A000261 -id:A000261 - cp's OEIS frontend

A383378 Expansion of e.g.f. exp(-3*x) / (1-x)^4.

1, 1, 5, 21, 129, 897, 7317, 67365, 692577, 7849953, 97199109, 1304688789, 18863836065, 292198665249, 4826470920021, 84669407740773, 1571901715253313, 30786460730863425, 634323280633460613, 13714611211502376597, 310448651226154786881, 7342298348439393120321

Offset: 0

Seiichi Manyama, Apr 24 2025

Column k=3 of A383341.
Cf. A000261, A383344, A383380.
Cf. A010843, A137775, A383382.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-3*x)/(1-x)^4))

a(n) = n! * Sum_{k=0..n} (-3)^(n-k) * binomial(k+3,3)/(n-k)!.
a(0) = a(1) = 1; a(n) = n*a(n-1) + 3*(n-1)*a(n-2).
a(n) = A137775(n+2)/(3*(n+1)).
a(n) ~ sqrt(2*Pi) * n^(n + 7/2) / (6*exp(n+3)). - Vaclav Kotesovec, Apr 25 2025

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

Jaap Spies, Dec 13 2003

Brualdi, Richard A. and Ryser, Herbert J., Combinatorial Matrix Theory, Cambridge NY (1991), Chapter 7.

Indranil Ghosh, Table of n, a(n) for n = 1..445
Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), pp. 197-210.

a(n) = A001909(n-1) + A001909(n), a(1)=5

Cf. A000255, A000153, A000261, A001909, A001910, A090010, A055790, A090012-A090016.

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 *)

a(n) = (n+3)*a(n-1) + (n-2)*a(n-2), a(1)=5, a(2)=25.

a(n) ~ exp(-1) * n! * n^4 / 24. - Vaclav Kotesovec, Nov 30 2017

Corrected by Jaap Spies, Jan 26 2004

A383380 Expansion of e.g.f. exp(-2*x) / (1-x)^4.

Original entry on oeis.org

1, 2, 8, 40, 248, 1808, 15136, 142784, 1496960, 17254144, 216740864, 2945973248, 43065951232, 673626675200, 11224114860032, 198447384666112, 3710328985124864, 73136238041563136, 1515739708283944960, 32947698735175172096, 749499782353468522496, 17806903161183314378752

Offset: 0

Seiichi Manyama, Apr 24 2025

Cf. A000261, A383344, A383378.
Cf. A000023, A052124, A087981, A383381.
Cf. A000255.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-2*x)/(1-x)^4))

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A000255.
a(n) = n! * Sum_{k=0..n} (-2)^(n-k) * binomial(k+3,3)/(n-k)!.
a(0) = 1, a(1) = 2; a(n) = (n+1)*a(n-1) + 2*(n-1)*a(n-2).
a(n) ~ sqrt(Pi) * n^(n + 7/2) / (3*sqrt(2)*exp(n+2)). - Vaclav Kotesovec, Apr 25 2025

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

Peter Luschny, Sep 20 2014

			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.

Cf. A000166, A000255, A000153, A000261, A001909, A001910, A176732 - A176736.

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

Gerhard Kirchner, Jul 19 2020

T(n,0) = !n (subfactorial) is the number of derangements or fixed-point-free permutations, see A000166(n) below: n persons are placed on n seats such that no person sits on a seat with the same number. The generalization of a permutation is a variation (n persons and n+k seats such that k seats remain free). In this sense, T(n,k) is the number of fixed-point-free variations. I am rather sure that such variations have been examined, but I cannot find a reference.
Some subsequences T(n,k) with k=const:
T(n,0) = A000166(n);   T(n,1) = A000255(n);   T(n,2) = A000153(n-1);
T(n,3) = A000261(n-1); T(n,4) = A001909(n-3); T(n,5) = A001910(n-4);
T(n,6) = A176732(n);   T(n,7) = A176733(n);   T(n,8) = A176734(n);
T(n,9) = A176735(n);   T(n,10) = A176736(n).

			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 ...
  .. ... .... .... ....

Gerhard Kirchner, Fixed-point-free variations

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));

T(n,k) = (n+k-1)*T(n-1,k) + (n-1)*T(n-2,k) for n >= 2, k >= 0 with T(0,k)=1 and T(1,k)=k.
For n=0, there is one empty variation. T(0,k) is used for the recurrence only, not in the table. For n=1, the person can be placed on seat number 2..k+1 (if k > 0).
You also find the recurrence in the formula section of A000166 (k=0) and in the name section of the other sequences listed above (1 <= k <= 10). Some sequences have a different offset.
T(n,k) = Sum_{r=0..n} (-1)^r*binomial(n,r)*(n+k-r)!/k!.
Proofs see link.

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A383378 Expansion of e.g.f. exp(-3*x) / (1-x)^4.

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A090014 Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line.

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A383380 Expansion of e.g.f. exp(-2*x) / (1-x)^4.

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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).

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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.

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