A001048 -id:A001048 - cp's OEIS frontend

A141052 Number of runs or rising sequences of length 2 among all permutations of n.

1, 4, 21, 130, 930, 7560, 68880, 695520, 7711200, 93139200, 1217462400, 17124307200, 257902444800, 4140968832000, 70614415872000, 1274546617344000, 24275666967552000, 486580401635328000, 10238462617743360000, 225651661258383360000, 5198503365971435520000

Offset: 2

Harlan J. Brothers, Jul 31 2008, Aug 24 2008

			a[3]=4 because of the 6 permutations of n=3, there are 4 ascending runs of length 2:
{1,3} in {1,3,2}
{1,3} in {2,1,3}
{2,3} in {2,3,1}
{1,2} in {3,1,2}
a[3]=4 because of the 6 permutations of n=3, there are 4 rising sequences of length 2:
{1,2} in {1,3,2}
{2,3} in {2,1,3}
{2,3} in {2,3,1}
{1,2} in {3,1,2}

Persi Diaconis, Mathematical developments from the analysis of riffle shuffling, p. 4.
Francis Edward Su, Rising Sequences in Card Shuffling
Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, American Mathematical Society, 1997, pp.120-131.

Column 2 of A122843.

Cf. A008292, A097900, A001286, A001048, A000142, A028387, A001710.

Table[n!(5n + 1)/4! + Floor[2/n](1/12), {n, 2, 10}]

a(n) = n!*(5n+1)/4! + floor(2/n)*(1/12), n>=2.
Recurrence: a(n) = (n+1)*a(n-1)+(n-1)!/6, n>=2, with a(2)=1 and a(3)=4.
E.g.f.: x^2*(x-2)*(x-6)/(24*(x-1)^2).

First example and typo in second example corrected by Harlan J. Brothers, Apr 29 2013

A196080 Numerators of the sum of the n-th partial sums of the expansions of e and 1/e.

Original entry on oeis.org

2, 2, 3, 3, 37, 37, 1111, 1111, 6913, 6913, 799933, 799933, 739138093, 739138093, 44841044309, 44841044309, 32285551902481, 32285551902481, 9879378882159187, 9879378882159187, 1251387991740163687

Offset: 0

Paul Curtz, Sep 27 2011

The n-th partial sums of the Taylor expansion of E are f(n) = A061354(n)/A061355(n) = 1, 2, 5/2, 8/3, 65/24, 163/60,.. .
The partial sums of an expansion of 1/e are essentially A000255(n-2)/A001048(n-1) preceded by 1 and 0, namely  g(n)= 1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760,... (Jolley's partial sums of 1/E in A068985 is the bisection 0, 1/3, 11/30, 103/280, 16687/45360,... of g(n).)
The current sequence are the numerators of f(n)+g(n), converging to E+1/E, namely 2, 2, 3, 3, 37/12, 37/12, 1111/360, 1111/360, 6913/2240 = 62217/21060, 6913/2240 = 62217/21060, 799933/259200 = 5599531/1814400,... The unreduced fractions are apparently given by duplicated A051396(n+1)/A002674(n).

			a(0)=1+1, a(1)=2+0, a(2)=(5+1)/2, a(3)=(8+1)/3.

Cf. A001113, A068985, A137204 (e+1/e).

a[n_] := (E*Gamma[n+1, 1] + (1/E)*Gamma[n+1, -1])/n! // FullSimplify // Numerator; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 02 2012 *)

Redefined by reduced fractions. - R. J. Mathar, Jul 02 2012

A350227 Triangular array read by rows. T(n,k) is the number of partial permutations on [n] with exactly k connected components, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 2, 0, 3, 4, 0, 8, 18, 8, 0, 30, 91, 72, 16, 0, 144, 540, 590, 240, 32, 0, 840, 3718, 5085, 2900, 720, 64, 0, 5760, 29232, 47516, 34230, 12040, 2016, 128, 0, 45360, 258732, 484092, 416857, 186480, 44576, 5376, 256, 0, 403200, 2547360, 5368184, 5340888, 2869314, 876960, 151872, 13824, 512

Offset: 0

Geoffrey Critzer, Dec 20 2021

			Triangle begins:
  1;
  0,   2;
  0,   3,   4;
  0,   8,  18,   8;
  0,  30,  91,  72,  16;
  0, 144, 540, 590, 240, 32;
  ...

Cf. A000079, A132393, A001048, A002720 (row sums).

nn = 9; Table[Take[(Range[0, nn]! CoefficientList[Series[1/(1 - x)^y Exp[y x/(1 - x)], {x, 0, nn}], {x, y}])[[i, All]], i], {i, 1, nn + 1}] // Grid

T(n,n) = 2^n = A000079(n) (counts the idempotent elements).
For n>=1, T(n,1) = (n-1)! + n! = A001048(n) (the component is a cycle or a directed path to a point with a self loop).
E.g.f.: exp(y*log(1/(1-x)))*exp(y*x/(1-x)).

A385577 Array read by ascending antidiagonals: A(n,m) = n*Pochhammer(n+1,m+1)/(m+2).

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 6, 8, 6, 0, 10, 20, 30, 24, 0, 15, 40, 90, 144, 120, 0, 21, 70, 210, 504, 840, 720, 0, 28, 112, 420, 1344, 3360, 5760, 5040, 0, 36, 168, 756, 3024, 10080, 25920, 45360, 40320, 0, 45, 240, 1260, 6048, 25200, 86400, 226800, 403200, 362880, 0

Offset: 0

Stefano Spezia, Jul 03 2025

			Array begins as:
   0,  0,   0,    0,     0,     0,      0, ...
   1,  2,   6,   24,   120,   720,   5040, ...
   3,  8,  30,  144,   840,  5760,  45360, ...
   6, 20,  90,  504,  3360, 25920, 226800, ...
  10, 40, 210, 1344, 10080, 86400, 831600, ...
  ...

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 20.

Paul W. Haggard and Bonnie L. Sadler, A Generalization of Triangular Numbers, International Journal of Mathematical Education in Science and Technology 25 (2): 195-202, (1994).

Cf. A000142, A001048, A002740, A006231, A007290, A068424, A238474.
Cf. A000217 (m=0), A033487 (m=2), A158874 (m=3).
Cf. A000004 (n=0).

A[n_,m_]:=n*Pochhammer[n+1,m+1]/(m+2); Table[A[n-m,m],{n,0,9},{m,0,n}]//Flatten

Sum_{m=0..n} A(n-m,m) = A006231(n+1).
A(n,1) = A007290(n+2).
A(1,n) = A000142(n+1).
A(2,n) = A001048(n+2).
A(3,n) = abs(A238474(n+1)).
A(n,n) = n!*A002740(n+2)

cp's OEIS Frontend

A141052 Number of runs or rising sequences of length 2 among all permutations of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A196080 Numerators of the sum of the n-th partial sums of the expansions of e and 1/e.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A350227 Triangular array read by rows. T(n,k) is the number of partial permutations on [n] with exactly k connected components, n>=0, 0<=k<=n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A385577 Array read by ascending antidiagonals: A(n,m) = n*Pochhammer(n+1,m+1)/(m+2).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula