A000460 -id:A000460 - cp's OEIS frontend

A151576 Number of permutations of 1..n arranged in a circle with exactly 3 adjacent element pairs in decreasing order.

0, 4, 55, 396, 2114, 9528, 38637, 146080, 526240, 1831644, 6217523, 20716164, 68059710, 221195824, 712856665, 2282058360, 7266358556, 23035517940, 72760054815, 229112753980, 719545590010, 2254604460264, 7050252659525, 22006821057936, 68581455012504, 213411502891468

Offset: 3

R. H. Hardin, May 21 2009

Exactly 2 adjacent element pairs in decreasing order gives A027540(n-1).

Andrew Howroyd, Table of n, a(n) for n = 3..500
Index entries for linear recurrences with constant coefficients, signature (16,-111,438,-1083,1740,-1817,1190,-444,72).

Column k=3 of A334218.
Related sequences: A151577-A151610.
Cf. A000460.

a(n)={n*(3^(n-1) - n*2^(n-1) + n*(n-1)/2)} \\ Andrew Howroyd, May 05 2020

From Andrew Howroyd, May 05 2020: (Start)
a(n) = n*A000460(n-1).
a(n) = n*(3^(n-1) - n*2^(n-1) + n*(n-1)/2).
a(n) = 16*a(n-1) - 111*a(n-2) + 438*a(n-3) - 1083*a(n-4) + 1740*a(n-5) - 1817*a(n-6) + 1190*a(n-7) - 444*a(n-8) + 72*a(n-9).
G.f.: x^4*(4 - 9*x - 40*x^2 + 131*x^3 - 98*x^4)/((1 - x)^4*(1 - 2*x)^3*(1 - 3*x)^2).
(End)

Terms a(18) and beyond from Andrew Howroyd, May 05 2020

A293616 Array of generalized Eulerian number triangles read by ascending antidiagonals, with m >= 0, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 0, 1, 0, 1, 10, 0, 7, 1, 0, 1, 15, 0, 25, 4, 0, 0, 1, 21, 0, 65, 10, 0, 1, 0, 1, 28, 0, 140, 20, 0, 15, 4, 0, 1, 36, 0, 266, 35, 0, 90, 30, 1, 0, 1, 45, 0, 462, 56, 0, 350, 120, 5, 0, 0, 1, 55, 0, 750, 84, 0, 1050, 350, 15, 0, 1, 0

Offset: 0

Peter Luschny, Oct 14 2017

			Array starts:
m\j| 0   1  2     3    4  5       6       7    8  9      10      11      12
---|----------------------------------------------------------------------------
m=0| 1,  0, 0,    0,   0, 0,      0,      0,   0, 0,      0,      0,      0, ...
m=1| 1,  1, 0,    1,   1, 0,      1,      4,   1, 0,      1,     11,     11, ...
m=2| 1,  3, 0,    7,   4, 0,     15,     30,   5, 0,     31,    146,     91, ...
m=3| 1,  6, 0,   25,  10, 0,     90,    120,  15, 0,    301,    896,    406, ...
m=4| 1, 10, 0,   65,  20, 0,    350,    350,  35, 0,   1701,   3696,   1316, ...
m=5| 1, 15, 0,  140,  35, 0,   1050,    840,  70, 0,   6951,  11886,   3486, ...
m=6| 1, 21, 0,  266,  56, 0,   2646,   1764, 126, 0,  22827,  32172,   8022, ...
m=7| 1, 28, 0,  462,  84, 0,   5880,   3360, 210, 0,  63987,  76692,  16632, ...
m=8| 1, 36, 0,  750, 120, 0,  11880,   5940, 330, 0, 159027, 165792,  31812, ...
m=9| 1, 45, 0, 1155, 165, 0,  22275,   9900, 495, 0, 359502, 331617,  57057, ...
   A000217, A001296,A000292,A001297,A027789,A000332,A001298,A293610,A293611, ...
.
m\j| ...    13  14      15       16       17      18      19 20
---|----------------------------------------------------------------
m=0| ...,    0, 0,       0,       0,       0,      0,      0, 0, ...  [A000007]
m=1| ...,    1, 0,       1,      26,      66,     26,      1, 0, ...  [A173018]
m=2| ...,    6, 0,      63,     588,     868,    238,      7, 0, ...  [A062253]
m=3| ...,   21, 0,     966,    5376,    5586,   1176,     28, 0, ...  [A062254]
m=4| ...,   56, 0,    7770,   30660,   24570,   4200,     84, 0, ...  [A062255]
m=5| ...,  126, 0,   42525,  129780,   84630,  12180,    210, 0, ...
m=6| ...,  252, 0,  179487,  446292,  245322,  30492,    462, 0, ...
m=7| ...,  462, 0,  627396, 1315776,  625086,  68376,    924, 0, ...
m=8| ...,  792, 0, 1899612, 3444012, 1440582, 140712,   1716, 0, ...
m=9| ..., 1287, 0, 5135130, 8198190, 3063060, 270270,   3003, 0, ...
          A000389, A112494, A293612, A293613,A293614,A000579.
.
The parameter m runs over the triangles and j indexes the triangles by reading them by rows. Let T(m, n) denote the row [T(m, n, k) for 0 <= k <= n] and T(m) denote the triangle [T(m, n) for n >= 0]. Then for instance T(2) is the triangle A062253, T(4, 2) is row 2 of A062255 (which is [65, 20, 0]) and T(4, 2, 1) = 20.

