A119697
a(n) = Fibonacci(n)*n*binomial(2*n,n)/(n+1).
Original entry on oeis.org
0, 1, 4, 30, 168, 1050, 6336, 39039, 240240, 1487772, 9237800, 57551494, 359444736, 2250244100, 14115694320, 88707831750, 558368324640, 3519726403710, 22215931214400, 140389620550410, 888125492826000, 5623962934819320, 35645449061816880, 226114365012465150
Offset: 0
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seq(binomial(2*n, n)*n*combinat[fibonacci](n)/(n+1), n=0..27);
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Table[Fibonacci[n]n Binomial[2n,n]/(n+1),{n,0,40}] (* Harvey P. Dale, Apr 29 2022 *)
A371986
Product of Lucas and Catalan numbers: a(n) = A000032(n)*A000108(n).
Original entry on oeis.org
2, 1, 6, 20, 98, 462, 2376, 12441, 67210, 369512, 2065908, 11698414, 66979864, 387050900, 2254552920, 13223768580, 78034377690, 462961545090, 2759796408600, 16522143563310, 99295449593340, 598836351581520, 3622983967834920, 21982916983078350, 133739841802846968
Offset: 0
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From Peter Luschny, Apr 15 2024: (Start)
a := n -> ((2 - 2*sqrt(5))^n + (2 + 2*sqrt(5))^n) * GAMMA(n + 1/2) / (sqrt(Pi) * GAMMA(n + 2)): seq(simplify(a(n)), n = 0..24);
# With g.f.:
assume(x>0); f := sqrt(1 - 4*x*(4*x + 1)):
gf := (sqrt(1 + f - 2*x) + sqrt(5)*sqrt(1 - f - 2*x) - sqrt(2))/(sqrt(8)*x):
ser := series(gf, x, 26): seq(simplify(coeff(ser, x, n)), n = 0..24);
# Recurrence:
a := proc(n) option remember: if n < 2 then return [2, 1][n + 1] fi;
2*(2*n - 1)*(n*a(n - 1) + (4*n - 6)*a(n - 2)) / (n*(n + 1)) end:
seq(a(n), n=0..24); (End)
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a[n_] := a[n] = Switch[n, 0, 2, 1, 1, _, 2*(2n - 1)*(n*a[n - 1] + (4n - 6)*a[n - 2])/(n*(n + 1))];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jun 17 2024, after Peter Luschny *)
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def A371986_gen(): # generator of terms
a, b, n = 2, 1, 2
while True:
yield a
a, b = b, (4*n - 2)*(n*b + (4*n - 6)*a) // (n*n + n)
n += 1
def A371986_list(len):
it = A371986_gen()
return [next(it) for _ in range(len)]
print(A371986_list(25)) # Peter Luschny, Apr 15 2024
A373614
a(n) = Fibonacci(n)^2 * Catalan(n).
Original entry on oeis.org
0, 1, 2, 20, 126, 1050, 8448, 72501, 630630, 5620472, 50807900, 465643906, 4313336832, 40331298100, 380115482760, 3607451824500, 34444346026230, 330647239219110, 3189220347667200, 30893105448487590, 300408447948394500, 2931423727834870320, 28696206742447216440, 281728667746183208850, 2773282854528632549376
Offset: 0
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gf := (2 * sqrt(-sqrt(16*x^2 - 12*x+1) - 6*x + 1) / (5*sqrt(10)*x) + 3*(1 -sqrt((-2*sqrt(16*x^2 - 12*x + 1) - 42*x + 7) / 5 + 6*x))/(10*x)) + (1 -sqrt(4*x + 1)) / (5*x): assume(x > 0): ser := series(gf, x, 30):
seq(coeff(ser, x, n), n = 0..24); # Peter Luschny, Jun 11 2024
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CoefficientList[Series[(2*Sqrt[-Sqrt[16*x^2 - 12*x + 1] - 6*x + 1]/(5*Sqrt[10]*x) + 3*(1 -Sqrt[(-2*Sqrt[16*x^2 - 12*x + 1] - 42*x + 7)/5 + 6*x])/(10*x)) + (1 -Sqrt[4*x + 1])/(5*x),{x,0,24},Assumptions->(x>0)],x] (* Stefano Spezia, Jun 11 2024 *)
(* A variant that does not need assumptions: *)
gf := ((2 Sqrt[1 - 2 x (Sqrt[5] + 3)] + Sqrt[2] (Sqrt[5] + 2) Sqrt[3 + Sqrt[5] - 8 x] + (Sqrt[5] + 3) (2 Sqrt[4 x + 1] - 5)) (Sqrt[5] - 3)) / (40 x);
Round[CoefficientList[Series[gf, {x, 0, 24}], x]] (* Peter Luschny, Jun 11 2024 *)
Original entry on oeis.org
0, 1, 6, 40, 294, 2310, 19008, 161733, 1411410, 12563408, 113624940, 1041158846, 9645100416, 90182859700, 849966450840, 8066498833800, 77019930780030, 739349587508730, 7131313919822400, 69079082238199110, 671733716498945100, 6554862704411317920, 64166669054324268120, 629964451984076275950
Offset: 0
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gf := (sqrt(-10*sqrt(16*x^2 - 12*x + 1) - 60*x + 35) - 5) / (10*x):
ser := series(gf, x, 32): seq(coeff(ser, x, n), n = 0..22);
# Peter Luschny, Jun 11 2024
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CoefficientList[Series[(Sqrt[(-2*Sqrt[16*x^2-12*x+1]-42*x+7)/5+6*x]-1)/(2*x),{x,0,23}],x] (* Stefano Spezia, Jun 11 2024 *)
Showing 1-4 of 4 results.