A077970
Expansion of 1/(1+x-2*x^2+2*x^3).
Original entry on oeis.org
1, -1, 3, -7, 15, -35, 79, -179, 407, -923, 2095, -4755, 10791, -24491, 55583, -126147, 286295, -649755, 1474639, -3346739, 7595527, -17238283, 39122815, -88790435, 201512631, -457339131, 1037945263, -2355648787, 5346217575, -12133405675, 27537138399, -62496384899, 141837473047
Offset: 0
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a:=[1,1,-3];; for n in [4..40] do a[n]:=-a[n-1]+2*a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Jun 24 2019
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R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1+x-2*x^2+2*x^3) )); // G. C. Greubel, Jun 24 2019
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CoefficientList[Series[1/(1+x-2x^2+2x^3),{x,0,40}],x] (* or *) LinearRecurrence[ {-1,2,-2},{1,-1,3},40] (* Harvey P. Dale, Sep 29 2018 *)
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Vec(1/(1+x-2*x^2+2*x^3)+O(x^40)) \\ Charles R Greathouse IV, Sep 26 2012
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(1/(1+x-2*x^2+2*x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 24 2019
A077863
Expansion of (1-x)^(-1)/(1-x-2*x^2-2*x^3).
Original entry on oeis.org
1, 2, 5, 12, 27, 62, 141, 320, 727, 1650, 3745, 8500, 19291, 43782, 99365, 225512, 511807, 1161562, 2636201, 5982940, 13578467, 30816750, 69939565, 158730000, 360242631, 817581762, 1855527025, 4211175812, 9557393387, 21690799062, 49227937461, 111724322360
Offset: 0
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CoefficientList[Series[(1-x)^(-1)/(1-x-2x^2-2x^3),{x,0,40}],x] (* or *) LinearRecurrence[{2,1,0,-2},{1,2,5,12},40] (* Harvey P. Dale, Sep 14 2016 *)
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my(x='x+O('x^40)); Vec((1-x)^(-1)/(1-x-2*x^2-2*x^3)) \\ Christian Krause, Jan 02 2023
A189187
Riordan matrix (1/(1-x-x^2-x^3),(x+x^2)/(1-x-x^2-x^3)).
Original entry on oeis.org
1, 1, 1, 2, 3, 1, 4, 7, 5, 1, 7, 17, 16, 7, 1, 13, 38, 46, 29, 9, 1, 24, 82, 122, 99, 46, 11, 1, 44, 174, 304, 303, 184, 67, 13, 1, 81, 362, 728, 857, 641, 309, 92, 15, 1, 149, 743, 1690, 2291, 2031, 1212, 482, 121, 17, 1, 274, 1509, 3827, 5869, 6004, 4260, 2108, 711, 154, 19, 1
Offset: 0
Triangle begins:
1
1,1
2,3,1
4,7,5,1
7,17,16,7,1
13,38,46,29,9,1
24,82,122,99,46,11,1
44,174,304,303,184,67,13,1
81,362,728,857,641,309,92,15,1
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Flatten[Table[Sum[Binomial[i+k,k]Sum[Binomial[i+k,j]Binomial[n-i-j,i+k],{j,0,n-k-2i}],{i,0,n}],{n,0,20},{k,0,n}]]
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create_list(sum(binomial(i+k,k)*sum(binomial(i+k,j)*binomial(n-i-j,i+k),j,0,n-k-2*i),i,0,n),n,0,8,k,0,n);
a(23) and a(40) corrected by
Georg Fischer, Feb 20 2021 and Apr 29 2022
Showing 1-3 of 3 results.
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