A233005 a(n) = floor(Pt(n)/n!), where Pt(n) is product of first n positive triangular numbers (A000217).

1, 1, 3, 7, 22, 78, 315, 1417, 7087, 38981, 233887, 1520268, 10641881, 79814109, 638512875, 5427359437, 48846234937, 464039231906, 4640392319062, 48724119350156, 535965312851718, 6163601097794765, 73963213173537187, 924540164669214843, 12019022140699792968

Offset: 1

Author

Alex Ratushnyak, Dec 03 2013

Keywords

• nonn

Examples

			a(4) = 7, because, the first four triangular numbers being 1, 3, 6, 10, their product is 180, which divided by 4! is 15/2 = 7.5.
a(5) = 22, because, the first five triangular numbers being 1, 3, 6, 10, 15, their product is 2700, which divided by 5! is 45/2 = 22.5.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..505

Crossrefs

Cf. A000142, A000217.

Cf. A006472 (triangular factorial, essentially equal to Pt(n)).

Cf. A067667 (Pt(n)/n! for n's of the form 2^k-1).

Programs

Mathematica

With[{nn=30},Floor[#[[1]]/#[[2]]]&/@Thread[{FoldList[Times,Accumulate[ Range[ nn]]],Range[nn]!}]] (* Harvey P. Dale, Apr 02 2017 *)

Python

f=t=1
for n in range(1,33):
  t*=n*(n+1)//2
  f*=n
  print(t//f, end=', ')

Formula

a(n) = floor((n+1)!/2^n). - Yifan Xie, Mar 05 2023