Volkan Yildiz has authored 4 sequences.
A192480
a(n) = n + A000108(n-1) for n > 1; a(0)=0, a(1)=1.
Original entry on oeis.org
0, 1, 3, 5, 9, 19, 48, 139, 437, 1439, 4872, 16807, 58798, 208025, 742914, 2674455, 9694861, 35357687, 129644808, 477638719, 1767263210, 6564120441, 24466267042, 91482563663, 343059613674, 1289904147349, 4861946401478, 18367353072179, 69533550916032
Offset: 0
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C := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A192480 := proc(n) if n <=1 then n; else n+C(n-1) ; end if; end proc:
seq(A192480(n),n=0..40) ; # R. J. Mathar, Jul 13 2011
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CoefficientList[Series[(2*x^2*(2 - x) + (1 - x)^2*(1 - Sqrt[1 - 4*x]))/(2*(1 - x)^2), {x,0,50}], x] (* G. C. Greubel, Mar 28 2017 *)
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x='x+O('x^50); concat([0], Vec((2*x^2*(2-x)+(1-x)^2*(1-sqrt(1-4*x)))/(2*(1-x)^2))) \\ G. C. Greubel, Mar 28 2017
A192481
a(n) = Sum_{i=1..n-1} (2^i*C(i)-a(i)) * (2^(n-i)*C(n-i)-a(n-i)), a(1)=1, where C(i)=A000108(i-1) are Catalan numbers.
Original entry on oeis.org
1, 1, 6, 29, 162, 978, 6156, 40061, 267338, 1819238, 12576692, 88079378, 623581332, 4455663876, 32090099352, 232711721757, 1697799727066, 12452943237342, 91774314536100, 679234371006982, 5046438870909244, 37623611703611452, 281391143518722728
Offset: 1
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C := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A192481 := proc(n) option remember; if n<=1 then n; else add( (2^i*C(i-1)-procname(i))*(2^(n-i)*C(n-i-1)-procname(n-i)), i=1..n-1) ; end if; end proc:
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CoefficientList[Series[(2 - Sqrt[1 - 8*x] - Sqrt[3 - 4*x - 2*Sqrt[1 - 8*x]])/2, {x,0,50}], x] (* G. C. Greubel, Feb 12 2017 *)
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x='x+O('x^50); Vec((2-sqrt(1-8*x)-sqrt(3-4*x-2*sqrt(1-8*x)))/2) \\ G. C. Greubel, Feb 12 2017
A192479
a(n) = 2^n*C(n-1) - A186997(n-1), where C(n) are the Catalan numbers (A000108).
Original entry on oeis.org
1, 3, 12, 61, 344, 2074, 13080, 85229, 569264, 3876766, 26817304, 187908802, 1330934032, 9513485076, 68539442800, 497178707325, 3628198048352, 26617955242806, 196205766112536, 1452410901340598, 10792613273706320
Offset: 1
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C := proc(n) binomial(2*n,n)/(n+1) ;end proc:
Yildf := proc(n) option remember; if n<=1 then 1; else add( (2^i*C(i-1)-procname(i))*procname(n-i),i=1..n-1) ; end if; end proc:
A192479 := proc(n) 2^n*C(n-1)-Yildf(n) ; end proc:
seq(A192479(n),n=1..30) ; # R. J. Mathar, Jul 13 2011
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a[1] = 1; a[n_] := 2^n*CatalanNumber[n-1] - Sum[Binomial[k, n-k-1]*Binomial[n+2*k-1, n+k-1]/(n+k), {k, 1, n-1}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Apr 02 2015 *)
A192482
a(n) = 2^n*C(n-1)-y(n), where y(n) = Sum_{i=1..n-1} (2^i*C(i-1)-y(i))*(2^(n-i)*C(n-i-1)-y(n-i)), y(0)=0, y(1)=1 and where C(i) is the i-th Catalan number.
Original entry on oeis.org
1, 3, 10, 51, 286, 1710, 10740, 69763, 464822, 3159450, 21821516, 152708078, 1080452972, 7716009724, 55545950568, 402649640163, 2936600795174, 21532660592418, 158645924209500, 1173875395710458, 8719519396134596, 64995349923442628, 486020221692290392
Offset: 1
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C:= n-> binomial(2*n, n)/(n+1):
y:= proc(n) option remember;
`if`(n<2, n, add((2^i *C(i-1) -y(i))*
(2^(n-i)*C(n-i-1)-y(n-i)), i=1..n-1))
end:
a:= n-> 2^n*C(n-1) -y(n):
seq(a(n), n=1..30); # Alois P. Heinz, Feb 06 2012
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c = CatalanNumber; y[n_] := y[n] = If[n<2, n, Sum[(2^i*c[i-1]-y[i])*(2^(n-i)*c[n-i-1] - y[n-i]), {i, 1, n-1}]]; a[n_] := 2^n*c[n-1]-y[n]; Table[ a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 28 2017, after Alois P. Heinz *)
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