A387249 -id:A387249 - cp's OEIS frontend

A387248 a(n) = 3/(n + 1) * Catalan(2*n).

3, 3, 14, 99, 858, 8398, 89148, 1002915, 11785890, 143291610, 1790214660, 22870640910, 297670187844, 3935861372604, 52749590350072, 715309969142307, 9800129095949682, 135490673691621794, 1888389218820071604, 26510079418051005210, 374589577468070301260, 5324240442532424176260, 76082624294738699098440

Offset: 0

Peter Bala, Aug 24 2025

Bisection of A007054.
Compare with Catalan(n) = 1/(n + 1) * binomial(2*n, n).
For r >= 2, there is a constant K_r such that K_r/(n + 1) * Catalan(r*n) is integral for all n.

Paolo Xausa, Table of n, a(n) for n = 0..800

Cf. A000108, A000984, A001448, A007054, A387249, A387250.

seq( 6/((2*n+1)*(2*n+2)) * binomial(4*n, 2*n), n = 0..22);

A387248[n_] := 3*CatalanNumber[2*n]/(n + 1); Array[A387248, 25, 0] (* Paolo Xausa, Sep 02 2025 *)

a(n) = 6/((2*n + 1)*(2*n + 2)) * binomial(4*n, 2*n).
a(n) = 4*Catalan(2*n) - Catalan(2*n+1) (showing a(n) to be an integer)
G.f.: A(x) = ((2 - f(x))*sqrt(2 + 2*f(x)) - 2)/(4*x), where f(x) = sqrt(1 - 16*x).
a(n) = 2*(4*n - 1)*(4*n - 3)/((n + 1)*(2*n + 1)) * a(n-1) with a(0) = 3.
a(n) ~ 3/(2*sqrt(2*Pi)) * 16^n/n^(5/2).
a(n) is odd iff n = 2^k - 1 for some k, so a(n) has the same parity as Catalan(n).
E.g.f.: 3*hypergeom([1/4, 3/4], [3/2, 2], 16*x). - Stefano Spezia, Aug 27 2025

A387250 a(n) = 105/(n + 1) * Catalan(4*n).

Original entry on oeis.org

105, 735, 50050, 5460315, 742511070, 114872107350, 19348562209860, 3461691866723475, 647897423565562310, 125577883051534761666, 25029394494457424675100, 5103876046438721064520350, 1060725331955983336553011500, 224018752093294694626068131340, 47967198494914114482847609250184

Offset: 0

Peter Bala, Aug 25 2025

For r >= 2, there is a constant K_r such that K_r/(n + 1) * Catalan(r*n) is integral for all n.

Cf. A000108, A387248, A387249.

seq( 105/((n + 1)*(4*n + 1)) * binomial(8*n, 4*n), n = 0..20);

a[n_]:=105/(n+1)*CatalanNumber[4n];Array[a,15,0] (* James C. McMahon, Aug 29 2025 *)

a(n) = 105/((n + 1)*(4*n + 1)) * binomial(8*n, 4*n).
a(n) = 2*(8*n - 1)*(8*n - 3)*(8*n - 5)*(8*n - 7)/((n + 1)*(2*n - 1)*(4*n + 1)*(4*n - 1)) * a(n-1) with a(0) = 105.
a(n) ~ 105/(8*sqrt(Pi)) * 256^n/n^(5/2).
E.g.f.: 105*hypergeom([1/8, 3/8, 5/8, 7/8], [1/2, 3/4, 5/4, 2], 256*x). - Stefano Spezia, Aug 27 2025

cp's OEIS Frontend

A387248 a(n) = 3/(n + 1) * Catalan(2*n).

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