A225479 -id:A225479 - cp's OEIS frontend

A052748 Expansion of e.g.f.: -(log(1-x))^3.

0, 0, 0, 6, 36, 210, 1350, 9744, 78792, 708744, 7036200, 76521456, 905507856, 11589357312, 159580302336, 2352940786944, 36994905688320, 617953469022720, 10929614667747840, 204073497562936320, 4011658382046919680, 82822558521844224000, 1791791417179298304000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Original name: A simple grammar.

Andrew Howroyd, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 704

Column k=3 of A225479.

Cf. A000399, A052517.

spec := [S,{B=Cycle(Z),S=Prod(B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
with(combinat):seq(stirling1(j, 3)*3!*(-1)^(j+1), j=0..50); # Leonid Bedratyuk, Aug 07 2012

a(n) = {3!*stirling(n,3,1)*(-1)^(n+1)} \\ Andrew Howroyd, Jul 27 2020

E.g.f.: log(1/(1-x))^3.
Recurrence: {a(1)=0, a(0)=0, a(2)=0, a(3)=6, (-1 - 3*n - 3*n^2 - n^3)*a(n+1) + (9*n + 7 + 3*n^2)*a(n+2) + (-6 - 3*n)*a(n+3) + a(n+4)}.
a(n) = stirling1(n, 3)*3!*(-1)^(n+1). - Leonid Bedratyuk, Aug 07 2012
a(n) = 6*A000399(n). - Andrew Howroyd, Jul 27 2020

Name changed and terms a(20) and beyond from Andrew Howroyd, Jul 27 2020

A052753 Expansion of e.g.f.: log(1-x)^4.

Original entry on oeis.org

0, 0, 0, 0, 24, 240, 2040, 17640, 162456, 1614816, 17368320, 201828000, 2526193824, 33936357312, 487530074304, 7463742249600, 121367896891776, 2089865973021696, 37999535417459712, 727710096185266176, 14642785817771802624, 308902349883623731200, 6818239581643475251200

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

G. C. Greubel, Table of n, a(n) for n = 0..448
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 709

Column k=4 of A225479.

Cf. A000454, A052517.

spec := [S,{B=Cycle(Z),S=Prod(B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[(Log[1-x])^4, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

x='x+O('x^30); concat(vector(4), Vec(serlaplace((log(1-x))^4))) \\ G. C. Greubel, Aug 30 2018

a(n) = {4!*stirling(n,4,1)*(-1)^n} \\ Andrew Howroyd, Jul 27 2020

E.g.f.: log(-1/(-1+x))^4.
Recurrence: {a(1)=0, a(0)=0, a(2)=0, (1+4*n+6*n^2+4*n^3+n^4)*a(n+1) + (-4*n^3-15-18*n^2-28*n)*a(n+2) + (6*n^2+24*n+25)*a(n+3) + (-4*n-10)*a(n+4)+a(n+5), a(3)=0, a(4)=24}.
a(n) ~ (n-1)! * 2*log(n)*(2*log(n)^2 + 6*gamma*log(n) - Pi^2 + 6*gamma^2), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 30 2013
a(n) = 24*A000454(n) = 4!*(-1)^n*Stirling1(n,4). - Andrew Howroyd, Jul 27 2020

New name using e.g.f., Vaclav Kotesovec, Sep 30 2013

A052767 Expansion of e.g.f.: -(log(1-x))^5.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 1800, 21000, 235200, 2693880, 32319000, 410031600, 5519487600, 78864820320, 1194924450720, 19166592681600, 324817601472000, 5803921108010880, 109115988701293440, 2154085473710580480, 44566174481427360000, 964537418717406213120, 21799797542483649131520

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Andrew Howroyd, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 724

Column k=5 of A225479.