Eric Weisstein's World of Mathematics, Nielsen Generalized Polylogarithm.

A000217(n) = T(n, 1, 0), A001296(n) = T(n, 2, 0), A000292(n) = T(n, 2, 1),
A001297(n) = T(n, 3, 0), A027789(n) = T(n, 3, 1), A000332(n) = T(n, 3, 2),
A001298(n) = T(n, 4, 0), A293610(n) = T(n, 4, 1), A293611(n) = T(n, 4, 2),
A000389(n) = T(n, 4, 3), A112494(n) = T(n, 5, 0), A293612(n) = T(n, 5, 1),
A293613(n) = T(n, 5, 2), A293614(n) = T(n, 5, 3), A000579(n) = T(n, 5, 4),
A144969(n) = T(n, 6, 0), A000580(n) = T(n, 6, 5), A000295(n) = T(1, n, 1),
A000460(n) = T(1, n, 2), A000498(n) = T(1, n, 3), A000505(n) = T(1, n, 4),
A000514(n) = T(1, n, 5), A001243(n) = T(1, n, 6), A001244(n) = T(1, n, 7),
A126646(n) = T(2, n, 0), A007820(n) = T(n, n, 0).
Cf. A173018, A062253, A062254, A062255, A293609.

A293616 := proc(m, n, k) option remember:
if m = 0 then m^n elif k < 0 or k > n then 0 elif n = 0 then 1 else
(k+m)*A293616(m,n-1,k) + (n-k)*A293616(m,n-1,k-1) + A293616(m-1,n,k) fi end:
for m in [$0..4] do for n in [$0..6] do print(seq(A293616(m, n, k), k=0..n)) od od;
# Sample uses:
A001298 := n -> A293616(n, 4, 0): A293614 := n -> A293616(n, 5, 3):
# Flatten:
a := proc(n) local w; w := proc(k) local t, s; t := 1; s := 1;
while t <= k do s := s + 1; t := t + s od; [s - 1, s - t + k] end:
seq(A293616(n - k, w(k)[1], w(k)[2]), k=0..n) end: seq(a(n), n = 0..11);

GenEulerianRow[0, n_] := Table[If[n==0 && j==0,1,0], {j,0,n}];
GenEulerianRow[m_, n_] := If[n==0,{1},Join[CoefficientList[x^(-m) (1 - x)^(n+m)
    PolyLog[-n-m, m, x] /. Log[1-x] -> 0, x], {0}]];
(* Sample use: *)
A173018Row[n_] := GenEulerianRow[1, n]; Table[A173018Row[n], {n, 0, 6}]

T(m, n, k) = (k + m)*T(m, n-1, k) + (n - k)*T(m, n-1, k-1) + T(m-1, n, k) with boundary conditions T(0, n, k) = 0^n; T(m, n, k) = 0 if k < 0 or k > n; and T(m, 0, k) = 0^k.

