A001564 -id:A001564 - cp's OEIS frontend

A094794 a(n) = (1/n!)*A001689(n).

44, 309, 1214, 3539, 8544, 18089, 34754, 61959, 104084, 166589, 256134, 380699, 549704, 774129, 1066634, 1441679, 1915644, 2506949, 3236174, 4126179, 5202224, 6492089, 8026194, 9837719, 11962724, 14440269, 17312534, 20624939, 24426264

Offset: 0

Benoit Cloitre, Jun 11 2004

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A001563, A001564, A001565, A001688, A001689, A023043, A096307.

Cf. A094791, A094792, A094793, A094795.

Table[n^5+10n^4+45n^3+100n^2+109n+44,{n,0,30}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{44,309,1214,3539,8544,18089},30]

a(n)=n^5+10*n^4+45*n^3+100*n^2+109*n+44 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = n^5 + 10*n^4 + 45*n^3 + 100*n^2 + 109*n + 44.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6), with a(0)=44, a(1)=309, a(2)=1214, a(3)=3539, a(4)=8544, a(5)=18089. - Harvey P. Dale, Jul 25 2012
G.f.: (x^5 + 10*x^3 + 20*x^2 + 45*x + 44) / (x-1)^6. - Colin Barker, Jun 15 2013
P-recursive: n*a(n) = (n+6)*a(n-1) - a(n-2) with a(0) = 44 and a(1) = 309. Cf. A094791 and A096307. - Peter Bala, Jul 25 2021

A094792 a(n) = (1/n!)*A001565(n).

Original entry on oeis.org

2, 11, 32, 71, 134, 227, 356, 527, 746, 1019, 1352, 1751, 2222, 2771, 3404, 4127, 4946, 5867, 6896, 8039, 9302, 10691, 12212, 13871, 15674, 17627, 19736, 22007, 24446, 27059, 29852, 32831, 36002, 39371, 42944, 46727, 50726, 54947, 59396, 64079

Offset: 0

Benoit Cloitre, Jun 11 2004

Number of injections from {1,2,3} to {1,2,...,n} with no fixed points. - Fiona T. Brunk (fbrunk(AT)mcs.st-and.ac.uk), May 23 2006

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A001563, A001564, A001565, A001688, A001689, A023043.

Cf. A094791, A094793, A094794, A094795.

with(combinat):seq(fibonacci(4, i)-1,i=1..40); # Zerinvary Lajos, Mar 20 2008

Table[n^3+3*n^2+5*n+2,{n,0,70}] (* Vladimir Joseph Stephan Orlovsky, May 04 2011 *)

a(n) = n^3 + 3*n^2 + 5*n + 2.
a(n) = Sum_{i=0..3} (-1)^i*binomial(3,i)*(n-i)!/(n-3)!. - Fiona T. Brunk (fbrunk(AT)mcs.st-and.ac.uk), May 23 2006
G.f.: (x^3+3*x+2) / (x-1)^4. - Colin Barker, Jun 15 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Fung Lam, Apr 17 2014
P-recursive: n*a(n) = (n+4)*a(n-1) - a(n-2) with a(0) = 2 and a(1) = 11. Cf. A094791. - Peter Bala, Jul 25 2021

A258198 a(n) = largest k for which A001563(k) = k*k! <= n.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4

Offset: 0

Antti Karttunen, May 23 2015

nonn

Number of nonzero terms of A001563 <= n.

Each n occurs A001564(n) times.

Antti Karttunen, Table of n, a(n) for n = 0..4320

Cf. A001563, A001564, A084556, A084557, A256450, A258199.

(define (A258198 n) (let loop ((k 1) (f 1)) (if (> (* k f) n) (- k 1) (loop (+ k 1) (* (+ k 1) f)))))

A094795 a(n) = (1/n!)*A023043(n).

Original entry on oeis.org

265, 2119, 9403, 30637, 81901, 190435, 398959, 770713, 1395217, 2394751, 3931555, 6215749, 9513973, 14158747, 20558551, 29208625, 40702489, 55744183, 75161227, 99918301, 131131645, 170084179, 218241343, 277267657, 349044001, 435685615

Offset: 0

Benoit Cloitre, Jun 11 2004

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A001563, A001564, A001565, A001688, A001689, A023043.

Cf. A094791, A094792, A094793, A094794.

