A000481 -id:A000481 - cp's OEIS frontend

A028245 a(n) = 5^(n-1) - 44^(n-1) + 63^(n-1) - 4*2^(n-1) + 1 (essentially Stirling numbers of second kind).

0, 0, 0, 0, 24, 360, 3360, 25200, 166824, 1020600, 5921520, 33105600, 180204024, 961800840, 5058406080, 26308573200, 135666039624, 694994293080, 3542142833040, 17980946172000, 90990301641624

Offset: 1

N. J. A. Sloane, Doug McKenzie mckfam4(AT)aol.com

For n>=2, a(n) is equal to the number of functions f: {1,2,...,n-1}->{1,2,3,4,5} such that Im(f) contains 4 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Mar 08 2007

Seiichi Manyama, Table of n, a(n) for n = 1..1431
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120).

Cf. A000481, A008277, A163626, A000225, A028243, A028244.

[5^(n-1) - 4*4^(n-1) + 6*3^(n-1) - 4*2^(n-1) + 1: n in [1..30]]; // G. C. Greubel, Nov 19 2017

24StirlingS2[Range[30],5] (* Harvey P. Dale, Jun 18 2013 *)
Table[5^(n - 1) - 4*4^(n - 1) + 6*3^(n - 1) - 4*2^(n - 1) + 1, {n, 21}] (* or *)
Rest@ CoefficientList[Series[-24 x^5/((x - 1) (4 x - 1) (3 x - 1) (2 x - 1) (5 x - 1)), {x, 0, 21}], x] (* Michael De Vlieger, Sep 24 2016 *)

for(n=1,30, print1(24*stirling(n,5,2), ", ")) \\ G. C. Greubel, Nov 19 2017

a(n) = 24*S(n, 5) = 24*A000481(n). - Emeric Deutsch, May 02 2004
G.f.: -24*x^5/((x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009; checked and corrected by R. J. Mathar, Sep 16 2009
E.g.f.: (Sum_{k=0..5} (-1)^(5-k)*binomial(5,k)*exp(k*x))/5. with a(0) = 0. - Wolfdieter Lang, May 03 2017

A049434 Stirling numbers of second kind: 8th column of Stirling2 triangle A008277.

Original entry on oeis.org

1, 36, 750, 11880, 159027, 1899612, 20912320, 216627840, 2141764053, 20415995028, 189036065010, 1709751003480, 15170932662679, 132511015347084, 1142399079991620, 9741955019900400, 82318282158320505, 690223721118368580, 5749622251945664950

Offset: 8

Wolfdieter Lang

See A000771.

Cf. A000225, A000392, A000453, A000481, A000770, A000771, A049435. a(n)= A008277(n, 8).

lst={};Do[f=StirlingS2[n, 8];AppendTo[lst, f], {n, 8, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x) (1 - 6 x) (1 - 7 x) (1 - 8 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)

G.f.: x^8/product_{k=1..8} (1-k*x).
E.g.f.: ((exp(x)-1)^8)/8!.
a(n) = det(|s(i+8,j+7)|, 1 <= i,j <= n-8), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

A049435 Stirling numbers of second kind: 10th column of Stirling2 triangle A008277.

Original entry on oeis.org

1, 55, 1705, 39325, 752752, 12662650, 193754990, 2758334150, 37112163803, 477297033785, 5917584964655, 71187132291275, 835143799377954, 9593401297313460, 108254081784931500, 1203163392175387500, 13199555372846848005, 143197070509423605675

Offset: 10

Wolfdieter Lang

See A000771.

Index entries for linear recurrences with constant coefficients, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

Cf. A000225, A000392, A000453, A000481, A000770, A000771, A049434, A049447. a(n) = A008277(n, 10).

lst={};Do[f=StirlingS2[n, 10];AppendTo[lst, f], {n, 10, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x) (1 - 6 x) (1 - 7 x) (1 - 8 x) (1 - 9 x) (1 - 10 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
StirlingS2[Range[10,35],10] (* Harvey P. Dale, Nov 07 2020 *)

G.f.: x^10/Product_{k=1..10} (1-k*x).
E.g.f.: ((exp(x)-1)^10)/10!.
a(n) = det(|s(i+10,j+9)|, 1 <= i,j <= n-10), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

A049447 Stirling numbers of second kind: 9th column of Stirling2 triangle A008277.

