A071688 -id:A071688 - cp's OEIS frontend

A119358 Number of n-element subsets of [2n] having an even sum.

1, 1, 2, 10, 38, 126, 452, 1716, 6470, 24310, 92252, 352716, 1352540, 5200300, 20056584, 77558760, 300546630, 1166803110, 4537543340, 17672631900, 68923356788, 269128937220, 1052049129144, 4116715363800, 16123803193628, 63205303218876, 247959261273752

Offset: 0

Paul Barry, May 16 2006

Old name was: Central coefficients of number triangle A119326.

			a(3) = 10: {1,2,3}, {1,2,5}, {1,3,4}, {1,3,6}, {1,4,5}, {1,5,6}, {2,3,5}, {2,4,6}, {3,4,5}, {3,5,6}. - _Alois P. Heinz_, Feb 04 2017

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A071688, A119326, A119359, A169888, A282011.

Column k=2 of A318557.

a:= proc(n) option remember; `if`(n<3, 1+n*(n-1)/2,
     ((4*n-10)*(5*n^2-10*n+4)*(a(n-1)+4*(n-2)*a(n-3)
      /(n-1))/(5*n^2-20*n+19)-4*(n-1)*a(n-2))/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 26 2018

Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1/2, 1/2, 1}, 1], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 04 2016 *)

G.f.: (1/sqrt(1-4x)+1/sqrt(1+4x^2))/2.
a(n) = Sum_{k=0..floor(n/2)} C(n,2k)^2.
a(n) = C(2n,n)/2+sin(Pi*(n+1)/2)*C(n,n/2)/2.
a(n) = A119326(2n,n).
a(n) = A071688(n) + A119359(n) for n>=1.
D-finite with recurrence n*(n-1)*(10*n-29)*a(n) +2*(n-1)*(5*n^2-74*n+164)*a(n-1) +4*(-40*n^3+310*n^2 -744*n+559)*a(n-2) +8*(n-2)*(5*n^2-74*n+164)*a(n-3) -16*(25*n-42)*(n-3)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Nov 05 2012
a(n) = hypergeom([(1-n)/2, (1-n)/2, -n/2, -n/2], [1/2, 1/2, 1], 1). - Vladimir Reshetnikov, Oct 04 2016
a(n) = A282011(2n,n). - Alois P. Heinz, Feb 04 2017

New name from Alois P. Heinz, Feb 04 2017

A071684 Number of plane trees with n edges and having an odd number of leaves.

Original entry on oeis.org

1, 1, 2, 7, 22, 66, 212, 715, 2438, 8398, 29372, 104006, 371516, 1337220, 4847208, 17678835, 64823110, 238819350, 883629164, 3282060210, 12233141908, 45741281820, 171529777432, 644952073662, 2430973304732, 9183676536076

Offset: 1

Sen-peng Eu, Jun 23 2002

Narayana transform (A001263) of [1, 0, 1, 0, 1, 0, 1, ...]. Example: a(4) = 7 = (1, 6, 6, 1) dot (1, 0, 1, 0) = (1 + 0 + 6 + 0). - Gary W. Adamson, Jan 04 2008

			a(3)=2 because among the 5 plane 3-trees there are 2 trees with odd number of leaves; a(4)=7 because among the 14 plane 4-trees there are 7 trees with odd number of leaves.

Robert Israel, Table of n, a(n) for n = 1..1668
Yu Hin Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
S. P. Eu, S. C. Liu and Y. N. Yeh, Odd or Even on Plane Trees, Discrete Math. 281 (2004), 189-196.

a(n) + A071688 = A000108: Catalan numbers.

Cf. A001263, A007595.

