A382436
Triangle read by rows, defined by the two-variable g.f. 1/(1 - (y + 1)*x - y*x^2 - (y^2 + y)*x^3).
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 9, 17, 9, 1, 1, 12, 36, 36, 12, 1, 1, 15, 64, 101, 64, 15, 1, 1, 18, 101, 227, 227, 101, 18, 1, 1, 21, 147, 440, 627, 440, 147, 21, 1, 1, 24, 202, 767, 1459, 1459, 767, 202, 24, 1, 1, 27, 266, 1235, 2994, 3999, 2994, 1235, 266, 27, 1
Offset: 0
Triangle begins:
1;
1, 1;
1, 3, 1;
1, 6, 6, 1;
1, 9, 17, 9, 1;
1, 12, 36, 36, 12, 1;
1, 15, 64, 101, 64, 15, 1;
1, 18, 101, 227, 227, 101, 18, 1;
1, 21, 147, 440, 627, 440, 147, 21, 1;
1, 24, 202, 767, 1459, 1459, 767, 202, 24, 1;
1, 27, 266, 1235, 2994, 3999, 2994, 1235, 266, 27, 1;
...
A077987
Expansion of 1/(1+2*x-x^2+2*x^3).
Original entry on oeis.org
1, -2, 5, -14, 37, -98, 261, -694, 1845, -4906, 13045, -34686, 92229, -245234, 652069, -1733830, 4610197, -12258362, 32594581, -86667918, 230447141, -612751362, 1629285701, -4332217046, 11519222517, -30629233482, 81442123573, -216551925662, 575804441861, -1531045056530
Offset: 0
-
a:=[1,-2,5];; for n in [4..40] do a[n]:=-2*a[n-1]+a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Jun 25 2019
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R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1+2*x-x^2+2*x^3) )); // G. C. Greubel, Jun 25 2019
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CoefficientList[Series[1/(1+2x-x^2+2x^3),{x,0,40}],x] (* or *) LinearRecurrence[{-2,1,-2},{1,-2,5},40] (* Harvey P. Dale, Dec 27 2013 *)
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Vec(1/(1+2*x-x^2+2*x^3)+O(x^40)) \\ Charles R Greathouse IV, Sep 27 2012
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(1/(1+2*x-x^2+2*x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 25 2019
A102035
Carrie's triangle, read by rows, where the terms are generated by the rule: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-3,k-1) + T(n-3,k), with T(0,0)=1.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 2, 6, 5, 1, 3, 11, 15, 7, 1, 4, 20, 36, 28, 9, 1, 6, 35, 78, 85, 45, 11, 1, 9, 59, 159, 221, 166, 66, 13, 1, 13, 98, 309, 522, 509, 287, 91, 15, 1, 19, 161, 579, 1153, 1382, 1018, 456, 120, 17, 1, 28, 261, 1056, 2421, 3444, 3141, 1840, 681, 153, 19, 1, 41, 419
Offset: 0
Generated by adding preceding terms in the triangle
at positions that form the letter 'C': T(n,k) =
T(n-3,k-1) + T(n-3,k) +
T(n-2,k-1) +
T(n-1,k-1) + T(n-1,k).
Rows begin:
[1],
[1,1],
[1,3,1],
[2,6,5,1],
[3,11,15,7,1],
[4,20,36,28,9,1],
[6,35,78,85,45,11,1],
[9,59,159,221,166,66,13,1],
[13,98,309,522,509,287,91,15,1],
[19,161,579,1153,1382,1018,456,120,17,1],...
Showing 1-3 of 3 results.
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