The On-Line Encyclopedia of Integer Sequences (OEIS)

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A120306

Numerator of the sum of all matrix elements of n X n matrix M[i,j]=CatalanNumber[i]/CatalanNumber[j], where CatalanNumber[k]=(2k)!/k!/(k+1)!=A000108[k].
(history; published version)

#4 by N. J. A. Sloane at Tue Jan 29 18:26:18 EST 2013

STATUS

editing

approved

#3 by N. J. A. Sloane at Tue Jan 29 18:26:15 EST 2013

COMMENTS

p divides a(p-1) for prime p=3,7,13,19,31,37,43,61,67..=A007645 Cuban primes: of form x^2+xy+y^2; or: primes of form x^2+3*y^2; or: primes == 0 or 1 mod 3.

p divides a(p-1) for primes p in A007645.

STATUS

approved

editing

#2 by Russ Cox at Sat Mar 31 13:20:27 EDT 2012

AUTHOR

_Alexander Adamchuk (alex(AT)kolmogorov.com), _, Jul 14 2006

Discussion

Sat Mar 31

13:20

OEIS Server: https://oeis.org/edit/global/879

#1 by N. J. A. Sloane at Fri Sep 29 03:00:00 EDT 2006

NAME

Numerator of the sum of all matrix elements of n X n matrix M[i,j]=CatalanNumber[i]/CatalanNumber[j], where CatalanNumber[k]=(2k)!/k!/(k+1)!=A000108[k].

DATA

1, 9, 68, 1364, 12064, 58303, 4517375, 1142991, 4251679307, 138473652271, 240881487689, 857560784067, 49571162119157, 12805922830496929, 167798784068528807, 365691567246838709, 46160923354240494523

OFFSET

1,2

COMMENTS

p divides a(p-1) for prime p=3,7,13,19,31,37,43,61,67..=A007645 Cuban primes: of form x^2+xy+y^2; or: primes of form x^2+3*y^2; or: primes == 0 or 1 mod 3.

FORMULA

a(n) = Numerator[Sum[Sum[(2i)!/(i!)^2/(i+1)/((2j)!/(j!)^2/(j+1)),{i,1,n}],{j,1,n}]].

MATHEMATICA

Numerator[Table[Sum[Sum[(2i)!/(i!)^2/(i+1)/((2j)!/(j!)^2/(j+1)), {i, 1, n}], {j, 1, n}], {n, 1, 20}]]

CROSSREFS

Cf. A000108, A014138, A007645.

KEYWORD

frac,nonn

AUTHOR

Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 14 2006

STATUS

approved