A112858 -id:A112858 - cp's OEIS frontend

A075840 Primes of the form (2*n)!/(n!)^2+1.

2, 3, 7, 71, 3433, 2704157, 35345263801, 2104098963721, 6892620648693261354601, 410795449442059149332177041, 1520803477811874490019821888415218657, 5949105755928259715106809205795376486501, 1480212998448786189993816895482588794876101

Offset: 1

Donald S. McDonald, Oct 14 2002

nonn

			7 is a term because C(4,2)+1 = 6+1 = 7 is prime.

New Zealand Science Monthly, Bulletin Board, Feb. 1999. Binomial(300,150)+185 = nextprime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..25

Cf. A092751 = n such that (2*n)!/(n!)^2+1 is prime, A112858 = primes of the form (2*n)!/(n!)^2-1.

Cf. A000984, n's are in A066699.

[a: n in [0..100] | IsPrime(a) where a is Factorial(2*n) div Factorial(n)^2+1]; // Vincenzo Librandi Mar 17 2015

a = Select[ Range[100], PrimeQ[Binomial[2#, # ] + 1] & ]; Binomial[2a, a] + 1
Select[Table[(2 n)! / (n!)^2 + 1, {n, 0, 80}], PrimeQ] (* Vincenzo Librandi, Mar 17 2015 *)

v=[]; for(n=0,100,x=bin(2*n,n)+1; if(isprime(x), v=concat(v,x),)); v

Edited by Robert G. Wilson v, Oct 15 2002
Definition corrected by Alexander Adamchuk, Nov 30 2007
Edited by N. J. A. Sloane, Nov 30 2007
a(13) from Vincenzo Librandi, Mar 17 2015

A110952 Triangle read by rows: T(n,k) = number of permutations of [n] where the first increasing run has length k and the last increasing run has length n-k-1, 0

Original entry on oeis.org

1, 3, 3, 6, 11, 6, 10, 26, 26, 10, 15, 50, 71, 50, 15, 21, 85, 155, 155, 85, 21, 28, 133, 295, 379, 295, 133, 28, 36, 196, 511, 799, 799, 511, 196, 36, 45, 276, 826, 1519, 1849, 1519, 826, 276, 45, 55, 375, 1266, 2674, 3829, 3829, 2674, 1266, 375, 55, 66, 495, 1860

Offset: 3

David Scambler, Nov 22 2006

Permutations of [n] with exactly 2 descents and the descents are adjacent. Adjusting for initial index: row sums are A045618; first diagonal is A000217, the triangular numbers; 2nd diagonal is A051925; and 3rd diagonal is A001701, generalized Stirling numbers.

			Triangle (beginning with n=3, k=1) is:
   1
   3  3
   6 11  6
  10 26 26 10
  15 50 71 50 15
  ...
For n=5, k = 2: T(5,2) = 11 = permutations of [5] with first run 2 long and last run 5-2-1 = 2 long, namely {14325, 15324, 15423, 24315, 25314, 25413, 34215, 35214, 35412, 45213, 45312}.

Joseph Adams, Table of n, a(n) for n = 3..4755

Cf. A045618, A000217, A051925, A001701, A112858.

T(n,k) = k*C(n,k+1) - C(n,k) + 1.

cp's OEIS Frontend

A075840 Primes of the form (2*n)!/(n!)^2+1.

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