Original entry on oeis.org

1, 45, 1155, 22275, 359502, 5135130, 67128490, 820784250, 9528822303, 106175395755, 1144614626805, 12011282644725, 123272476465204, 1241963303533920, 12320068811796900, 120622574326072500, 1167921451092973005, 11201516780955125625, 106563273280541795575

Offset: 9

Wolfdieter Lang

See A000771.

Cf. A000225, A000392, A000453, A000481, A000770, A000771, A049434.

lst={};Do[f=StirlingS2[n, 9];AppendTo[lst, f], {n, 9, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x) (1 - 6 x) (1 - 7 x) (1 - 8 x) (1 - 9 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
StirlingS2[Range[9,30],9] (* Harvey P. Dale, Dec 12 2022 *)

a(n)= A008277(n, 9).
G.f.: x^9/product_{k=1..9} (1-k*x).
E.g.f.: ((exp(x)-1)^9)/9!.
a(n) = det(|s(i+9,j+8)|, 1 <= i,j <= n-9), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

A320528 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 5 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132, 123050, 688850, 3752350, 20032446, 105372624, 548066568, 2826316248, 14478890712, 73794322750, 374602205590, 1895629599050, 9568906539786, 48208435317284, 242500368793628, 1218342441784468, 6115097961883092, 30669103347259650, 153720181809997530, 770100204404335350, 3856500105221902326

Offset: 1

Robert A. Russell, Oct 14 2018

Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern.

			For a(6)=6, the chiral pairs are AABCDE-ABCDEE, ABACDE-ABCDED, ABCADE-ABCDEC, ABCDAE-ABCDEB, ABBCDE-ABCDDE, and ABCBDE-ABCDCE.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-48,-36,551,-683,-1542,3546,80,-4280,2400).

Col. 5 of A320525.

Cf. A000481 (oriented), A056329 (unoriented), A304975 (achiral).

I:=[0,0,0,0,0,6,64,508,3428,21132]; [n le 10 select I[n] else 13*Self(n-1)-48*Self(n-2)-36*Self(n-3)+551*Self(n-4)-683*Self(n-5) -1542*Self(n-6)+3546*Self(n-7)+80*Self(n-8)-4280*Self(n-9) +2400*Self(n-10): n in [1..30]]; // G. C. Greubel, Oct 20 2018

k=5; Table[(StirlingS2[n,k] - If[EvenQ[n], 3StirlingS2[n/2+2,5] - 11StirlingS2[n/2+1,5] + 6StirlingS2[n/2,5], StirlingS2[(n+5)/2,5] - 3StirlingS2[(n+3)/2,5]])/2, {n,30}]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k = 5; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400}, {0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132}, 30]

m=30; v=concat([0,0,0,0,0,6,64,508,3428,21132], vector(m-10)); for(n=11, m, v[n] = 13*v[n-1]-48*v[n-2]-36*v[n-3]+551*v[n-4]-683*v[n-5] -1542*v[n-6] +3546*v[n-7] +80*v[n-8] -4280*v[n-9] +2400*v[n-10]); v \\ G. C. Greubel, Oct 20 2018

a(n) = (S2(n,k) - A(n,k))/2, where k=5 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
G.f.: (x^5 / Product_{k=1..5} (1 - k*x) - x^5 (1 + x) (1 - 3 x^2) (1 + 2 x - 2 x^2) / Product_{k=1..5} (1 - k*x^2)) / 2.
a(n) = (A000481(n) - A304975(n)) / 2 = A000481(n) - A056329(n) = A056329(n) - A304975(n).
a(n) = 13*a(n-1) - 48*a(n-2) - 36*a(n-3) + 551*a(n-4) - 683*a(n-5) - 1542*a(n-6) + 3546*a(n-7) + 80*a(n-8) - 4280*a(n-9) + 2400*a(n-10) for n>10. - Colin Barker, May 12 2020

A346920 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^5 / 5!).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 140, 1050, 6951, 42777, 260590, 1809060, 17418401, 229768539, 3402511476, 50013258750, 706670789371, 9659104177101, 130958047050698, 1834295186003784, 27849428308615221, 472297857494304303, 8856291348143365456, 176841068643273207426

Offset: 0

Ilya Gutkovskiy, Aug 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..469

Cf. A000670, A330047, A346894, A346895.

