A000629 -id:A000629 - cp's OEIS frontend

A005460 a(n) = (3n+4)(n+3)!/24.

1, 7, 50, 390, 3360, 31920, 332640, 3780000, 46569600, 618710400, 8821612800, 134399865600, 2179457280000, 37486665216000, 681734237184000, 13071512982528000, 263564384219136000, 5575400435404800000, 123469776914964480000, 2856835183101419520000

Offset: 0

N. J. A. Sloane

Essentially Stirling numbers of second kind - third external diagonal of Worpitzky triangle A028246.

R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.
R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
R. Austin, R. K. Guy, and R. Nowakowski, Unpublished notes, 1987
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
John K. Sikora, On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles, arXiv:1806.00887 [math.NT], 2018.

Cf. A028246.

[(3*n+4)*Factorial(n+3)/24: n in [0..20]]; // Vincenzo Librandi, Oct 08 2011

Table[StirlingS2[n+3, n+1]*n!, {n,0,20}]

a(n)=(3*n+4)*(n+3)!/24 \\ Charles R Greathouse IV, Jun 30 2017

[factorial(n)*stirling_number2(n+3,n+1) for n in range(21)] # G. C. Greubel, Nov 22 2022

E.g.f.: (1+2*x)/(1-x)^5.

a(n) = S2(n+3, n+1)*n! = n!*A001296(n+1). - Olivier Gérard, Sep 13 2016

A124212 Expansion of e.g.f. exp(x)/sqrt(2-exp(2*x)).

Original entry on oeis.org

1, 2, 8, 56, 560, 7232, 114368, 2139776, 46223360, 1132124672, 30999600128, 938366468096, 31114518056960, 1121542540992512, 43664751042265088, 1826043989622358016, 81635676596544143360

Offset: 0

Karol A. Penson, Oct 19 2006

Robert Israel, Table of n, a(n) for n = 0..60

Cf. A000629, A000984, A014307, A124214, A145901, A176785, A186695, A229558.

Cf. A276371, A365777.

  N:= 60; # to get a(n) for n <= N
S:= series(exp(x)/sqrt(2-exp(2*x)), x, N+1):
seq(coeff(S,x,j), j=0..N); # Robert Israel, May 19 2014

CoefficientList[Series[E^x/Sqrt[2-E^(2*x)]-1, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 03 2013 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=0,n,A=1+intformal(A+A^3)); n!*polcoeff(A,n)} \\ Paul D. Hanna, Oct 04 2008

E.g.f. satisfies: A'(x) = A(x) + A(x)^3 with A(0)=1. [From Paul D. Hanna, Oct 04 2008]
G.f.: 1/G(0) where G(k) = 1 - x*(4*k+2)/( 1 - 2*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013
G.f.: 2/G(0), where G(k)= 1 + 1/(1 - x*(8*k+4)/(x*(8*k+4) - 1 + 4*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
a(n) ~ 2^(n+1/2)*n^n/(log(2)^(n+1/2)*exp(n)). - Vaclav Kotesovec, Jun 03 2013
From Peter Bala, Aug 30 2016: (Start)
a(n) = 1/sqrt(2) * Sum_{k >= 0} (1/8)^k*binomial(2*k,k)*(2*k + 1)^n = 1/sqrt(2) * Sum_{k >= 0} (-1/2)^k*binomial(-1/2,k)*(2*k + 1)^n. Cf. A176785, A124214 and A229558.
a(n) = Sum_{k = 0..n} (1/4)^k*binomial(2*k,k)*A145901(n,k).
a(n) = Sum_{k = 0..n} ( Sum_{i = 0..k} (-1)^(k-i)/4^k* binomial(2*k,k)*binomial(k,i)*(2*i + 1)^n ). (End)
a(n) = 2^n * A014307(n). - Seiichi Manyama, Nov 18 2023

Definition corrected by Robert Israel, May 19 2014

A062208 a(n) = Sum_{m>=0} binomial(m,3)^n*2^(-m-1).

Original entry on oeis.org

1, 1, 63, 16081, 10681263, 14638956721, 35941784497263, 143743469278461361, 874531783382503604463, 7687300579969605991710001, 93777824804632275267836362863, 1537173608464960118370398000894641, 32970915649974341628739088902163732463

Offset: 0

Angelo Dalli, Jun 13 2001

Number of alignments of n strings of length 3.

Conjectures: a(2*n) = 3 (mod 60) and a(2*n+1) = 1 (mod 60); for fixed k, the sequence a(n) (mod k) eventually becomes periodic with exact period a divisor of phi(k), where phi(k) is Euler's totient function A000010. - Peter Bala, Feb 04 2018

Alois P. Heinz, Table of n, a(n) for n = 0..100
J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522

Cf. A000670, A055203, A001850, A126086.
See A062204 for further references, formulas and comments.
Cf. A001850, A062204, A062205, A299041.
Row n=3 of A262809.

