A090014
Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line.
Original entry on oeis.org
5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365, 1893369505, 27756952355, 435287980375, 7269934161905, 128812336516885, 2413131201408695, 47652865538001595, 989254278781162325
Offset: 1
- Brualdi, Richard A. and Ryser, Herbert J., Combinatorial Matrix Theory, Cambridge NY (1991), Chapter 7.
-
f[x_] := x*HypergeometricPFQ[{1, 5}, {}, x/(x+1)]/(x+1); Total /@ Partition[ CoefficientList[ Series[f[x], {x, 0, 18}], x], 2, 1] // Rest (* Jean-François Alcover, Nov 12 2013, after A001909 and Mark van Hoeij *)
t={5,25};Do[AppendTo[t,(n+3)*t[[-1]]+(n-2)*t[[-2]]],{n,3,17}];t (* Indranil Ghosh, Feb 21 2017 *)
A247490
Square array read by antidiagonals: A(k, n) = (-1)^(n+1)* hypergeom([k, -n+1], [], 1) for n>0 and A(k,0) = 0 (n>=0, k>=1).
Original entry on oeis.org
0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 2, 3, 2, 0, 1, 3, 7, 11, 9, 0, 1, 4, 13, 32, 53, 44, 0, 1, 5, 21, 71, 181, 309, 265, 0, 1, 6, 31, 134, 465, 1214, 2119, 1854, 0, 1, 7, 43, 227, 1001, 3539, 9403, 16687, 14833, 0, 1, 8, 57, 356, 1909, 8544, 30637, 82508, 148329, 133496
Offset: 0
k\n
[1], 0, 1, 0, 1, 2, 9, 44, 265, 1854, ... A000166
[2], 0, 1, 1, 3, 11, 53, 309, 2119, 16687, ... A000255
[3], 0, 1, 2, 7, 32, 181, 1214, 9403, 82508, ... A000153
[4], 0, 1, 3, 13, 71, 465, 3539, 30637, 296967, ... A000261
[5], 0, 1, 4, 21, 134, 1001, 8544, 81901, 870274, ... A001909
[6], 0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319, ... A001910
[7], 0, 1, 6, 43, 356, 3333, 34754, 398959, 4996032, ... A176732
[8], 0, 1, 7, 57, 527, 5441, 61959, 770713, 10391023, ... A176733
The referenced sequences may have a different offset or other small deviations.
-
A := (k,n) -> `if`(n<2,n,hypergeom([k,-n+1],[],1)*(-1)^(n+1));
seq(print(seq(round(evalf(A(k,n),100)), n=0..8)), k=1..8);
-
from mpmath import mp, hyp2f0
mp.dps = 25; mp.pretty = True
def A247490(k, n):
if n < 2: return n
if k == 1 and n == 2: return 0 # (failed to converge)
return int((-1)^(n+1)*hyp2f0(k, -n+1, 1))
for k in (1..8): print([k], [A247490(k, n) for n in (0..8)])
A336246
Array read by upwards antidiagonals: T(n,k) is the number of ways to place n persons on different seats such that each person number p, 1 <= p <= n, differs from the seat number s(p), 1 <= s(p) <= n+k, k >= 0.
Original entry on oeis.org
0, 1, 1, 2, 3, 2, 9, 11, 7, 3, 44, 53, 32, 13, 4, 265, 309, 181, 71, 21, 5, 1854, 2119, 1214, 465, 134, 31, 6, 14833, 16687, 9403, 3539, 1001, 227, 43, 7, 133496, 148329, 82508, 30637, 8544, 1909, 356, 57, 8, 1334961, 1468457, 808393, 296967, 81901, 18089, 3333, 527, 73, 9
Offset: 1
For k=1, the n-tuples of seat numbers are:
- for n=1: 2 => T(1,1) = 1.
- for n=2: 21, 23, 31 => T(2,1) = 3,
21: person 1 sits on seat 2 and vice versa.
A counterexample is 13 because person 1 would sit on seat 1.
- for n=3: 214,231,234,241,312,314,341,342,412,431,432 => T(3,1) = 11.
Array begins:
0 1 2 3 4 ...
1 3 7 13 21 ...
2 11 32 71 134 ...
9 53 181 465 1001 ...
44 309 1214 3539 8544 ...
.. ... .... .... ....
Cf.
A000166,
A000255,
A000153,
A000261,
A001909,
A001910,
A176732,
A176733,
A176734,
A176735,
A176736.
-
block(nr: 0, k: -1, mmax: 55,
/*First mmax terms are returned, recurrence used*/
a: makelist(0, n, 1, mmax),
while nr
-
block(n: 1, k: 0, mmax: 55,
/*First mmax terms are returned, explicit formula used*/
a: makelist(0, n, 1, mmax),
for m from 1 thru mmax do (su: 0,
for r from 0 thru n do su: su+(-1)^r*binomial(n,r)*(n+k-r)!/k!,
a[m]: su, if n=1 then (n: k+2, k: 0) else (n: n-1, k: k+1)),
return(a));
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