A387248 -id:A387248 - cp's OEIS frontend

A387249 a(n) = 10/(n + 1) * Catalan(3*n).

10, 25, 440, 12155, 416024, 16158075, 682341000, 30582833775, 1433226830360, 69533550916004, 3468169547356640, 176946775343535925, 9199844912200348840, 486018122664268428850, 26029619941269629306160, 1410698658798280045783575, 77251704848334920869407000, 4269325372507953547350453420

Offset: 0

Peter Bala, Aug 24 2025

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..500

Cf. A000108, A000984, A007272, A066802, A387248, A387250.

seq( 10/((n+1)*(3*n+1)) * binomial(6*n, 3*n), n = 0..20);

A387249[n_] := 10*CatalanNumber[3*n]/(n + 1); Array[A387249, 20, 0] (* Paolo Xausa, Sep 02 2025 *)

a(n) = 10/((n + 1)*(3*n + 1)) * binomial(6*n, 3*n).
a(n) = (3*n + 2)/2 * (16*Catalan(3*n) - 8*Catalan(3*n+1) + Catalan(3n+2)) (shows a(n) to be an integer since Catalan(n) is odd iff n = 2^k - 1 for some k, so Catalan(3*n+2) is always even).
a(n) = (3*n + 2)/2 * A007272(3*n).
a(n) = 8*(2*n - 1)*(6*n - 1)*(6*n - 5)/((n + 1)*(3*n + 1)*(3*n - 1)) * a(n-1) with a(0) = 10.
a(n) ~ 10/(sqrt(27*Pi)) * 64^n/n^(5/2).
E.g.f.: 10*hypergeom([1/6, 1/2, 5/6], [2/3, 4/3, 2], 64*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

A387249 a(n) = 10/(n + 1) * Catalan(3*n).

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