).
a(n) = a(n-1)-a(n-4), a(0) = a(1) = 1, a(2) = a(3) = 0.
a(n) = A099530(n) - A099530(n-2).

A198862 Sum of the n-th antidiagonal in the triangle A192011.

Original entry on oeis.org

-1, 2, 2, 2, 3, 1, -1, -3, -6, -7, -6, -3, 3, 10, 16, 19, 16, 6, -10, -29, -45, -51, -41, -12, 33, 84, 125, 137, 104, 20, -105, -242, -346, -366, -261, -19, 327, 693, 954, 973, 646, -47, -1001, -1974, -2620, -2573, -1572, 402, 3022

Offset: 0

Paul Curtz, Oct 30 2011

The current sequence and its successive differences are:
-1, 2,  2,  2,  3,  1, -1 ,-3, -6, -7, ...
3, 0, 0, 1, -2, -2, -2, -3, -1, 1, 3, 6, 7, 6, 3, -3, -10, -16, ...
-3, 0, 1, -3, 0, 0, -1, 2, 2, 2, 3, 1, -1, -3, -6, -7, -6, ...
3, 1, -4, 3, 0, -1, 3, 0, 0, 1, -2, -2, -2, -3, -1, 1, 3, 6, 7, ...
-2, -5, 7, -3, -1, 4, -3, 0, 1, -3, 0, 0, -1, 2, 2, 2, 3, 1, ...
-3, 12, -10, 2, 5, -7, 3, 1, -4, 3, 0, -1, 3, 0, 0, 1, -2, ...
15, -22, 12, 3, -12, 10, -2, -5, 7, -3, -1, 4, -3, 0, ...
-37, 34, -9, -15, 22, -12, -3, 12, -10, 2, 5, -7, 3, 1, -4, ...
Each row obeys the same linear recurrence and is a version of the row 4 lines farther up in the same array shifted right by 12 places.

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

A198862 := proc(n)
        add( A192011(n-k,k),k=0..floor(n/2)) ;
end proc:
seq(A198862(n),n=0..80) ; # R. J. Mathar, Nov 03 2011

a(n) = Sum_{k=0..floor(n/2)} A192011(n-k,k).
a(n) = a(n-1) - a(n-4), n > 3.
From R. J. Mathar, Nov 02 2011: (Start)
G.f.: (-1 + 3*x) / (1 - x + x^4).
a(n) = 3*A099530(n-1) - A099530(n). (End)

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A099531 Expansion of (1+x)^3/((1+x)^3+x^4).

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A290989 Expansion of x^6*(1 + x^3)/(1 - 4*x + 5*x^2 - x^3 - 2*x^4 + x^6 + x^7 - 2*x^8 + x^9).

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A193884 Expansion of o.g.f. (1-x^2)/(1-x+x^4).

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A198862 Sum of the n-th antidiagonal in the triangle A192011.

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A290989 Expansion of x^6(1 + x^3)/(1 - 4x + 5x^2 - x^3 - 2x^4 + x^6 + x^7 - 2*x^8 + x^9).