LinearRecurrence[{7,-21,35,-35,21,-7,1},{265,2119,9403,30637,81901,190435,398959},30] (* Harvey P. Dale, Aug 29 2023 *)

a(n) = n^6 + 15*n^5 + 100*n^4 + 355*n^3 + 694*n^2 + 689*n + 265.
G.f.: -(265 + 264*x + 135*x^2 + 40*x^3 + 15*x^4 + x^6)/(x-1)^7. - R. J. Mathar, Nov 15 2019
P-recursive: n*a(n) = (n+7)*a(n-1) - a(n-2) with a(0) = 265 and a(1) = 2119. Cf. A094791. - Peter Bala, Jul 25 2021

A136123 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k maximal strings of increasing consecutive integers (0<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 11, 12, 1, 53, 56, 11, 309, 321, 87, 3, 2119, 2175, 693, 53, 16687, 17008, 5934, 680, 11, 148329, 150504, 55674, 8064, 309, 1468457, 1485465, 572650, 96370, 5805, 53, 16019531, 16170035, 6429470, 1200070, 95575, 2119

Offset: 0

Emeric Deutsch and Vladeta Jovovic, Dec 17 2007

Row n has 1+floor(n/2) terms. Row sums are the factorials (A000142). Column 0 yields A000255. Column 1 yields A001277. Column 2 yields A001278. Column 3 yields A001279. Column 4 yields A001280. Sum(k*T(n,k),k>=0)=(n-2)!*(n^2 - 3n + 3)=A001564(n-2).

			T(3,0)=3 because we have 132, 213 and 321; T(6,3)=3 because we have 125634, 341256, 563412.
Triangle starts:
    1;
    1;
    1,   1;
    3,   3;
   11,  12,  1;
   53,  56, 11;
  309, 321, 87, 3;
  ...

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264, Table 7.6.1.

Cf. A000142, A000255, A001277, A001278, A001279, A001280, A001564, A010029 (rows reversed).

G:=Sum(factorial(n)*(((1-t)*x^2-x)/((1-t)*x^2-1))^n, n=0..infinity): Gser:= simplify(series(G,x=0,13)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,x,n)) end do: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form
# alternative
A136123 := proc(n,k)
    add( x^i*( ((1-y)*x-1)/((1-y)*x^2-1) )^i*i!,i=0..n+1) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,k) ;
end proc:
seq(seq( A136123(n,k),k=0..floor(n/2)),n=0..12) ; # R. J. Mathar, Jul 01 2022

T[n_, k_] := Sum[x^i*(((1-y)*x-1)/((1-y)*x^2-1))^i*i!, {i, 0, n+1}] //
   SeriesCoefficient[#, {x, 0, n}]& //
   SeriesCoefficient[#, {y, 0, k}]&;
Table[Table[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 09 2023, after R. J. Mathar *)

G.f.: G(x,t) = Sum_{n>=0} n!*(((1-t)*x^2 - x)/((1-t)*x^2-1))^n. - Vladeta Jovovic

A306512 Number A(n,k) of permutations p of [n] having no index i with |p(i)-i| = k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 9, 1, 1, 2, 3, 5, 44, 1, 1, 2, 6, 9, 21, 265, 1, 1, 2, 6, 14, 34, 117, 1854, 1, 1, 2, 6, 24, 53, 176, 792, 14833, 1, 1, 2, 6, 24, 78, 265, 1106, 6205, 133496, 1, 1, 2, 6, 24, 120, 362, 1554, 8241, 55005, 1334961

Offset: 0

Alois P. Heinz, Feb 20 2019

			A(4,0) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
A(4,1) = 5: 1234, 1432, 3214, 3412, 4231.
A(4,2) = 9: 1234, 1243, 1324, 2134, 2143, 2341, 4123, 4231, 4321.
Square array A(n,k) begins:
     1,   1,    1,    1,    1,    1,    1,    1, ...
     0,   1,    1,    1,    1,    1,    1,    1, ...
     1,   1,    2,    2,    2,    2,    2,    2, ...
     2,   2,    3,    6,    6,    6,    6,    6, ...
     9,   5,    9,   14,   24,   24,   24,   24, ...
    44,  21,   34,   53,   78,  120,  120,  120, ...
   265, 117,  176,  265,  362,  504,  720,  720, ...
  1854, 792, 1106, 1554, 2119, 2790, 3720, 5040, ...

Alois P. Heinz, Antidiagonals n = 0..45, flattened
Wikipedia, Permutation

Columns k=0-3 give: A000166, A078480, A306523, A324365.
A(n+2j,n+j) (j=0..5) give: A000142, A001564, A001688, A023043, A023045, A023047.
A(2n,n) gives A306535.
Cf. A306506.

A:= proc(n, k) option remember; `if`(k>=n, n!, LinearAlgebra[
      Permanent](Matrix(n, (i, j)-> `if`(abs(i-j)=k, 0, 1))))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
b:= proc(s, k) option remember; (n-> `if`(n=0, 1, add(
      `if`(abs(i-n)=k, 0, b(s minus {i}, k)), i=s)))(nops(s))
    end:
A:= (n, k)-> `if`(k>=n, n!, b({$1..n}, k)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := If[k > n, n!, Permanent[Table[If[Abs[i-j] == k, 0, 1], {i, 1, n}, {j, 1, n}]]]; A[0, 0] = 1;
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 05 2021, from first Maple program *)
b[s_, k_] := b[s, k] = With[{n = Length[s]}, If[n == 0, 1, Sum[
     If[Abs[i-n] == k, 0, b[s ~Complement~ {i}, k]], {i, s}]]];
A[n_, k_] := If[k >= n, n!, b[Range@n, k]];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Sep 01 2021, from second Maple program *)

A(n,k) = n! - A306506(n,k).