Cf. A000482, A052517.

spec := [S,{B=Cycle(Z),S=Prod(B,B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[-(Log[1-x])^5,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 14 2019 *)

a(n) = {5!*stirling(n,5,1)*(-1)^(n+1)} \\ Andrew Howroyd, Jul 27 2020

E.g.f.: log(-1/(-1+x))^5.
Recurrence: a(1)=0, a(0)=0, a(2)=0, a(4)=0, a(3)=0, (-1-5*n-10*n^2-10*n^3-5*n^4-n^5)*a(n+1) + (31+5*n^4+70*n^2+30*n^3+75*n)*a(n+2) + (-125*n-90-60*n^2-10*n^3)*a(n+3) + (10*n^2+65+50*n)*a(n+4) + (-15-5*n)*a(n+5) + a(n+6)=0, a(5)=120.
a(n) = 120*A000482(n) = 5!*Stirling1(n,5)*(-1)^(n+1). - Andrew Howroyd, Jul 27 2020

Definition clarified by Harvey P. Dale, Oct 14 2019

Terms a(20) and beyond from Andrew Howroyd, Jul 27 2020

A052779 Expansion of e.g.f.: (log(1-x))^6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 15120, 231840, 3265920, 45556560, 649479600, 9604465200, 148370508000, 2402005525920, 40797624067200, 726963917097600, 13580328282393600, 265689107448756480, 5437099866285377280, 116229410301685651200, 2591985252922277184000, 60218914823672258142720

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Original name: a simple grammar.

Andrew Howroyd, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 736

Column k=6 of A225479.

Cf. A001233, A052517.

spec := [S,{B=Cycle(Z),S=Prod(B,B,B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

a(n) = {6!*stirling(n,6,1)*(-1)^n} \\ Andrew Howroyd, Jul 27 2020

E.g.f.: log(-1/(-1+x))^6.
Recurrence: {a(1)=0, a(0)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=0, a(6)=720, (1+15*n^2+6*n+6*n^5+15*n^4+20*n^3+n^6)*a(n+1) + (-63-186*n-225*n^2-6*n^5-45*n^4-140*n^3)*a(n+2) + (540*n+120*n^3+375*n^2+15*n^4+301)*a(n+3) + (-390*n-20*n^3-350-150*n^2)*a(n+4) + (140+15*n^2+90*n)*a(n+5) + (-21-6*n)*a(n+6) + a(n+7)}.
a(n) = 720*A001233(n) = 6!*(-1)^n*Stirling1(n,6). - Andrew Howroyd, Jul 27 2020

Name changed and terms a(20) and beyond from Andrew Howroyd, Jul 27 2020

A225475 Triangle read by rows, k!*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 3, 4, 2, 15, 23, 18, 6, 105, 176, 172, 96, 24, 945, 1689, 1900, 1380, 600, 120, 10395, 19524, 24278, 20880, 12120, 4320, 720, 135135, 264207, 354662, 344274, 241080, 116760, 35280, 5040, 2027025, 4098240, 5848344, 6228096, 4993296, 2956800, 1229760

Offset: 0

Peter Luschny, May 19 2013

The Stirling-Frobenius cycle numbers are defined in A225470.

			[n\k][ 0,    1,    2,    3,   4,   5]
[0]    1,
[1]    1,    1,
[2]    3,    4,    2,
[3]   15,   23,   18,    6,
[4]  105,  176,  172,   96,  24,
[5]  945, 1689, 1900, 1380, 600, 120.

Vincenzo Librandi, Rows n = 0..50, flattened
Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

Cf. A028338, A225479 (m=1), A048594.