Let h(m, n) = x^(-m)*(1 - x)^(n + m)*PolyLog(-n - m, m, x) and p(m, n) the polynomial given by the expansion of h(m, n) after replacing log(1 - x) by 0. Then T(m, n, k) is the k-th coefficient of p(m, n) for 0 <= k < n.

A048470 a(n) = (n+1)*(2^(n+1) - n)/2.

Original entry on oeis.org

1, 3, 9, 26, 70, 177, 427, 996, 2268, 5075, 11209, 24510, 53170, 114597, 245655, 524168, 1113976, 2359143, 4980565, 10485570, 22019886, 46137113, 96468739, 201326316, 419430100, 872414907, 1811938977, 3758096006, 7784627818

Offset: 0

Clark Kimberling

a(n) = T(0,n) + T(1,n-1) + ... + T(n,0), array T given by A047858.

Sum of the consecutive integers from 2^n-n up to and including 2^n. - J. M. Bergot, Jun 27 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

[(n+1)*(2^(n+1)-n)/2: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011

a(n)=(n+1)*(2<Charles R Greathouse IV, Jun 28 2013

G.f.: (-1 + 5*x^3 - 7*x^2 + 4*x)/((2*x-1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 13 2009

a(n) = 3^n - Eulerian(n,2) = A000244(n) - A000460(n). - Peter Luschny, May 13 2016

Edited by T. D. Noe, Dec 11 2006

A085852 Triangle T(n, k) read by rows; given by [0, 1, 0, 2, 0, 3, 0, 4, ...] DELTA [1, 0, 2, 0, 2, 0, 3, 0, 2, 0, 4, 0, 2, 0, ...] (A000005 interspersed with 0's) where DELTA is Deléham's operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 11, 11, 1, 0, 1, 26, 62, 26, 1, 0, 1, 57, 266, 258, 57, 1, 0, 1, 120, 991, 1792, 903, 120, 1, 0, 1, 247, 3405, 10363, 9483, 2829, 247, 1, 0, 1, 502, 11140, 53818, 80342, 42906, 8212, 502, 1, 0, 1, 1013, 35348, 260996

Offset: 0

N. J. A. Sloane, Aug 16 2003

			Triangle begins:
1,
0, 1,
0, 1, 1,
0, 1, 4, 1,
0, 1, 11, 11, 1,
0, 1, 26, 62, 26, 1,
0, 1, 57, 266, 258, 57, 1,
0, 1, 120, 991, 1792, 903, 120, 1,
0, 1, 247, 3405, 10363, 9483, 2829, 247, 1,
0, 1, 502, 11140, 53818, 80342, 42906, 8212, 502, 1,
...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000295 (3rd column), A000460 (4th column), A000498 (5th column).

m = 13;
(* DELTA is defined in A084938 *)
DELTA[LinearRecurrence[{0, 2, 0, -1}, {0, 1, 0, 2}, m], Table[ {DivisorSigma[0, n], 0}, {n, 1, m}] // Flatten, m] // Flatten (* Jean-François Alcover, Feb 19 2020 *)

Incorrect comment removed by Philippe Deléham, Sep 25 2015

A095889 Number of permutations of [n] with exactly 3 descents which avoid the pattern 4321.

Original entry on oeis.org

10, 148, 1260, 8160, 44790, 220180, 1001000, 4295168, 17633122, 69921460, 269652100, 1016524352, 3760334958, 13690636212, 49172875520, 174559419680, 613383654170, 2136135472020, 7380351046300, 25319087913760, 86308819112550, 292526411823828, 986294675689560

Offset: 5

Mike Zabrocki, Nov 10 2004

Colin Barker, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (22,-216,1248,-4710,12180,-22004,27752,-23937,13446,-4428,648).

Cf. A000295, A000460.

