cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A116854 First differences of the rows in the triangle of A116853, starting with 0.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 11, 3, 4, 6, 53, 11, 14, 18, 24, 309, 53, 64, 78, 96, 120, 2119, 309, 362, 426, 504, 600, 720, 16687, 2119, 2428, 2790, 3216, 3720, 4320, 5040, 148329, 16687, 18806, 21234, 24024, 27240, 30960, 35280, 40320, 1468457, 148329, 165016, 183822, 205056, 229080, 256320, 287280, 322560, 362880
Offset: 1

Views

Author

Gary W. Adamson, Feb 24 2006

Keywords

Comments

Row n contains the first differences of row n of A116853, starting with T(n,1) = A116853(n,1) - 0.
As in A116853, 0! = 1 is omitted here. - Georg Fischer, Mar 23 2019

Examples

			First few rows of the triangle are:
[1]    1;
[2]    1,   1;
[3]    3,   1,   2;
[4]   11,   3,   4,   6;
[5]   53,  11,  14,  18,  24;
[6]  309,  53,  64,  78,  96, 120;
[7] 2119, 309, 362, 426, 504, 600, 720;
...
For example, row 4 (11, 3, 4, 6) are first differences along row 4 of A116853: ((0), 11, 14, 18, 24).

Links

Crossrefs

Cf. A000255, A116853.
Cf. A000142 (row sums), A033815 (central terms), A047920, A068106 (with 0!), A055790 (column k=3), A277609 (k=4), A277563 (k=5), A280425 (k=6).

Programs

  • Haskell
    a116854 n k = a116854_tabl !! (n-1) !! (k-1)
a116854_row n = a116854_tabl !! (n-1)
a116854_tabl = [1] : zipWith (:) (tail $ map head tss) tss
               where tss = a116853_tabl
-- Reinhard Zumkeller, Aug 31 2014
  • Maple
    A116853 := proc(n,k) option remember ; if n = k then n! ; else procname(n,k+1)-procname(n-1,k) ; end if; end proc:
A116854 := proc(n,k) if k = 1 then A116853(n,1) ; else A116853(n,k) -A116853(n,k-1) ; end if; end proc:
seq(seq(A116854(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Mar 27 2010
  • Mathematica
    rows = 10;
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[({0}~Join~Table[t[n, k], {k, 1, n}]) // Differences, {n, 1, rows}] // Flatten (* Jean-François Alcover, Dec 21 2019 *)

Formula

T(n,k) = A116853(n,k) - A116853(n,k-1) if k>1.
T(n,1) = A116853(n,1) = A000255(n-1).
Sum_{k=1..n} T(n,1) = n! = A000142(n).

Extensions

Definition made concrete and sequence extended by R. J. Mathar, Mar 27 2010

A246606 Central terms of the triangle A116853.

Original entry on oeis.org

1, 4, 78, 3216, 229080, 25022880, 3884393520, 812752093440, 220448163358080, 75225258805132800, 31537353006189676800, 15933924342019634227200, 9548252826112300306406400, 6695627848564821490753228800, 5431772705577464891946292992000, 5047432593984519350928894369792000
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 31 2014

Keywords

Comments

a(n) = A116853(2*n,n).

Links

Crossrefs

Cf. A000142, A116853.

Programs

  • Haskell
    a246606 n = a116853 (2 * n - 1) n
  • Maple
    a := n -> (2*n - 1)!*hypergeom([1 - n], [1 - 2*n], -1):
seq(simplify(a(n)), n=1..15); # Peter Luschny, Nov 04 2018

Formula

a(n) = (2*n - 1)!*hypergeom([1 - n], [1 - 2*n], -1). - Peter Luschny, Nov 04 2018
Conjecture: D-finite with recurrence +(-2*n+3)*a(n) +4*(n-1)*(2*n^2-4*n+1)*a(n-1) +(n-1)*(n-2)*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jul 01 2022
a(n) ~ sqrt(Pi) * 2^(2*n) * n^(2*n - 1/2) / exp(2*n + 1/2). - Vaclav Kotesovec, Mar 08 2023

A180191 Number of permutations of [n] having at least one succession. A succession of a permutation p is a position i such that p(i+1)-p(i) = 1.

Original entry on oeis.org

0, 1, 3, 13, 67, 411, 2921, 23633, 214551, 2160343, 23897269, 288102189, 3760013027, 52816397219, 794536751217, 12744659120521, 217140271564591, 3916221952414383, 74539067188152941, 1493136645424092773, 31400620285465593339, 691708660911435955579
Offset: 1

Views

Author

Emeric Deutsch, Sep 07 2010

Keywords

Comments

a(n) = A180190(n,1).
a(n+2) = p(n+2) where p(x) is the unique degree-n polynomial such that p(k) = k! for k = 1, ..., n+1. - Michael Somos, Jan 05 2012
From Jon Perry, Jan 04 2013: (Start)
Number of permutations of {1,...,n-1,n+1} with at least one indexed point p(k)=k with 1<=k<=n. Note that this means p(k)=n+1 is never an indexed point as k
For n>1, a(n) is the number of permutations of [n+1] that have a fixed point and contain 12; for example the a(3)=3 such permutations of {1,2,3,4} are 1234, 1243, and 3124.
(End)
For n > 0: row sums of triangle A116853. - Reinhard Zumkeller, Aug 31 2014

Examples

			x^2 + 3*x^3 + 13*x^4 + 67*x^5 + 411*x^6 + 2921*x^7 + 23633*x^8 + ...
a(3) = 3 because we have 123, 312, and 231; the permutations 132, 213, and 321 have no successions.
a(4) = 13 since p(x) = (3*x^2 - 7*x + 6) / 2 interpolates p(1) = 1, p(2) = 2, p(3) = 6, and p(4) = 13. - _Michael Somos_, Jan 05 2012

Links

Crossrefs

Cf. A000142, A000166, A000255, A002467, A028417, A180190.
Cf. A116853, A276975.
Column k=1 of A306234, A306461, and of A324362(n-1).

Programs

  • Haskell
    a180191 n = if n == 1 then 0 else sum $ a116853_row (n - 1)
-- Reinhard Zumkeller, Aug 31 2014
  • Maple
    d[0] := 1: for n to 50 do d[n] := n*d[n-1]+(-1)^n end do: seq(factorial(n)-d[n]-d[n-1], n = 1 .. 22);
  • Mathematica
    f[n_] := Sum[ -(-1)^k (n - k)! Binomial[n - 1, k], {k, 1, n}]; Array[f, 20] (* Robert G. Wilson v, Oct 16 2010 *)
  • PARI
    {a(n) = if( n<2, 0, n--; subst( polinterpolate( vector( n, k, k!)), x, n+1))} /* Michael Somos, Jan 05 2012 */

Formula

a(n) = n! - d(n) - d(n-1), where d(j) = A000166(j) are the derangement numbers.
a(n) = n! - A000255(n-1) = A002467(n) - A000166(n-1). - Jon Perry, Jan 05 2013
a(n) = (n-1)! [x^(n-1)] (1-exp(-x))/(1-x)^2. - Alois P. Heinz, Feb 23 2019
Showing 1-3 of 3 results.

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