A375285
Expansion of 1/((1 - x - x^5)^2 - 4*x^6).
Original entry on oeis.org
1, 2, 3, 4, 5, 8, 17, 36, 69, 120, 196, 320, 547, 980, 1786, 3216, 5661, 9804, 16932, 29472, 51820, 91602, 161767, 284424, 498103, 871150, 1525380, 2676544, 4703158, 8265354, 14514236, 25464576, 44656997, 78324398, 137430720, 241225072, 423451668, 743244866
Offset: 0
- Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,2,2,0,0,0,-1).
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my(N=40, x='x+O('x^N)); Vec(1/((1-x-x^5)^2-4*x^6))
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a(n) = sum(k=0, n\5, binomial(2*n-8*k+2, 2*k+1))/2;
A182891
Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps of weight 2 at level 0. The members of L_n are paths of weight n that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.
Original entry on oeis.org
1, 1, 1, 1, 3, 2, 7, 3, 1, 15, 8, 3, 35, 21, 6, 1, 83, 50, 16, 4, 197, 123, 45, 10, 1, 473, 308, 117, 28, 5, 1145, 769, 304, 83, 15, 1, 2787, 1926, 798, 232, 45, 6, 6819, 4843, 2085, 636, 140, 21, 1, 16759, 12204, 5433, 1744, 416, 68, 7, 41345, 30813, 14154, 4749, 1200, 222, 28, 1
Offset: 0
T(3,1)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them, namely hH and Hh, have exactly one H-step at level 0.
Triangle starts:
1;
1;
1,1;
3,2;
7,3,1;
15,8,3;
35,21,6,1;
- M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
- E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.
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G:=1/(z^2-t*z^2+sqrt((1+z+z^2)*(1-3*z+z^2))): Gser:=simplify(series(G,z=0,18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,k),k=0..floor(n/2)) od; # yields sequence in triangular form
A375273
Expansion of 1/(1 - 2*x - 3*x^2 - 4*x^3 + 4*x^4).
Original entry on oeis.org
1, 2, 7, 24, 73, 238, 763, 2436, 7821, 25050, 80255, 257200, 824081, 2640582, 8461187, 27111644, 86872853, 278363058, 891946503, 2858027016, 9157854361, 29344123550, 94026132235, 301283944500, 965391362461, 3093362593162, 9911930522767, 31760378496864
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(1/(1-2*x-3*x^2-4*x^3+4*x^4))
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a(n) = sum(k=0, n\2, 2^k*binomial(2*n-2*k+2, 2*k+1))/2;
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