SFCO[n_, k_, m_] := SFCO[n, k, m] = If[ k > n || k < 0, Return[0], If[ n == 0 && k == 0, Return[1], Return[ k*SFCO[n - 1, k - 1, m] + (m*n - 1)*SFCO[n - 1, k, m]]]]; Table[ SFCO[n, k, 2], {n, 0, 8}, {k, 0, n}] // Flatten  (* Jean-François Alcover, Jul 02 2013, translated from Sage *)

@CachedFunction
def SF_CO(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return k*SF_CO(n-1, k-1, m) + (m*n-1)*SF_CO(n-1, k, m)
for n in (0..8): [SF_CO(n, k, 2) for k in (0..n)]

For a recurrence see the Sage program.
T(n, 0) ~ A001147; T(n, 1) ~ A004041.
T(n, n) ~ A000142; T(n, n-1) ~ A001563.
T(n,k) = A028338(n,k)*A000142(k). - Philippe Deléham, Jun 24 2015

A344498 a(n) = |Stirling1(n, floor(n/2))| * floor(n/2)!.

Original entry on oeis.org

1, 0, 1, 2, 22, 100, 1350, 9744, 162456, 1614816, 32319000, 410031600, 9604465200, 148370508000, 3986353491120, 72622987557120, 2202727143576960, 46243059751848960, 1563325251963995520, 37165349757066935040, 1385918755006365216000, 36804377751967949760000

Offset: 0

Peter Luschny, May 22 2021

nonn

Cf. A132393, A225479 (middle column), A344397.

a := n -> abs(Stirling1(n, floor(n/2))) * floor(n/2)! :
seq(a(n), n = 0..21);

a[n_] := Abs @ StirlingS1[n, Floor[n/2]] * Floor[n/2]!; Array[a, 22, 0] (* Amiram Eldar, May 22 2021 *)

a(n) = floor(n/2)! * [x^floor(n/2)] Pochhammer(x, n).

A372347 a(n) = Sum_{j=0..n} p(n - j, j) where p(n, x) = Sum_{k=0..n} k! * Stirling1(n, k) * x^k.

Original entry on oeis.org

1, 1, 2, 4, 12, 52, 334, 2866, 31902, 439510, 7372150, 147351714, 3460114654, 94073798158, 2926942982790, 103161703653178, 4084845678671086, 180433041383154870, 8836346732709839206, 477142911818397135058, 28265453383985064929934

Offset: 0

Peter Luschny, Apr 28 2024

nonn

Cf. A225479.

p := n -> local k; add(k!*Stirling1(n, k)*x^k, k = 0..n):
a := n -> local j; add(subs(x=j, p(n - j)), j = 0..n):
seq(a(n), n = 0..21);

A376873 a(n) = n! * |Stirling1(2*n, n)|.

Original entry on oeis.org

1, 1, 22, 1350, 162456, 32319000, 9604465200, 3986353491120, 2202727143576960, 1563325251963995520, 1385918755006365216000, 1500893038955163069216000, 1949720475921117012670233600, 2992360962617823634351113600000, 5356716752093284789859604692736000

Offset: 0

Peter Luschny, Oct 29 2024

nonn

Cf. A132393, A187646, A225479.

a := n -> n! * abs(Stirling1(2*n, n)):
seq(a(n), n = 0..14);

Array[#!*Abs[StirlingS1[2 #, #]] &, 14] (* Michael De Vlieger, Oct 29 2024 *)

a(n) = n!*abs(stirling(2*n, n, 1)); \\ Michel Marcus, Oct 29 2024

from sympy.functions.combinatorial.numbers import factorial, stirling
def A376873(n): return factorial(n)*stirling(n<<1,n,kind=1) # Chai Wah Wu, Oct 29 2024

a(n) = n!*A187646(n).

a(n) = A225479(2*n, n).

A225474 Triangle read by rows, k!2^ks_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 2, 3, 8, 8, 15, 46, 72, 48, 105, 352, 688, 768, 384, 945, 3378, 7600, 11040, 9600, 3840, 10395, 39048, 97112, 167040, 193920, 138240, 46080, 135135, 528414, 1418648, 2754192, 3857280, 3736320, 2257920, 645120, 2027025, 8196480, 23393376, 49824768, 79892736

Offset: 0

Peter Luschny, May 19 2013

The Stirling-Frobenius cycle numbers are defined in A225470.