Vec(2*x^5*(5 - 36*x + 82*x^2 - 36*x^3 - 87*x^4 + 80*x^5) / ((1 - x)^4*(1 - 2*x)^3*(1 - 3*x)^4) + O(x^30)) \\ Colin Barker, Nov 03 2017

G.f.: 2*x^5*(5 - 36*x + 82*x^2 - 36*x^3 - 87*x^4 + 80*x^5) / ((1 - x)^4*(1 - 2*x)^3*(1 - 3*x)^4).

a(n) = (n*(27 + 81*2^n - 115*3^n + 3*(27*2^n+2*3^n)*n + (-27+3^n)*n^2)) / 162. - Colin Barker, May 03 2019

A328088 a(n) = Sum_{k=4..n} ( binomial(n,k)(k-2)(2^k-2*k-2) ) - (2^n-n-1).

Original entry on oeis.org

1, 94, 683, 3520, 15461, 61826, 232543, 838276, 2930585, 10014406, 33633299, 111448904, 365403853, 1187875594, 3834883271, 12309375244, 39320806145, 125090127950, 396537120379, 1253145232336, 3949433330741, 12416933938834, 38953666980143, 121962851990420, 381179210953321, 1189376848680406, 3705576521235683

Offset: 4

N. J. A. Sloane, Oct 17 2019

Robert Israel, Table of n, a(n) for n = 4..2087
J. B. Remmel et al., The combinatorial properties of the Benoumhani polynomials for the Whitney numbers of Dowling lattices, Discrete Math., 342 (2019), 2966-2983. See page 2981, formula for coefficient of m in B_{m+1}(n,n-2).
Index entries for linear recurrences with constant coefficients, signature (14,-82,260,-481,518,-300,72).

For the constant term see A000460.

f:= n -> 3^(n-1)*(2*n-6) + 2^(n-1)*(-n^2+n+6) - n - 1:
map(f, [$4..40]); # Robert Israel, Oct 18 2019

Vec(x^4*(1 + 80*x - 551*x^2 + 1406*x^3 - 1772*x^4 + 1128*x^5 - 288*x^6) / ((1 - x)^2*(1 - 2*x)^3*(1 - 3*x)^2) + O(x^40)) \\ Colin Barker, Oct 19 2019

a(n) = 3^(n-1)*(2*n-6) + 2^(n-1)*(-n^2+n+6) - n - 1. - Robert Israel, Oct 18 2019
From Colin Barker, Oct 19 2019: (Start)
G.f.: x^4*(1 + 80*x - 551*x^2 + 1406*x^3 - 1772*x^4 + 1128*x^5 - 288*x^6) / ((1 - x)^2*(1 - 2*x)^3*(1 - 3*x)^2).
a(n) = -1 + 3*2^n - 2*3^n + (1/6)*(-6 + 3*2^n + 4*3^n)*n - 2^(-1+n)*n^2  for n>3.
a(n) = 14*a(n-1) - 82*a(n-2) + 260*a(n-3) - 481*a(n-4) + 518*a(n-5) - 300*a(n-6) + 72*a(n-7) for n>10.
(End)
E.g.f.: x^2/2 + 2*x^3/3 + exp(2*x)*(3 - 2*x^2 + (-3 + x)*cosh(x) + (-1 + 3*x)*sinh(x)). - Stefano Spezia, Oct 19 2019

cp's OEIS Frontend

A151576 Number of permutations of 1..n arranged in a circle with exactly 3 adjacent element pairs in decreasing order.

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A293616 Array of generalized Eulerian number triangles read by ascending antidiagonals, with m >= 0, n >= 0 and 0 <= k <= n.

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Maple

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A048470 a(n) = (n+1)*(2^(n+1) - n)/2.

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Magma

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A085852 Triangle T(n, k) read by rows; given by [0, 1, 0, 2, 0, 3, 0, 4, ...] DELTA [1, 0, 2, 0, 2, 0, 3, 0, 2, 0, 4, 0, 2, 0, ...] (A000005 interspersed with 0's) where DELTA is Deléham's operator defined in A084938.

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Examples

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Mathematica

Extensions

A095889 Number of permutations of [n] with exactly 3 descents which avoid the pattern 4321.

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Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A328088 a(n) = Sum_{k=4..n} ( binomial(n,k)*(k-2)*(2^k-2*k-2) ) - (2^n-n-1).

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Maple

PARI

Formula

A328088 a(n) = Sum_{k=4..n} ( binomial(n,k)(k-2)(2^k-2*k-2) ) - (2^n-n-1).