A008542 -id:A008542 - cp's OEIS frontend

A318181 Expansion of e.g.f. exp((1 - exp(-6*x))/6).

1, 1, -5, 19, 1, -1103, 15211, -123821, 120865, 19464193, -474727877, 7017193075, -50549088671, -931708750607, 49742453940331, -1276858353426317, 21239149342811329, -100057086073774463, -9091588769200298501, 454849803186974314579, -13529950476868715792063, 262961916344710204693681

Offset: 0

Ilya Gutkovskiy, Aug 20 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..447
Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A005012, A008542, A009235, A014182, A317996, A318179, A318180.

seq(n!*coeff(series(exp((1-exp(-6*x))/6),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 09 2019

nmax = 21; CoefficientList[Series[Exp[(1 - Exp[-6 x])/6], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-6)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 21}]
a[n_] := a[n] = Sum[(-6)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
Table[(-6)^n BellB[n, -1/6], {n, 0, 21}] (* Peter Luschny, Aug 20 2018 *)

a(n) = Sum_{k=0..n} (-6)^(n-k)*Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-6)^(k-1)*binomial(n-1,k-1)*a(n-k).
a(n) = (-6)^n*BellPolynomial_n(-1/6). - Peter Luschny, Aug 20 2018

A053101 a(n) = ((6n+8)(!^6))/8(!^6), related to A034689 (((6n+2)(!^6))/2 sextic, or 6-factorials).

Original entry on oeis.org

1, 14, 280, 7280, 232960, 8852480, 389509120, 19475456000, 1090625536000, 67618783232000, 4598077259776000, 340257717223424000, 27220617377873920000, 2340973094497157120000, 215369524693738455040000

Offset: 0

Wolfdieter Lang

Row m=7 of the array A(7; m,n) := ((6*n+m)(!^6))/m(!^6), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..344

Cf. A047058, A008542(n+1), A034689(n+1), A034723(n+1), A034724(n+1), A034787(n+1), A034788(n+1), A053100, this sequence, A053102, A053103 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(7/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 13, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 6*x)^(7/3), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(7/3))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((6*n+8)(!^6))/8(!^6)= A034689(n+2)/8.

E.g.f.: 1/(1-6*x)^(7/3).

A051151 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -6, 1, 72, -18, 1, -1296, 396, -36, 1, 31104, -10800, 1260, -60, 1, -933120, 355104, -48600, 3060, -90, 1, 33592320, -13716864, 2104704, -158760, 6300, -126, 1, -1410877440, 609700608, -102114432, 8772624, -423360, 11592, -168

Offset: 1

Wolfdieter Lang

a(n,m) = R_n^m(a=0, b=6) in the notation of the given 1961 and 1962 references.
a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x-6*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
This is the signed Stirling1 triangle A008275 with diagonal d >= 0 (main diagonal d = 0) scaled with 6^d.

			Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
        1;
       -6,      1;
       72,    -18,      1;
    -1296,    396,    -36,    1;
    31104, -10800,   1260,  -60,   1;
  -933120, 355104, -48600, 3060, -90, 1;
   ...
3rd row o.g.f.: E(3,x) = 72*x - 18*x^2 + x^3.

Wolfdieter Lang, First 10 rows.
D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356. [The numbers R_n^m(a,b) are first introduced.]
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77. [Special cases of the numbers R_n^m(a,b) are tabulated.]

First (m=1) column sequence is: A047058(n-1).
Row sums (signed triangle): A008543(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A008542(n).
Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051141 (b=3), A051142 (b=4), A051150 (b=5).

a(n, m) = a(n-1, m-1) - 6*(n-1)*a(n-1, m), n >= m >= 1; a(n, m) := 0, n < m; a(n, 0) := 0 for n >= 1; a(0, 0) = 1.
E.g.f. for the m-th column of the signed triangle: ((log(1 + 6*x)/6)^m)/m!.
a(n, m) = S1(n, m)*6^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).

Various sections edited by Petros Hadjicostas, Jun 08 2020

A053103 a(n) = ((6n+10)(!^6))/10(!^6), related to A034724 (((6n+4)(!^6))/4 sextic, or 6-factorials).