Cf. A000481, A327506, A346924.

nmax = 24; CoefficientList[Series[1/(1 - (Exp[x] - 1)^5/5!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 5] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^5/5!))) \\ Michel Marcus, Aug 07 2021

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (5*k)!*x^(5*k)/(120^k*prod(j=1, 5*k, 1-j*x)))) \\ Seiichi Manyama, May 09 2022

a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/120^k); \\ Seiichi Manyama, May 09 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,5) * a(n-k).
a(n) ~ n! / (5*(1 + 120^(-1/5)) * log(1 + 120^(1/5))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
From Seiichi Manyama, May 09 2022: (Start)
G.f.: Sum_{k>=0} (5*k)! * x^(5*k)/(120^k * Product_{j=1..5*k} (1 - j * x)).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/120^k. (End)

A056329 Number of reversible string structures with n beads using exactly five different colors.

Original entry on oeis.org

0, 0, 0, 0, 1, 9, 76, 542, 3523, 21393, 123680, 690550, 3756151, 20042589, 105394296, 548123982, 2826435403, 14479204833, 73794961960, 374603884910, 1895632969351, 9568915372269, 48208452866816

Offset: 1

Marks R. Nester

nonn

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

Number of set partitions for an unoriented row of n elements using exactly five different elements. An unoriented row is equivalent to its reverse. - Robert A. Russell, Oct 14 2018

			For a(6)=9, the color patterns are ABCDEA, ABCDBA, ABCCDE, AABCDE, ABACDE, ABCADE, ABCDAE, ABBCDE, and ABCBDE. The first three are achiral. - _Robert A. Russell_, Oct 14 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Index entries for linear recurrences with constant coefficients, signature (13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400).

Column 5 of A284949.
Cf. A056312.
Cf. A000481 (oriented), A320528 (chiral), A304975 (achiral).

k=5; Table[(StirlingS2[n,k] + If[EvenQ[n], 3StirlingS2[n/2+2,5] - 11StirlingS2[n/2+1,5] + 6StirlingS2[n/2,5], StirlingS2[(n+5)/2,5] - 3StirlingS2[(n+3)/2,5]])/2, {n,30}] (* Robert A. Russell, Oct 14 2018 *)
Ach[n_, k_] := Ach[n, k] = If[n < 2, Boole[n == k && n >= 0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]]
k=5; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 30}] (* Robert A. Russell, Oct 14 2018 *)
LinearRecurrence[{13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400}, {0, 0, 0, 0, 1, 9, 76, 542, 3523, 21393}, 30] (* Robert A. Russell, Oct 14 2018 *)

a(n) = A056324(n) - A056323(n).
Empirical g.f.: -x^5*(70*x^5 - 102*x^4 + 22*x^3 + 7*x^2 - 4*x + 1) / ((x-1)*(2*x-1)*(2*x+1)*(3*x-1)*(4*x-1)*(5*x-1)*(2*x^2-1)*(5*x^2-1)). - Colin Barker, Nov 25 2012
From Robert A. Russell, Oct 14 2018: (Start)
a(n) = (S2(n,k) + A(n,k))/2, where k=5 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
a(n) = (A000481(n) + A304975(n)) / 2 = A000481(n) - A320528(n) = A320528(n) + A304975(n). (End)

A346977 Expansion of e.g.f. log( 1 + (exp(x) - 1)^5 / 5! ).

Original entry on oeis.org

1, 15, 140, 1050, 6951, 42399, 239800, 1164570, 2553551, -54771717, -1384474728, -23286667950, -339924740609, -4554547609233, -56481301888144, -630768487283886, -5665064764515849, -18095553874845909, 924820173031946752, 35413415495503624986

Offset: 5

Ilya Gutkovskiy, Aug 09 2021

sign

Cf. A000481, A327506, A346955, A346974, A346975, A346976.