A(n,n+i) = n! for i >= 0.

A023044 7th differences of factorial numbers.

Original entry on oeis.org

1854, 16687, 165016, 1781802, 20886576, 264398280, 3597143040, 52370755920, 812752093440, 13397819541120, 233845982899200, 4309095479673600, 83609603781580800, 1704092533657113600, 36403110891295948800

Offset: 0

David W. Wilson

nonn

Paolo Xausa, Table of n, a(n) for n = 0..400
Milan Janjić, Enumerative Formulas for Some Functions on Finite Sets.
Index entries for sequences related to factorial numbers.

Cf. A000142, A001563, A001564, A001565, A001688, A001689, A023043, A023045, A023046, A023047.

Differences[Range[0, 25]!, 7] (* Paolo Xausa, May 26 2025 *)

A061312 Triangle T[n,m]: T[n,-1] = 0; T[0,0] = 0; T[n,0] = n*n!; T[n,m] = T[n,m-1] - T[n-1,m-1].

Original entry on oeis.org

0, 1, 1, 4, 3, 2, 18, 14, 11, 9, 96, 78, 64, 53, 44, 600, 504, 426, 362, 309, 265, 4320, 3720, 3216, 2790, 2428, 2119, 1854, 35280, 30960, 27240, 24024, 21234, 18806, 16687, 14833, 322560, 287280, 256320, 229080, 205056, 183822, 165016, 148329

Offset: 0

Wouter Meeussen, Jun 06 2001

Appears in the (n,k)-matching problem A076731. [Johannes W. Meijer, Jul 27 2011]

			0,
1, 1,
4, 3, 2,
18, 14, 11, 9,
96, 78, 64, 53, 44,
600, 504, 426, 362, 309, 265,
4320, 3720, 3216, 2790, 2428, 2119, 1854,
35280, 30960, 27240, 24024, 21234, 18806, 16687, 14833,

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A061018.
From Johannes W. Meijer, Jul 27 2011: (Start)
Columns: A001563, A001564, A001565, A001688, A001689, A023044, A023045, A023046, A023047; A000166, A000255, A055790;
The row sums equal A193465. (End)

[[(&+[(-1)^j*Binomial(k+1,j)*Factorial(n-j+1): j in [0..k+1]]): k in [0..n]]: n in [0..20]]; // G. C. Greubel, Aug 13 2018

A061312 := proc(n,m): add(((-1)^j)*binomial(m+1,j)*(n+1-j)!, j=0..m+1) end: seq(seq(A061312(n,m), m=0..n), n=0..7); # Johannes W. Meijer, Jul 27 2011

T[n_, k_]:= Sum[(-1)^j*Binomial[k + 1, j]*(n + 1 - j)!, {j, 0, k + 1}]; Table[T[n, k], {n, 0, 100}, {k, 0, n}] // Flatten  (* G. C. Greubel, Aug 13 2018 *)

for(n=0,20, for(k=0,n, print1(sum(j=0,k+1, (-1)^j*binomial(k+1,j) *(n-j+1)!), ", "))) \\ G. C. Greubel, Aug 13 2018

T[n,m] = T[n,m-1]-T[n-1,m-1] with T[n,-1] = 0 and T[n,0] = A001563(n) = n*n!

T(n,m) = sum(((-1)^j)*binomial(m+1,j)*(n+1-j)!, j=0..m+1) [Johannes W. Meijer, Jul 27 2011]

A094791 Triangle read by rows giving coefficients of polynomials arising in successive differences of (n!)_{n>=0}.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 3, 5, 2, 1, 6, 17, 20, 9, 1, 10, 45, 100, 109, 44, 1, 15, 100, 355, 694, 689, 265, 1, 21, 196, 1015, 3094, 5453, 5053, 1854, 1, 28, 350, 2492, 10899, 29596, 48082, 42048, 14833, 1, 36, 582, 5460, 32403, 124908, 309602, 470328, 391641, 133496

Offset: 0

Benoit Cloitre, Jun 11 2004

Let D_0(n)=n! and D_{k+1}(n)=D_{k}(n+1)-D_{k}(n), then D_{k}(n)=n!*P_{k}(n) where P_{k} is a polynomial with integer coefficients of degree k.