			[n\k][ 0,    1,    2,     3,    4,    5]
[0]    1,
[1]    1,    2,
[2]    3,    8,    8,
[3]   15,   46,   72,    48,
[4]  105,  352,  688,   768,  384,
[5]  945, 3378, 7600, 11040, 9600, 3840.

Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

Cf. A028338, A225479 (m=1), A048594, A161198, A225475.

Cf. A000079, A000142, A000165, A001147, A014479, A028338.

SFCSO[n_, k_, m_] := SFCSO[n, k, m] = If[k>n || k<0, 0, If[n == 0 && k == 0, 1, m*k*SFCSO[n-1, k-1, m] + (m*n-1)*SFCSO[n-1, k, m]]]; Table[SFCSO[n, k, 2], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 05 2014, translated from Sage *)

@CachedFunction
def SF_CSO(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return m*k*SF_CSO(n-1, k-1, m) + (m*n-1)*SF_CSO(n-1, k, m)
for n in (0..8): [SF_CSO(n, k, 2) for k in (0..n)]

For a recurrence see the Sage program.
T(n, 0) ~ A001147; T(n, n) ~ A000165; T(n, n-1) ~ A014479.
T(n,k) = A028338(n,k) * A000165(k) = A225475(n,k) * A000079(k) = A161198(n,k) * A000142(k). - Philippe Deléham, Jun 25 2015

A356654 Triangle read by rows. T(n, k) = k! * Sum_{j=k..n} Lah(n, j) * Stirling2(j, k), where Lah(n, k) = A271703(n, k).

Original entry on oeis.org

1, 0, 1, 0, 3, 2, 0, 13, 18, 6, 0, 73, 158, 108, 24, 0, 501, 1510, 1590, 720, 120, 0, 4051, 15962, 23040, 15960, 5400, 720, 0, 37633, 186270, 345786, 325920, 168000, 45360, 5040, 0, 394353, 2385182, 5469492, 6579384, 4594800, 1884960, 423360, 40320

Offset: 0

Peter Luschny, Sep 01 2022

The same construction with Stirling1 in place of Stirling2 gives A225479, the ordered Stirling cycle numbers.

			Triangle T(n, k) begins:
[0] 1;
[1] 0,      1;
[2] 0,      3,       2;
[3] 0,     13,      18,       6;
[4] 0,     73,     158,     108,      24;
[5] 0,    501,    1510,    1590,     720,     120;
[6] 0,   4051,   15962,   23040,   15960,    5400,     720;
[7] 0,  37633,  186270,  345786,  325920,  168000,   45360,   5040;
[8] 0, 394353, 2385182, 5469492, 6579384, 4594800, 1884960, 423360, 40320;

Cf. A271703, A048993, A225479, A000262 (column 1), A052838 (column 2), A084358 (row sums).

L := (n, k) -> `if`(n = k, 1, binomial(n-1, k-1) * n! / k!):
T := (n, k) -> k! * add(L(n, j) * Stirling2(j, k), j = k..n):
seq(seq(T(n, k), k = 0..n), n = 0..9);

T[n_, k_] := k! * Sum[Binomial[n, j] * FactorialPower[n - 1, n - j] * StirlingS2[j, k], {j, k, n}]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Sep 01 2022 *)

cp's OEIS Frontend

A052748 Expansion of e.g.f.: -(log(1-x))^3.

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A052753 Expansion of e.g.f.: log(1-x)^4.

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A052767 Expansion of e.g.f.: -(log(1-x))^5.

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A052779 Expansion of e.g.f.: (log(1-x))^6.

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A225475 Triangle read by rows, k!*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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A344498 a(n) = |Stirling1(n, floor(n/2))| * floor(n/2)!.

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A372347 a(n) = Sum_{j=0..n} p(n - j, j) where p(n, x) = Sum_{k=0..n} k! * Stirling1(n, k) * x^k.

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A376873 a(n) = n! * |Stirling1(2*n, n)|.

A225474 Triangle read by rows, k!2^ks_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.