A320825
Expansion of -(10*x^2 - 6*x + 1)*sqrt(1 - 4*x)/(3*x - 1)^2.
Original entry on oeis.org
-1, 2, 1, 0, -3, -10, -17, 28, 435, 2710, 13489, 60392, 254211, 1028250, 4046977, 15621932, 59463209, 224047866, 838012755, 3118339056, 11563677321, 42790868982, 158180470803, 584617335420, 2161733579313, 8001589660746, 29660171058675, 110136151678696, 409773163539325
Offset: 0
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m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(-(10*x^2-6*x+1)*Sqrt(1-4*x)/(1-3*x)^2)); // G. C. Greubel, Oct 27 2018
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c := n -> catalan(n)*(n-3)*(n+1)/((2*n-3)*(2*n-1)):
h := n -> hypergeom([1, -n], [5/2 - n], 3/4): A320825 := n -> c(n)*h(n):
seq(simplify(A320825(n)), n=0..28);
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CoefficientList[Series[-(10x^2 - 6x + 1) Sqrt[1 - 4x]/(3x - 1)^2, {x, 0, 28}], x]
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x='x+O('x^40); Vec(-(10*x^2-6*x+1)*sqrt(1-4*x)/(1-3*x)^2) \\ G. C. Greubel, Oct 27 2018
A320826
Expansion of x*(1 - 4*x)^(3/2)/(3*x - 1)^2.
Original entry on oeis.org
0, 1, 0, -3, -14, -51, -168, -521, -1542, -4365, -11740, -29439, -65670, -112273, -28344, 1018689, 6961550, 34606929, 151831044, 623095683, 2453975622, 9402575805, 35339538912, 130994480547, 480676041954, 1750847208621, 6343667488692, 22899720430251, 82466180250590
Offset: 0
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m:=30; R:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!(x*(1-4*x)^(3/2)/(1-3*x)^2)); // G. C. Greubel, Oct 27 2018
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c := n -> (-4)^(n-1)*binomial(3/2, n-1):
h := n -> hypergeom([2, 1 - n], [7/2 - n], 3/4):
A320826 := n -> c(n)*h(n): seq(simplify(A320826(n)), n=0..28);
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CoefficientList[Series[(x (1 - 4 x)^(3/2))/(3 x - 1)^2, {x, 0, 28}], x]
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x='x+O('x^30); concat([0], Vec(x*(1-4*x)^(3/2)/(1-3*x)^2)) \\ G. C. Greubel, Oct 27 2018
A291292
Necklace Catalan numbers.
Original entry on oeis.org
1, 1, 1, 3, 10, 34, 116, 396, 1353, 4631, 15895, 54757, 189465, 658835, 2303381, 8098783, 28642314, 101894922, 364614216, 1312191768, 4748561094, 17275277322, 63163858146, 232041604038, 856219298484, 3172442815476, 11799466553232, 44041859928944, 164924424558532, 619454123593948
Offset: 0
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Concatenation([1,1,1,3],List([4..30],n->3^(n-2)+(Sum([0..n-4],i->(3^i)*(2*(n-i-3))/((n-i-1)*(n-i))*Binomial(2*(n-i-2),n-i-2))))); # Muniru A Asiru, Oct 05 2018
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a:=n->`if`(n<=2,1,`if`(n=2,3,3^(n-2)+add((3^i)*(2*(n-i-3))/((n-i-1)*(n-i))*binomial(2*(n-i-2),n-i-2),i=0..n-4))); seq(a(n),n=0..30); # Muniru A Asiru, Oct 05 2018
# Alternative:
ogf := x -> 3/2 + (x - sqrt(1 - 4*x))*(2*x - 1)/(6*x - 2):
ser := series(ogf(x),x,32):
seq(coeff(ser, x, n), n=0..29); # Peter Luschny, Oct 25 2018
# Derivation of the recurrence (requires Maple 2022):
FormalPowerSeries:-FindRE(3/2 + (x - sqrt(1 - 4*x))*(2*x - 1)/(6*x - 2),x,a(n)); # Georg Fischer, Oct 21 2022
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Flatten[{1, 1, Table[3^(n - 2) + Sum[3^i*2*(n - i - 3)/((n - i - 1)*(n - i)) * Binomial[2*(n - i - 2), n - i - 2], {i, 0, n - 4}], {n, 2, 30}]}] (* Vaclav Kotesovec, Oct 22 2018 *)
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a(n) = if (n<=2, 1, if (n==2, 3, 3^(n-2) + sum(i=0, (n-4), (3^i)*(2*(n-i-3))/((n-i-1)*(n-i))*binomial(2*(n-i-2), n-i-2)))); \\ Michel Marcus, Oct 05 2018
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