The horizontal reversal of this triangle arises as a binomial convolution of the derangements coefficients der(n,i) (numbers of permutations of size n with i derangements = A098825(n,i) = number of permutations of size n with n-i rencontres = A008290(n,n-i), see formula section). - Olivier Gérard, Jul 31 2011

			D_3(n) = n!*(n^3 + 3*n^2 + 5*n + 2).
D_4(n) = n!*(n^4 + 6*n^3 + 17*n^2 + 20*n + 9).
Table begins:
  1
  1  0
  1  1   1
  1  3   5   2
  1  6  17  20    9
  1 10  45 100  109   44
  1 15 100 355  694  689  265
  ...

Successive differences of factorial numbers: A001563, A001564, A001565, A001688, A001689, A023043.
Rencontres numbers A008290. Partial derangements A098825.
Row sum is A000255. Signed version in A126353.
Cf. A094792, A094793, A094794, A094795.

with(LREtools): A094791_row := proc(n)
delta(x!,x,n); simplify(%/x!); seq(coeff(%,x,n-j),j=0..n) end:
seq(print(A094791_row(n)),n=0..9); # Peter Luschny, Jan 09 2015

d[0][n_] := n!; d[k_][n_] := d[k][n] = d[k - 1][n + 1] - d[k - 1][n] // FullSimplify;
row[k_] := d[k][n]/n! // FullSimplify // CoefficientList[#, n]& // Reverse;
Array[row, 10, 0] // Flatten (* Jean-François Alcover, Aug 02 2019 *)

T(n, n) = A000166(n).
T(2, k) = A000217(k).
Sum_{k=0..n} T(n,n-k)*x^k = Sum_{i=0..n} der(n,i)*binomial( n+x, i) (an analog of Worpitzky's identity). - Olivier Gérard, Jul 31 2011
The n-th row polynomial R(n,x) = Sum {k = 0..n} T(n,k)*x^k is P-recursive in the variable x: x*R(n,x) = (x+n+1)*R(n,x-1) - R(n,x-2). - _Peter Bala, Jul 25 2021

Edited and T(0,0) corrected according to the author's definition by Olivier Gérard, Jul 31 2011

A116853 Difference triangle of factorial numbers read by upward diagonals.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 11, 14, 18, 24, 53, 64, 78, 96, 120, 309, 362, 426, 504, 600, 720, 2119, 2428, 2790, 3216, 3720, 4320, 5040, 16687, 18806, 21234, 24024, 27240, 30960, 35280, 40320

Offset: 1

Gary W. Adamson, Feb 24 2006

This is a subsequence of Euler's difference table A068106 and of A047920 (in a different ordering), since 0! = 1 was left out here. - Georg Fischer, Mar 23 2019

			Starting with 1, 2, 6, 24, 120 ... we take the first difference row (A001563), second, third, etc. Reorient into a flush left format, getting:
[1]    1;
[2]    1,   2;
[3]    3,   4,   6;
[4]   11,  14,  18,  24;
[5]   53,  64,  78,  96, 120;
[6]  309, 362, 426, 504, 600, 720;
...

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A000142 (factorial numbers).
Cf. A000255 (first column and inverse binomial transform of A000142).
N-th forward differences of A000142: A001563 (1st), A001564 (2nd), A001565 (3rd), A001688 (4th), A001689 (5th).
Cf. A047920 (with 0!, different order), A068106 (with 0!), A180191 (row sums), A246606 (central terms).

a116853 n k = a116853_tabl !! (n-1) !! (k-1)
a116853_row n = a116853_tabl !! (n-1)
a116853_tabl = map reverse $ f (tail a000142_list) [] where
   f (u:us) vs = ws : f us ws where ws = scanl (-) u vs
-- Reinhard Zumkeller, Aug 31 2014

rows = 8;
rr = Range[rows]!;
dd = Table[Differences[rr, n], {n, 0, rows-1}];
T = Array[t, {rows, rows}];
Do[Thread[Evaluate[Diagonal[T, -k+1]] = dd[[k, ;;rows-k+1]]], {k, rows}];
Table[t[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 21 2019 *)

Take successive difference rows of factorial numbers n! starting with n=1. Reorient into a triangle format.

cp's OEIS Frontend

A094794 a(n) = (1/n!)*A001689(n).

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A094792 a(n) = (1/n!)*A001565(n).

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A258198 a(n) = largest k for which A001563(k) = k*k! <= n.

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A094795 a(n) = (1/n!)*A023043(n).

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A136123 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k maximal strings of increasing consecutive integers (0<=k<=floor(n/2)).

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A306512 Number A(n,k) of permutations p of [n] having no index i with |p(i)-i| = k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A023044 7th differences of factorial numbers.

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A061312 Triangle T[n,m]: T[n,-1] = 0; T[0,0] = 0; T[n,0] = n*n!; T[n,m] = T[n,m-1] - T[n-1,m-1].

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