A083385 -id:A083385 - cp's OEIS frontend

A083410 a(n) = A083385(n)/n.

1, 4, 22, 154, 1306, 12994, 148282, 1908274, 27333706, 431220034, 7428550042, 138737478994, 2792050329706, 60231133487074, 1386484468239802, 33921605427779314, 878976357571495306, 24046780495646314114, 692622345890928153562, 20950628198687114521234, 663992311200423614606506

Offset: 1

N. J. A. Sloane, Jun 08 2003

nonn

From Michael Somos, Mar 04 2004: (Start)
Stirling transform of A052849(n+1)=[4,12,48,240,...] is 4*a(n)=[4,16,88,616,...].
Stirling transform of A001710(n+1)=[1,3,12,160,...] is a(n)=[1,4,22,154,...].
Stirling transform of A001563(n+1)=[4,18,96,600,...] is a(n+1)=[4,22,154,...]. (End)

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

A005649(n)=2*a(n), if n>0.
Pairwise sums of A091346.
Cf. A090665.

b:= proc(n, m) option remember;
     `if`(n=0, (m+1)!, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0)/2:
seq(a(n), n=1..23);  # Alois P. Heinz, Feb 14 2025

a[n_] := (-1)^n (PolyLog[-n - 1, 2] - PolyLog[-n, 2])/8;
Array[a, 21] (* Jean-François Alcover, Sep 10 2018, from A005649 *)

a(n)=if(n<0,0,n!*polcoeff(subst((1/(1-y)^2-1)/2,y,exp(x+x*O(x^n))-1),n))

E.g.f.: (1/(2-exp(x))^2-1)/2. - Michael Somos, Mar 04 2004
G.f.: 1/Q(0), where Q(k) = 1 - x*(3*k+4) - 2*x^2*(k+1)*(k+3)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
a(n) ~ n! * n / (8 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018
a(n) = Sum_{k=1..n} k * A090665(n,k). - Alois P. Heinz, Feb 20 2025

A261781 Number T(n,k) of compositions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order and all k letters occur at least once in the composition; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 4, 16, 13, 0, 8, 66, 132, 75, 0, 16, 248, 924, 1232, 541, 0, 32, 892, 5546, 13064, 13060, 4683, 0, 64, 3136, 30720, 114032, 195020, 155928, 47293, 0, 128, 10888, 162396, 893490, 2327960, 3116220, 2075948, 545835

Offset: 0

Alois P. Heinz, Aug 31 2015

From Vaclav Kotesovec, Oct 14 2017: (Start)
Conjecture: For k > 0 the recurrence order for column k is equal to k*(k+1)/2.
Column k > 0 is asymptotic to c(k) * d(k)^n, where c(k) and d(k) are constants (dependent only on k).
k                           c(k)                            d(k)
1  A131577(n) ~ 0.50000000000000000000000000 * 2.00000000000000000000000000^n.
2  A293579(n) ~ 0.60355339059327376220042218 * 3.41421356237309504880168872^n.
3  A293580(n) ~ 0.64122035031051210658648604 * 4.84732210186307263951891624^n.
4  A293581(n) ~ 0.66065168848540565019767995 * 6.28521350788324520158143964^n.
5  A293582(n) ~ 0.67250239588725756267924287 * 7.72502395887257562679242875^n.
6  A293583(n) ~ 0.68048292906885160660288253 * 9.16579514882621927923459043^n.
7  A293584(n) ~ 0.68622254929933439577377124 * 10.6071156901906815408327973^n.
8  A293585(n) ~ 0.69054873168854973836384871 * 12.0487797070167958138215794^n.
9  A293586(n) ~ 0.69392626461456654033893782 * 13.4906727630621977261008808^n.
10 A293587(n) ~ 0.69663630864564830007443110 * 14.9327261729129660014886221^n.
---
Conjecture: d(k+1) - d(k) tends to 1/log(2).
d(2) - d(1) = 1.414213562373095048801688724209698...
d(3) - d(2) = 1.433108539489977590717227522340838...
d(4) - d(3) = 1.437891406020172562062523400686067...
d(5) - d(4) = 1.439810450989330425210989107036901...
d(6) - d(5) = 1.440771189953643652442161677346934...
d(7) - d(6) = 1.441320541364462261598206961226199...
d(8) - d(7) = 1.441664016826114272988782079622148...
d(9) - d(8) = 1.441893056045401912279301345910755...
d(10)- d(9) = 1.442053409850768275387741352145193...
1 / log(2)  = 1.442695040888963407359924681001892...
(End)