Original entry on oeis.org

1, 16, 352, 9856, 335104, 13404160, 616591360, 32062750720, 1859639541760, 119016930672640, 8331185147084800, 633170071178444800, 51919945836632473600, 4568955233623657676800, 429481791960623821619200

Offset: 0

Wolfdieter Lang

Row m=10 of the array A(7; m,n) := ((6*n+m)(!^6))/m(!^6), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..343

Cf. A047058, A008542(n+1), A034689(n+1), A034723(n+1), A034724(n+1), A034787(n+1), A034788(n+1), A053100, A053101, A053102, this sequence (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(8/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 15, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 6*x)^(16/6), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 16 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(8/3))) \\ G. C. Greubel, Aug 16 2018

a(n) = ((6*n+10)(!^6))/10(!^6) = A034724(n+2)/10.

E.g.f.: 1/(1-6*x)^(8/3).

A113133 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x(n-1)a(n-1) + Sum_{j=2..n-2} (j-1)a(j)a(n-j), n>=4 and for x = 6.

Original entry on oeis.org

1, 1, 6, 72, 1332, 33264, 1040256, 38926656, 1692061488, 83688313536, 4638320578944, 284692939944192, 19169186341398912, 1404935464314299904, 111348880778746460160, 9489756817594314049536, 865470841829802331976448

Offset: 0

Philippe Deléham and Paul D. Hanna, Oct 28 2005

nonn

			a(2) = 6.
a(3) = 2*6^2 = 72.
a(4) = 6*3*72 + 1*6*6 = 1332.
a(5) = 6*4*1332 + 1*6*72 + 2*72*6 = 33264.
a(6) = 6*5*33264 + 1*6*1332 + 2*72*72 + 3*1332*6 = 1040256.
G.f.: A(x) = 1 + x + 6*x^2 + 72*x^3 + 1332*x^4 + 33264*x^5
+...
= x/series_reversion(x + x^2 + 7*x^3 + 91*x^4 + 1729*x^5
+...).

Cf. A008542, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113132(x=5), A113134(x=7), A113135(x=8).

x=6;a[0]=a[1]=1;a[2]=x;a[3]=2x^2;a[n_]:=a[n]=x*(n-1)*a[n-1]+Sum[(j-1)*a[j ]*a[n-j], {j, 2, n-2}];Table[a[n], {n, 0, 17}](Robert G. Wilson v)

a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,6*j+1))))))[n+1]

a(n,x=6)=if(n<0,0,if(n==0 || n==1,1,if(n==2,x,if(n==3,2*x^2,x*(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j))))))

a(n+1) = Sum{k, 0<=k<=n} 6^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of sextuple factorial numbers (A008542).
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of sextuple factorial numbers (A008542).

A227544 Expansion of e.g.f. 1/(1 - sin(6*x))^(1/6).

Original entry on oeis.org

1, 1, 7, 55, 721, 11761, 240247, 5801095, 162512161, 5171130721, 184337942887, 7275081518935, 314918762166001, 14834964193292881, 755507853144691927, 41362173671901329575, 2422478811455080626241, 151132171549872325122241, 10006051653759338150151367, 700695219796759105368529015

Offset: 0

Paul D. Hanna, Dec 20 2013

nonn

Generally, for e.g.f. 1/(1-sin(p*x))^(1/p) we have a(n) ~ n! * 2^(n+3/p) * p^n / (Gamma(2/p) * n^(1-2/p) * Pi^(n+2/p)). - Vaclav Kotesovec, Jan 03 2014

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 55*x^3/3! + 721*x^4/4! + 11761*x^5/5! + ...
where A(x)^3 = 1 + 3*x + 27*x^2/2! + 297*x^3/3! + 4617*x^4/4! + 87723*x^5/5! + ...
and 1/A(x)^3 = 1 - 3*x - 9*x^2/2! + 27*x^3/3! + 81*x^4/4! - 243*x^5/5! + ...
which illustrates 1/A(x)^3 = cos(3*x) - sin(3*x).
O.g.f.: 1/(1-x - 6*1*1*x^2/(1-7*x - 6*2*4*x^2/(1-13*x - 6*3*7*x^2/(1-19*x - 6*4*10*x^2/(1-25*x - 6*5*13*x^2/(1-...)))))), a continued fraction.

