A105291 -id:A105291 - cp's OEIS frontend

A109892 a(n) = least integer of the form (n!+1)(n!+2)...(n!+k)/n!.

2, 6, 84, 20475, 234531275, 199200973555045, 16481425431663122588749, 173392935733620216899469862542865, 300717095810709134168380432250652303057474577

Offset: 1

Amarnath Murthy, Jul 13 2005

nonn

Equivalently, binomial(n!+n,n). Proof: (n!+1)(n!+2)...(n!+k) == k! mod n! == 0 mod n! if and only if k >= n (for n >= 2). - Paul D Hanna and Robert Israel, Aug 31 2010.
Note that k <= n. Subsidiary sequence to be investigated: n such that k < n.
This is just a coincidence, but k=2,6,84 are also such that floor(exp(1)*10^k) is a prime, cf. A064118. - M. F. Hasler, Aug 31 2013

			a(4)=25*26*27*28/24=20475.

Cf. A105291.

A109892 := proc(n) local k,fn; k := 1; fn := n! ; while mul(fn+i,i=1..k) mod fn <> 0 do k := k+1; od ; RETURN(mul(fn+i,i=1..k)/fn) ; end: seq(A109892(n),n=1..10) ; # R. J. Mathar, Aug 15 2007

Table[(n+n!)!/(n!*(n!)!), {n, 1, 9}] (* Jean-François Alcover, Mar 04 2014, after first comment *)

Corrected and extended by R. J. Mathar, Aug 15 2007

A180397 T(n,m) = binomial(m!,n).

Original entry on oeis.org

1, 2, 0, 6, 1, 0, 24, 15, 0, 0, 120, 276, 20, 0, 0, 720, 7140, 2024, 15, 0, 0, 5040, 258840, 280840, 10626, 6, 0, 0, 40320, 12698280, 61949040, 8214570, 42504, 1, 0, 0, 362880, 812831040, 21324644880, 11104365420, 190578024, 134596, 0, 0, 0, 3628800

Offset: 1

R. H. Hardin, Sep 01 2010

T(n,m) = number of n X m arrays with each row a permutation of 1..m and rows lexicographically in strictly increasing order.

			Table starts
1 2 6 24 120 720 5040 40320 362880 3628800
0 1 15 276 7140 258840 12698280 812831040 65840765760
0 0 20 2024 280840 61949040 21324644880 10923907290240
0 0 15 10626 8214570 11104365420 26853059065140
0 0 6 42504 190578024 1590145128144
0 0 1 134596 3652745460
0 0 0 346104
0 0 0
0 0
0

R. H. Hardin, Table of n, a(n) for n = 1..1275

See A105291 for another version.
First row gives A000142.
Main diagonal gives A086687.

t[n_, m_] := Binomial[m!, n]; Table[t[m - n + 1, n], {m, 9}, {n, m, 1, -1}] // Flatten (* to display table in Comment *) Table[ t[m, n], {m, 10}, {n, 8}] // TableForm (* Robert G. Wilson v, Sep 02 2010 *)

T(n,m) = binomial(m!,n).

cp's OEIS Frontend

A109892 a(n) = least integer of the form (n!+1)(n!+2)...(n!+k)/n!.

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