A282466
a(n) = n*a(n-1) + n!, with n>0, a(0)=5.
Original entry on oeis.org
5, 6, 14, 48, 216, 1200, 7920, 60480, 524160, 5080320, 54432000, 638668800, 8143027200, 112086374400, 1656387532800, 26153487360000, 439378587648000, 7825123418112000, 147254595231744000, 2919482409811968000, 60822550204416000000, 1328364496464445440000
Offset: 0
- C. Mariconda and A. Tonolo, Calcolo discreto, Apogeo (2012), page 240 (Example 9.57 gives the recurrence).
Cf. sequences with formula (n + k)*n!:
A052521 (k=-5),
A282822 (k=-4),
A052520 (k=-3),
A052571 (k=-2),
A062119 (k=-1),
A001563 (k=0),
A000142 (k=1),
A001048 (k=2),
A052572 (k=3),
A052644 (k=4), this sequence (k=5).
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A282466:= func< n | (n+5)*Factorial(n) >; // G. C. Greubel, May 14 2025
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RecurrenceTable[{a[0] == 5, a[n] == n a[n - 1] + n!}, a, {n, 0, 30}] (* or *)
Table[(n + 5) n!, {n, 0, 30}]
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def A282466(n): return (n+5)*factorial(n) # G. C. Greubel, May 14 2025
A324225
Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.
Original entry on oeis.org
1, 1, 2, 1, 2, 4, 6, 4, 2, 6, 12, 18, 24, 18, 12, 6, 24, 48, 72, 96, 120, 96, 72, 48, 24, 120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120, 720, 1440, 2160, 2880, 3600, 4320, 5040, 4320, 3600, 2880, 2160, 1440, 720, 5040, 10080, 15120, 20160, 25200, 30240, 35280, 40320, 35280, 30240, 25200, 20160, 15120, 10080, 5040
Offset: 1
The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have (signed) displacement lists [p(i)-i, i=1..3]: [0,0,0], [0,1,-1], [1,-1,0], [1,1,-2], [2,-1,-1], [2,0,-2], representing the indices of falling diagonals of 1's in the permutation matrices
[1 ] [1 ] [ 1 ] [ 1 ] [ 1] [ 1]
[ 1 ] [ 1] [1 ] [ 1] [1 ] [ 1 ]
[ 1] [ 1 ] [ 1] [1 ] [ 1 ] [1 ] , respectively. Indices -2 and 2 occur twice, -1 and 1 occur four times, and 0 occurs six times. So row n=3 is [2, 4, 6, 4, 2].
Triangle T(n,k) begins:
: 1 ;
: 1, 2, 1 ;
: 2, 4, 6, 4, 2 ;
: 6, 12, 18, 24, 18, 12, 6 ;
: 24, 48, 72, 96, 120, 96, 72, 48, 24 ;
: 120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120 ;
- Alois P. Heinz, Rows n = 1..100, flattened
- Nadir Samos Sáenz de Buruaga, Rafał Bistroń, Marcin Rudziński, Rodrigo Miguel Chinita Pereira, Karol Życzkowski, and Pedro Ribeiro, Fidelity decay and error accumulation in quantum volume circuits, arXiv:2404.11444 [quant-ph], 2024. See p. 18.
- Wikipedia, Permutation
- Wikipedia, Permutation matrix
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b:= proc(s, c) option remember; (n-> `if`(n=0, c,
add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1-n..n-1))(b({$1..n}, 0)):
seq(T(n), n=1..8);
# second Maple program:
egf:= k-> (t-> x^t/t*hypergeom([2, t], [t+1], x))(abs(k)+1):
T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
# third Maple program:
T:= (n, k)-> (t-> `if`(t
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T[n_, k_] := With[{t = Abs[k]}, If[tJean-François Alcover, Mar 25 2021, after 3rd Maple program *)
A280556
a(n) = Sum_{k=1..n} k^2 * (k+1)!.
Original entry on oeis.org
0, 2, 26, 242, 2162, 20162, 201602, 2177282, 25401602, 319334402, 4311014402, 62270208002, 958961203202, 15692092416002, 271996268544002, 4979623993344002, 96035605585920002, 1946321606541312002, 41359334139002880002, 919636959090769920002, 21356013827774545920002
Offset: 0
- Seiichi Manyama, Table of n, a(n) for n = 0..446
- Mathematical Reflections, Problem J256, Issue 1, 2013, p 4.
- Mathematical Reflections, Solution to Problem J256, Issue 2, 2013, p 4.
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A280556:=n->add(k^2*(k+1)!, k=1..n): seq(A280556(n), n=0..30); # Wesley Ivan Hurt, Jan 05 2017
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Table[Sum[k^2 (k+1)!,{k,n}],{n,0,20}] (* Harvey P. Dale, Jun 05 2017 *)
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a(n) = sum(k=1, n, k^2*(k+1)!)
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