Cf. A001586 (p=2), A007788 (p=3), A144015 (p=4), A230134 (p=5), A235128 (p=7), A230114 (p=8).

Cf. A008542, A136630.

CoefficientList[Series[1/(1-Sin[6*x])^(1/6), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jan 03 2014 *)

{a(n)=local(X=x+x*O(x^n)); n!*polcoeff((cos(3*X)-sin(3*X))^(-1/3), n)}
for(n=0,20,print1(a(n),", "))

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(intformal(A^3/subst(A^3, x, -x)))); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a008542(n) = prod(k=0, n-1, 6*k+1);
a(n) = sum(k=0, n, a008542(k)*(6*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

E.g.f. A(x) satisfies:
(1) A(x) = (cos(3*x) - sin(3*x))^(-1/3).
(2) A(x)^3/A(-x)^3 = 1/cos(6*x) + tan(6*x).
(3) A(x) = exp( Integral A(x)^3/A(-x)^3 dx ).
O.g.f.: 1/G(0) where G(k) = 1 - (6*k+1)*x - 6*(k+1)*(3*k+1)*x^2/G(k+1) [continued fraction formula from A144015 due to Sergei N. Gladkovskii].
a(n) ~ n! * 2^(2*n+1/2) * 3^n / (Gamma(1/3) * n^(2/3) * Pi^(n+1/3)). - Vaclav Kotesovec, Jan 03 2014
a(n) = Sum_{k=0..n} A008542(k) * (6*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A053102 a(n) = ((6n+9)(!^6))/9(!^6), related to A034723 (((6n+3)(!^6))/3 sextic, or 6-factorials).

Original entry on oeis.org

1, 15, 315, 8505, 280665, 10945935, 492567075, 25120920825, 1431892487025, 90209226682575, 6224436641097675, 466832748082325625, 37813452594668375625, 3289770375736148679375, 305948644943461827181875

Offset: 0

Wolfdieter Lang

Row m=9 of the array A(7; m,n) := ((6*n+m)(!^6))/m(!^6), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..344

Cf. A047058, A008542(n+1), A034689(n+1), A034723(n+1), A034724(n+1), A034787(n+1), A034788(n+1), A053100, A053101, this sequence, A053103 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(15/6))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 14, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 6*x)^(15/6), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(15/6))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((6*n+9)(!^6))/9(!^6) = A034723(n+2)/9.

E.g.f.: 1/(1-6*x)^(15/6).

A144773 10-fold factorials: Product_{k=0..n-1} (10*k+1).

Original entry on oeis.org

1, 1, 11, 231, 7161, 293601, 14973651, 913392711, 64850882481, 5252921480961, 478015854767451, 48279601331512551, 5359035747797893161, 648443325483545072481, 84946075638344404495011, 11977396665006561033796551, 1808586896415990716103279201, 291182490322974505292627951361

Offset: 0

Philippe Deléham, Sep 21 2008

G. C. Greubel, Table of n, a(n) for n = 0..320

Essentially a duplicate of A045757.
Cf. k-fold factorials: A000142 ("1-fold"), A001147 (2-fold), A007559 (3), A007696 (4), A008548 (5), A008542 (6), A045754 (7), A045755 (8), A045756 (9), A256268 (combined table).
Cf. A048994, A132393.