			A(3,2) = 16: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a2ab, 1a2bb, 1b2aa, 1b2ab, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,    3;
  0,  4,   16,    13;
  0,  8,   66,   132,     75;
  0, 16,  248,   924,   1232,    541;
  0, 32,  892,  5546,  13064,  13060,   4683;
  0, 64, 3136, 30720, 114032, 195020, 155928, 47293;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
E. Munarini, M. Poneti, and S. Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8, Table 2.

Columns k=0..10 give A000007, A131577, A293579, A293580, A293581, A293582, A293583, A293584, A293585, A293586, A293587.
Row sums give A120733.
Main diagonal gives A000670.
T(2n,n) gives A261784.
T(n+1,n)/2 gives A083385.
Cf. A261719 (same for partitions), A261780.

A:= proc(n, k) option remember; `if`(n=0, 1,
      add(A(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

A[n_, k_] := A[n, k] = If[n==0, 1,
    Sum[A[n-j, k]*Binomial[j+k-1, k-1], {j, 1, n}]];
T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A261780(n,k-i).

A083384 a(n) = nSum(((k-1)/2)k!*Stirling_2(n,k),k=1..n).

Original entry on oeis.org

0, 2, 27, 316, 3825, 49866, 706923, 10899512, 182218005, 3289724710, 63865092159, 1327750936788, 29447495757225, 694257067232834, 17343019158929235, 457695211932767344, 12726295039220109885, 371902424983010438238, 11396594412860395106151, 365458808048854606362380

Offset: 1

N. J. A. Sloane, Jun 07 2003

nonn

Cf. A000670, A083385, A083411, A083410.

[n*&+[(k-1)/2*Factorial(k)*StirlingSecond(n, k): k in [0..n]]: n in [1..25]]; //  Vincenzo Librandi, Sep 01 2018

a[n_] := n Sum[1/2 (k-1) k! StirlingS2[n, k], {k, 1, n}];
Array[a, 20] (* Jean-François Alcover, Sep 01 2018 *)
Rest[Range[0, 19]! CoefficientList[Series[x (Exp[x] - 1) Exp[x] / (2 - Exp[x])^3, {x, 0, 19}], x]] (* Vincenzo Librandi, Sep 01 2018 *)

Equals A083385(n) - n*A000670(n).
E.g.f.: x*(exp(x)-1)*exp(x)/(2-exp(x))^3. - Vladeta Jovovic, Sep 14 2003
a(n) ~ n! * n^2 / (8 * (log(2))^(n+2)). - Vaclav Kotesovec, Feb 18 2017

cp's OEIS Frontend

A083410 a(n) = A083385(n)/n.

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A261781 Number T(n,k) of compositions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order and all k letters occur at least once in the composition; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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A083384 a(n) = nSum(((k-1)/2)k!*Stirling_2(n,k),k=1..n).

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cp's OEIS Frontend

A083410 a(n) = A083385(n)/n.

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Programs

Maple

Mathematica

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Formula

A261781 Number T(n,k) of compositions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order and all k letters occur at least once in the composition; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A083384 a(n) = n*Sum(((k-1)/2)*k!*Stirling_2(n,k),k=1..n).

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Author

Keywords

Crossrefs

Programs

Magma

Mathematica

Formula

A083384 a(n) = nSum(((k-1)/2)k!*Stirling_2(n,k),k=1..n).