A098470
Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 5th column from the center.
Original entry on oeis.org
1, 6, 28, 112, 414, 1452, 4917, 16236, 52624, 168168, 531531, 1665456, 5182008, 16031952, 49366674, 151419816, 462919401, 1411306358, 4292487562, 13029127584, 39478598170, 119439969220, 360881425710, 1089126806040
Offset: 5
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# Assuming offset 0:
a := n -> simplify(GegenbauerC(n, -n-5, -1/2)):
seq(a(n), n=0..25); # Peter Luschny, May 09 2016
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Table[GegenbauerC[n, -n - 5, -1/2], {n,0,50}] (* G. C. Greubel, Feb 28 2017 *)
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x='x + O('x^50); Vec(32*x^5/(sqrt((1+x)*(1-3*x))*(1-x-sqrt((1+x)*(1-3*x)))^5)) \\ G. C. Greubel, Feb 28 2017
A098521
E.g.f. exp(x)*BesselI(2,2*sqrt(2)*x)/2.
Original entry on oeis.org
0, 0, 1, 3, 14, 50, 195, 721, 2716, 10116, 37845, 141295, 528330, 1975766, 7395479, 27698685, 103821240, 389410568, 1461605481, 5489516955, 20630539910, 77579118330, 291893775019, 1098848179561, 4138773239892, 15596070165900, 58797332264125, 221762856917511, 836756771788098
Offset: 0
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m:=40; R:=PowerSeriesRing(Rationals(), m); [0,0] cat Coefficients(R!((1-2*x-3*x^2-(1-x)*Sqrt(1-2*x-7*x^2))/(8*x^2*Sqrt(1-2*x-7*x^2)))); // G. C. Greubel, Aug 17 2018
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CoefficientList[Series[(1-2*x-3*x^2-(1-x)*Sqrt[1-2*x-7*x^2]) / (8*x^2*Sqrt[1-2*x-7*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 28 2012 *)
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x='x+O('x^66); concat([0,0],Vec((1-2*x-3*x^2-(1-x)*sqrt(1-2*x-7*x^2))/(8*x^2*sqrt(1-2*x-7*x^2)))) \\ Joerg Arndt, May 11 2013
A375248
Expansion of (1 - x)/(1 - 2*x - 3*x^2)^(7/2).
Original entry on oeis.org
1, 6, 35, 168, 756, 3192, 12936, 50688, 193479, 722722, 2651649, 9581936, 34176324, 120526056, 420852204, 1456709328, 5002984791, 17062825626, 57827993685, 194871361608, 653285629920, 2179701604080, 7241015510820, 23958512912880, 78978801164445
Offset: 0
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a[n_]:=(1+n)(2+n)(3+n)(4+n)(5+n)Hypergeometric2F1[(1-n)/2,-n/2,3,4]/120; Array[a,25,0] (* Stefano Spezia, Aug 07 2024 *)
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my(N=30, x='x+O('x^N)); Vec((1-x)/(1-2*x-3*x^2)^(7/2))
A114576
Triangle read by rows: T(n,k) is number of Motzkin paths of length n having k UH's, where U=(1,1), H=(1,0) (0<=k<=floor(n/3)).
Original entry on oeis.org
1, 1, 2, 3, 1, 6, 3, 11, 10, 23, 26, 2, 47, 70, 10, 102, 176, 45, 221, 449, 160, 5, 493, 1121, 539, 35, 1105, 2817, 1680, 196, 2516, 7031, 5082, 868, 14, 5763, 17604, 14856, 3486, 126, 13328, 43996, 42660, 12810, 840, 30995, 110147, 120338, 44640, 4410, 42
Offset: 0
T(4,1)=3 because we have H(UH)D, (UH)DH and (UH)HD, where U=(1,1), H=(1,0), D=(1,-1) (the UH's are shown between parentheses).
Triangle begins:
1;
1;
2;
3,1;
6,3;
11,10;
23,26,2;
47,70,10;
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G:=(1-z-sqrt(1-2*z-3*z^2-4*z^3*t+4*z^3))/2/z^2/(1-z+t*z): Gser:=simplify(series(G,z=0,20)): P[0]:=1: for n from 1 to 16 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 16 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/3)) od; # yields sequence in triangular form
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