G:=((1+4*x^2)^(1/2)-(1-4*x)^(1/2)-2*x)/4/x: Gser:=series(G,x=0,30): seq(coeff(Gser,x,n),n=1..26); # Emeric Deutsch, Feb 17 2007

a[n_] := If[EvenQ[n], Binomial[2n, n]/(2n + 2), Binomial[2n, n]/(2n + 2) - (-1)^((n + 1)/2)Binomial[n - 1, (n - 1)/2]/(n + 1)]
Table[(CatalanNumber[n] + 2^n Binomial[1/2, (n + 1)/2])/2, {n, 20}] (* Vladimir Reshetnikov, Oct 03 2016 *)

a(2*n) = (1/(4*n + 2))*binomial(4*n, 2*n);
a(2*n-1) = (1/(4*n))*binomial(4*n-2, 2*n-1) - (-1)^n*(1/(2*n))*binomial(2*n-2, n-1), with n>0.
G.f.: (1/4)*((1+4*x^2)^(1/2) - (1-4*x)^(1/2)-2*x)/x. - Vladeta Jovovic, Apr 19 2003
a(0)=0; a(n) = Sum_{k = 0..floor(n/2)} (1/n)*C(n,2*k+1)*C(n,2*k) for n>0. - Paul Barry, Jan 25 2007
a(n) = Sum_{k=1..n} (1/n)*C(n,k)*C(n,k-1)*(1-(-1)^k)/2. - Paul Barry, Dec 16 2008
Conjecture: n*(n+1)*(10*n-37)*a(n) + 2*n*(5*n^2-42*n+91)*a(n-1) + 4*(-40*n^3+270*n^2-560*n+357)*a(n-2) + 8*(n-3)*(5*n^2-42*n+91)*a(n-3) - 16*(n-4)*(25*n-51)*(2*n-7)*a(n-4) = 0. - R. J. Mathar, Jul 05 2018
a(n) = (A000108(n) + 2^n * binomial(1/2, (n+1)/2))/2. - Vladimir Reshetnikov, Oct 03 2016
32*n*(2*n+1)*a(n) - 48*(n+2)*(n+1)*a(n+1) + 8*(n^2-n-9)*a(n+2) - 4*(2*n^2+10*n+9)*a(n+3) - 2*(n+5)*(n+6)*a(n+4) + (n+5)*(n+6)*a(n+5) = 0. - Robert Israel, Jul 05 2018

Edited by Robert G. Wilson v, Jun 25 2002

A119359 Central coefficients of number triangle A119326.

Original entry on oeis.org

0, 1, 1, 7, 31, 106, 386, 1499, 5755, 21886, 83854, 323302, 1248534, 4828916, 18719364, 72711123, 282867795, 1101981430, 4298723990, 16788997874, 65641296578, 256895812108, 1006307847324, 3945185527582, 15478851119966

Offset: 0

Paul Barry, May 16 2006

a(n) = A119326(2n,n+1). A119358(n)-a(n) = A071688(n).

Table[HypergeometricPFQ[{-1/2 - n/2, 1/2 - n/2, 1 - n/2, -n/2}, {1/2, 1/2, 1}, 1] - KroneckerDelta[n], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 04 2016 *)
Table[(2^n Binomial[1/2, (n+1)/2]  + Binomial[n, n/2] Cos[Pi n/2] + n CatalanNumber[n])/2 - KroneckerDelta[n], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 06 2016 *)

G.f.: (1/sqrt(1-4x)+(1/sqrt(1+4x^2)-1)-c(x)+x*c(-x^2))/2, c(x) the g.f. of A000108;
a(n) = (C(2n,n+1)+C((n-1)/2)*sin(Pi*n/2)-2*0^n-2C(n-1,n/2)*sin(Pi*(n-1)/2))/2.
a(n) = hypergeom([-1/2-n/2, 1/2-n/2, 1-n/2, -n/2], [1/2, 1/2, 1], 1) - 0^n. - Vladimir Reshetnikov, Oct 04 2016

cp's OEIS Frontend

A119358 Number of n-element subsets of [2n] having an even sum.

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A071684 Number of plane trees with n edges and having an odd number of leaves.

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A119359 Central coefficients of number triangle A119326.

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