R:=PowerSeriesRing(Rationals(), 15); Coefficients(R!(Laplace( (1-10*x)^(-1/10) ))); // G. C. Greubel, Mar 03 2020

G(x):=(1-10*x)^(-1/10): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..14); # Zerinvary Lajos, Apr 03 2009

b = 10; Table[FullSimplify[b^n*Gamma[n + 1/b]/Gamma[1/b]], {n, 0, 14}] (* Michael De Vlieger, Sep 14 2016 *)
Join[{1},FoldList[Times,10 Range[0,15]+1]] (* Harvey P. Dale, Oct 24 2022 *)

Vec(serlaplace( (1-10*x)^(-1/10) +O('x^15) )) \\ G. C. Greubel, Mar 03 2020

[10^n*rising_factorial(1/10,n) for n in (0..15)] # G. C. Greubel, Mar 03 2020

a(n) = Sum_{k = 0..n} (-10)^(n - k) * A048994(n, k).
a(n) = Sum_{k = 0..n} 10^(n - k) * A132393(n, k).
E.g.f.: (1 - 10*x)^(-1/10).
a(n) = A045757(n), n>0.
a(n) = (-9)^n * Sum_{k = 0..n} (10/9)^k * s(n + 1,n + 1 - k), where s(n, k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/Q(0), where Q(k) = 1 - (10*k+1)*x/( 1 - 10*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 09 2014
a(n) = 10^n * Gamma(n + 1/10) / Gamma(1/10). - Artur Jasinski Aug 23 2016
a(n) ~ sqrt(2*Pi)*10^n*n^(n-2/5)/(Gamma(1/10)*exp(n)). - Ilya Gutkovskiy, Sep 11 2016
D-finite with recurrence: a(n) - (10*n-9)*a(n-1) = 0. - R. J. Mathar, Jan 20 2020
Sum_{n>=0} 1/a(n) = 1 + (e/10^9)^(1/10)*(Gamma(1/10) - Gamma(1/10, 1/10)). - Amiram Eldar, Dec 22 2022

A088996 Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 7, 6, 0, 6, 29, 46, 24, 0, 24, 146, 329, 326, 120, 0, 120, 874, 2521, 3604, 2556, 720, 0, 720, 6084, 21244, 39271, 40564, 22212, 5040, 0, 5040, 48348, 197380, 444849, 598116, 479996, 212976, 40320

Offset: 0

Philippe Deléham, Dec 01 2003, Aug 17 2007

			Triangle begins:
  1;
  0,    1;
  0,    1,     2;
  0,    2,     7,      6;
  0,    6,    29,     46,     24;
  0,   24,   146,    329,    326,    120;
  0,  120,   874,   2521,   3604,   2556,    720;
  0,  720,  6084,  21244,  39271,  40564,  22212,   5040;
  0, 5040, 48348, 197380, 444849, 598116, 479996, 212976, 40320;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Trevor Hyde, Liminal reciprocity and factorization statistics, arXiv:1803.08438 [math.NT], 2018.

Variant: A059364, diagonals give A000007, A000142, A067318.

Cf. A001147 (row sums), A048994, A084938.

A088996:= func< n,k | (&+[(-1)^j*Binomial(j,n-k)*StirlingFirst(n,n-j): j in [0..n]]) >;
[A088996(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 23 2022

A059364 := (n, k) -> add(abs(Stirling1(n, n - j))*binomial(j, n - k), j = 0..n);
seq(seq(A059364(n, k), k = 0..n), n = 0..8);  # Peter Luschny, Aug 27 2025

T[n_, k_]:= T[n, k]= Sum[(-1)^(n-i)*Binomial[i, k] StirlingS1[n+1, n+1-i], {i, 0, n}]; {{1}}~Join~Table[Abs@ T[n, k], {n,0,10}, {k,n+1,0,-1}] (* Michael De Vlieger, Jun 19 2018 *)

def A088996(n,k): return add((-1)^(n-i)*binomial(i,k)*stirling_number1(n+1,n+1-i) for i in (0..n))
for n in (0..10): [A088996(n,k) for k in (0..n)]  # Peter Luschny, May 12 2013

