A098461
Expansion of E.g.f.: 1/sqrt(1-2*x-3*x^2).
Original entry on oeis.org
1, 1, 6, 42, 456, 6120, 101520, 1980720, 44634240, 1139080320, 32488646400, 1023985670400, 35345049062400, 1325988036172800, 53721616851302400, 2337607853957376000, 108727934847307776000, 5383304681800421376000, 282682783375630589952000
Offset: 0
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Table[(n!/2^n) Sum[Binomial[n, k] Binomial[2 (n - k), n] 3^k, {k, 0, Floor[n/2]}], {n, 0, 17}] (* Michael De Vlieger, Sep 14 2016 *)
A307695
Expansion of 1/(sqrt(1-4*x)*sqrt(1-16*x)).
Original entry on oeis.org
1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300, 225759595348, 3444812388280, 52781007848284, 811510465220920, 12513859077134008, 193460383702061200, 2997463389599395270, 46532910920993515900, 723626591914643806180, 11270311875128088314200
Offset: 0
Cf.
A000984 (c=0,d=4,e=1),
A026375 (c=1,d=5,e=1),
A081671 (c=2,d=6,e=1),
A098409 (c=3,d=7,e=1),
A098410 (c=4,d=8,e=1),
A104454 (c=5,d=9,e=1).
Cf.
A084605 (c=-3,d=5,e=2),
A098453 (c=-2,d=6,e=2),
A322242 (c=-1,d=7,e=2),
A084771 (c=1,d=9,e=2),
A248168 (c=3,d=11,e=2).
Cf.
A322246 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3).
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a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 13 2021 *)
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N=66; x='x+O('x^N); Vec(1/sqrt(1-20*x+64*x^2))
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{a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))}
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{a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))}
A110135
Square array of expansions of 1/sqrt(1-4x-4*k*x^2), read by antidiagonals.
Original entry on oeis.org
1, 2, 1, 6, 2, 1, 20, 8, 2, 1, 70, 32, 10, 2, 1, 252, 136, 44, 12, 2, 1, 924, 592, 214, 56, 14, 2, 1, 3432, 2624, 1052, 304, 68, 16, 2, 1, 12870, 11776, 5284, 1632, 406, 80, 18, 2, 1, 48620, 53344, 26840, 9024, 2332, 520, 92, 20, 2, 1, 184756, 243392, 137638, 50304
Offset: 0
As a square array, rows start
1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, ...
6, 8, 10, 12, 14, 16, ...
20, 32, 44, 56, 68, 80, ...
70, 136, 214, 304, 406, 520, ...
252, 592, 1052, 1632, 2332, 3152, ...
As a number triangle, rows start
1;
2, 1;
6, 2, 1;
20, 8, 2, 1;
70, 30, 10, 2, 1;
252, 136, 44, 12, 2, 1;
A294409
a(n) = n! * [x^n] exp(n*x)*BesselI(0,2*n*x).
Original entry on oeis.org
1, 1, 12, 189, 4864, 159375, 6578496, 323652399, 18572378112, 1216112914971, 89530000000000, 7319100286183983, 657910135976361984, 64494528072860946073, 6847518630093139525632, 782782183702056884765625, 95860848315529046085599232, 12520224284071636768582166787, 1737254440584625641929018966016
Offset: 0
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seq(coeff((1+n*x+n^2*x^2)^n,x,n),n=0..100); # Robert Israel, Oct 30 2017
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Table[n! SeriesCoefficient[Exp[n x] BesselI[0, 2 n x], {x, 0, n}], {n, 0, 18}]
Table[CoefficientList[Series[(1 + n x + n^2 x^2)^n, {x, 0, n}], x][[-1]], {n, 0, 18}]
Table[SeriesCoefficient[1/Sqrt[(1 + n x) (1 - 3 n x)], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[n^n Sum[Binomial[n, k] Binomial[k, n - k],{k, 0, n}], {n, 1, 18}]]
Join[{1}, Table[n^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {1}, 4], {n, 1, 18}]]
A376827
T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).
Original entry on oeis.org
1, 1, 1, 3, 6, 3, 7, 21, 21, 7, 19, 76, 114, 76, 19, 51, 255, 510, 510, 255, 51, 141, 846, 2115, 2820, 2115, 846, 141, 393, 2751, 8253, 13755, 13755, 8253, 2751, 393, 1107, 8856, 30996, 61992, 77490, 61992, 30996, 8856, 1107
Offset: 0
[0] 1;
[1] 1, 1;
[2] 3, 6, 3;
[3] 7, 21, 21, 7;
[4] 19, 76, 114, 76, 19;
[5] 51, 255, 510, 510, 255, 51;
[6] 141, 846, 2115, 2820, 2115, 846, 141;
[7] 393, 2751, 8253, 13755, 13755, 8253, 2751, 393;
[8] 1107, 8856, 30996, 61992, 77490, 61992, 30996, 8856, 1107;
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T := (n, k) -> binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4): seq(seq(simplify(T(n, k)), k = 0..n), n = 0..8);
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A376827[n_, k_] := Binomial[n, k]*Hypergeometric2F1[(1-n)/2, -n/2, 1, 4];
Table[A376827[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Oct 21 2024 *)
Showing 1-5 of 5 results.
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