Cf. A346947, A346955.

nmax = 24; CoefficientList[Series[Log[1 + (Exp[x] - 1)^5/5!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
a[n_] := a[n] = StirlingS2[n, 5] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 5] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 5, 24}]

a(n) = Stirling2(n,5) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,5) * k * a(k).
a(n) ~ -(n-1)! * 2^(1+n) * 5^n * cos(n*arctan((2*arctan(sqrt(10 - 2*sqrt(5))/(1 + sqrt(5) + 2^(7/5)/15^(1/5)))) / log(1 + 3^(1/5)*5^(7/10)/2^(2/5) + 15^(1/5)/2^(2/5) + 2^(6/5)*15^(2/5)))) / (100*arctan(sqrt(10 - 2*sqrt(5))/(1 + sqrt(5) + 2^(7/5)/15^(1/5)))^2 + (5*log(1 + 3^(1/5)*5^(7/10)/2^(2/5) + 15^(1/5)/2^(2/5) + 2^(6/5)*15^(2/5)))^2)^(n/2). - Vaclav Kotesovec, Aug 10 2021
a(n) = Sum_{k=1..floor(n/5)} (-1)^(k-1) * (5*k)! * Stirling2(n,5*k)/(k * 120^k). - Seiichi Manyama, Jan 23 2025

A249163 Triangle read by rows: the positive terms of A163626.

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 50, 24, 1, 180, 360, 1, 602, 3360, 720, 1, 1932, 25200, 20160, 1, 6050, 166824, 332640, 40320, 1, 18660, 1020600, 4233600, 1814400, 1, 57002, 5921520, 46070640, 46569600, 3628800, 1, 173052, 33105600, 451725120, 898128000, 239500800

Offset: 0

Paul Curtz, Dec 15 2014

We have two possibilities: with or without 0's.
Without 0's:
1,
1,
1,   2,
1,  12,
1,  50,  24,
1, 180, 360,
etc.
Sum of every row: A000670(n).
First two terms of successive columns: 1, 1, 2, 12, 24, 360, ... = A211374.
With 0's:
1,   0,    0,   0,
1,   0,    0,   0,
1,   2,    0,   0,
1,  12,    0,   0,
1,  50,   24,   0,
1, 180,  360,   0,
1, 602, 3360, 720,
etc.
The columns are essentially A000012, A028243, A028246, A228909, A228911, A228913, from Stirling numbers of the second kind S(n,3), S(n,5), S(n,7), S(n,9), S(n,11), ... .

Cf. A163626, A000670, A211374; also A000012, A000392, A000481, A000771, A049447, A028243, A028246, A091137, A228909, A163626, A228911, A228913 and Worpitzky numbers for the second Bernoulli numbers A164555(n)/A027642(n).

Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[Derivative[n][y][x], y[x]] // Rest; Table[ Select[row[n], Positive] , {n, 0, 12}] // Flatten
(* or, simply: *) Table[(-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten // Select[#, Positive]& (* Jean-François Alcover, Dec 16 2014 *)

A346955 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^5 / 5! ).

Original entry on oeis.org

1, 15, 140, 1050, 6951, 42651, 253660, 1594230, 12463451, 134921787, 1806513072, 25539589530, 355175465191, 4797717669123, 63797550625676, 860468790181686, 12275324511112971, 192498455326842819, 3353266112959628272, 63379650000684213834

Offset: 5

Ilya Gutkovskiy, Aug 08 2021

nonn

Cf. A000481, A000629, A003704, A327506, A346390, A346920, A346954.

nmax = 24; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^5/5!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
a[n_] := a[n] = StirlingS2[n, 5] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 5] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 5, 24}]

a(n) = Stirling2(n,5) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,5) * k * a(k).
a(n) ~ (n-1)! / (log(120^(1/5) + 1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/5)} (5*k)! * Stirling2(n,5*k)/(k * 120^k). - Seiichi Manyama, Jan 23 2025

cp's OEIS Frontend

A028245 a(n) = 5^(n-1) - 4*4^(n-1) + 6*3^(n-1) - 4*2^(n-1) + 1 (essentially Stirling numbers of second kind).

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A049434 Stirling numbers of second kind: 8th column of Stirling2 triangle A008277.

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A049435 Stirling numbers of second kind: 10th column of Stirling2 triangle A008277.

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A049447 Stirling numbers of second kind: 9th column of Stirling2 triangle A008277.

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A320528 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 5 colors (subsets).

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A346920 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^5 / 5!).

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A056329 Number of reversible string structures with n beads using exactly five different colors.

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A346977 Expansion of e.g.f. log( 1 + (exp(x) - 1)^5 / 5! ).

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A028245 a(n) = 5^(n-1) - 44^(n-1) + 63^(n-1) - 4*2^(n-1) + 1 (essentially Stirling numbers of second kind).