T(n, k) given by [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938. [Original name.]
Sum_{k=0..n} (-1)^k*T(n,k) = (-1)^n.
From Vladeta Jovovic, Dec 15 2004: (Start)
E.g.f.: (1-y-y*x)^(-1/(1+x)).
Sum_{k=0..n} T(n, k)*x^k = Product_{k=1..n} (k*x+k-1). (End)
T(n, k) = n*T(n-1, k-1) + (n-1)*T(n-1, k); T(0, 0) = 1, T(0, k) = 0 if k > 0, T(n, k) = 0 if k < 0. - Philippe Deléham, May 22 2005
Sum_{k=0..n} T(n,k)*x^(n-k) = A019590(n+1), A000012(n), A000142(n), A001147(n), A007559(n), A007696(n), A008548(n), A008542(n), A045754(n), A045755(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, respectively. Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000007(n), A001147(n), A008544(n), A008545(n), A008546(n), A008543(n), A049209(n), A049210(n), A049211(n), A049212(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Aug 10 2007
T(n, k) = Sum_{j=0..n} (-1)^j*binomial(j, n-k)*StirlingS1(n, n-j). - G. C. Greubel, Feb 23 2022

New name using a formula of G. C. Greubel by Peter Luschny, Aug 27 2025

A347014 Expansion of e.g.f.: exp(x) / (1 - 6*x)^(1/6).

Original entry on oeis.org

1, 2, 10, 116, 2140, 52856, 1627192, 59930480, 2568599056, 125553289760, 6892279877536, 419788155021632, 28090704069553600, 2048487353985408896, 161687913401407530880, 13733087614786273308416, 1248892148354210466595072, 121073054127693143488709120

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Binomial transform of A008542.

In general, for k >= 1, if e.g.f. = exp(x) / (1 - k*x)^(1/k), then a(n) ~ n! * exp(1/k) * k^n / (Gamma(1/k) * n^(1 - 1/k)). - Vaclav Kotesovec, Aug 14 2021

Cf. A000522, A008542, A056547, A084262, A346258, A347012, A347013.

g:= proc(n) option remember; `if`(n<2, 1, (6*n-5)*g(n-1)) end:
a:= n-> add(binomial(n, k)*g(k), k=0..n):
seq(a(n), n=0..17);  # Alois P. Heinz, Aug 10 2021

nmax = 17; CoefficientList[Series[Exp[x]/(1 - 6 x)^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 17}]
Table[HypergeometricU[1/6, n + 7/6, 1/6]/6^(1/6), {n, 0, 17}]

a[n]:=if n<2 then n+1 else (6*n-4)*a[n-1]-6*(n-1)*a[n-2];
makelist(a[n],n,0,50); /* Tani Akinari, Sep 08 2023 */

a(n) = Sum_{k=0..n} binomial(n,k) * A008542(k).
a(n) ~ n! * exp(1/6) * 6^n / (Gamma(1/6) * n^(5/6)). - Vaclav Kotesovec, Aug 14 2021
a(n+2) = (6*n+8)*a(n+1) - 6*(n+1)*a(n). - Tani Akinari, Sep 08 2023

cp's OEIS Frontend

A318181 Expansion of e.g.f. exp((1 - exp(-6*x))/6).

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A053101 a(n) = ((6*n+8)(!^6))/8(!^6), related to A034689 (((6*n+2)(!^6))/2 sextic, or 6-factorials).

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A051151 Generalized Stirling number triangle of first kind.

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A053103 a(n) = ((6*n+10)(!^6))/10(!^6), related to A034724 (((6*n+4)(!^6))/4 sextic, or 6-factorials).

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A113133 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x*(n-1)*a(n-1) + Sum_{j=2..n-2} (j-1)*a(j)*a(n-j), n>=4 and for x = 6.

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A227544 Expansion of e.g.f. 1/(1 - sin(6*x))^(1/6).

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A053102 a(n) = ((6*n+9)(!^6))/9(!^6), related to A034723 (((6*n+3)(!^6))/3 sextic, or 6-factorials).

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A053101 a(n) = ((6n+8)(!^6))/8(!^6), related to A034689 (((6n+2)(!^6))/2 sextic, or 6-factorials).

A053103 a(n) = ((6n+10)(!^6))/10(!^6), related to A034724 (((6n+4)(!^6))/4 sextic, or 6-factorials).

A113133 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x(n-1)a(n-1) + Sum_{j=2..n-2} (j-1)a(j)a(n-j), n>=4 and for x = 6.

A053102 a(n) = ((6n+9)(!^6))/9(!^6), related to A034723 (((6n+3)(!^6))/3 sextic, or 6-factorials).