A000629 := proc(n) local k ; sum( k^n/2^k,k=0..infinity) ; end: A062208 := proc(n) local a,stir,ni,n1,n2,n3,stir2,i,j,tmp ; a := 0 ; if n = 0 then RETURN(1) ; fi ; stir := combinat[partition](n) ; stir2 := {} ; for i in stir do if nops(i) <= 3 then tmp := i ; while nops(tmp) < 3 do tmp := [op(tmp),0] ; od: tmp := combinat[permute](tmp) ; for j in tmp do stir2 := stir2 union { j } ; od: fi ; od: for ni in stir2 do n1 := op(1,ni) ; n2 := op(2,ni) ; n3 := op(3,ni) ; a := a+combinat[multinomial](n,n1,n2,n3)*(A000629(3*n1+2*n2+n3)-1/2-2^(3*n1+2*n2+n3)/4)*(-3)^n2*2^n3 ; od: a/(2*6^n) ; end: seq(A062208(n),n=0..14) ; # R. J. Mathar, Apr 01 2008
a:=proc(n) options operator, arrow: sum(binomial(m, 3)^n*2^(-m-1),m=0.. infinity) end proc: seq(a(n),n=0..12); # Emeric Deutsch, Mar 22 2008

a[n_] = Sum[2^(-1-m)*((m-2)*(m-1)*m)^n, {m, 0, Infinity}]/6^n; a /@ Range[0, 12] (* Jean-François Alcover, Jul 13 2011 *)
With[{r = 3}, Flatten[{1, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, r]^k, {i, 0, j}], {j, 0, k*r}], {k, 1, 15}]}]] (* Vaclav Kotesovec, Mar 22 2016 *)

From Vaclav Kotesovec, Mar 22 2016: (Start)
a(n) ~ 3^(2*n + 1/2) * n!^3 / (Pi * n * 2^(n+3) * (log(2))^(3*n+1)).
a(n) ~ sqrt(Pi)*3^(2*n+1/2)*n^(3*n+1/2) / (2^(n+3/2)*exp(3*n)*(log(2))^(3*n+1)).
(End)
a(n) = Sum_{k = 3..3*n} Sum_{i = 0..k} (-1)^(k-i)*binomial(k,i)* binomial(i,3)^n. Row sums of A299041. - Peter Bala, Feb 04 2018

New definition from Vladeta Jovovic, Mar 01 2008

Edited by N. J. A. Sloane, Sep 19 2009 at the suggestion of Max Alekseyev

A123227 Expansion of e.g.f.: 2exp(2x) / (3 - exp(2*x)).

Original entry on oeis.org

1, 3, 12, 66, 480, 4368, 47712, 608016, 8855040, 145083648, 2641216512, 52891055616, 1155444326400, 27344999497728, 696933753434112, 19031293222127616, 554336947975618560, 17155693983744196608, 562168282464340672512, 19444889661250162262016

Offset: 0

Philippe Deléham, Oct 06 2006

G. C. Greubel, Table of n, a(n) for n = 0..200
Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.
Eric Weisstein's MathWorld, Polylogarithm.
Eric Weisstein's MathWorld, Lerch Transcendent.

Cf. A000629, A201339, A122704, A009362, A123125, A028246.

a := n -> 2^(n+1)*polylog(-n, 1/3):
seq(round(evalf(a(n),32)), n=0..19); # Peter Luschny, Nov 03 2015
seq(expand(2^(n+1)*polylog(-n,1/3)), n=0..100); # Robert Israel, Nov 03 2015

CoefficientList[Series[2*Exp[2*x]/(3-Exp[2*x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 24 2013 *)
Round@Table[(-1)^(n+1) (LerchPhi[Sqrt[3], -n, 0] + LerchPhi[-Sqrt[3], -n, 0]), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)

{a(n)=n!*polcoeff(2*exp(2*x+x*O(x^n))/(3 - exp(2*x+x*O(x^n))), n)} /* Paul D. Hanna */

{a(n)=polcoeff(sum(m=0, n, 3^m*m!*x^m/prod(k=1, m, 1+2*k*x+x*O(x^n))), n)} /* Paul D. Hanna */

{Stirling2(n, k)=if(k<0||k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0, n, (-2)^(n-k)*3^k*Stirling2(n, k)*k!)} /* Paul D. Hanna */

my(x='x+O('x^20)); Vec(serlaplace(2*exp(2*x)/(3-exp(2*x)))) \\ Joerg Arndt, May 06 2013

@CachedFunction
def BB(n, k, x):  # Modified Cardinal B-splines
    if n == 1: return 0 if (x < 0) or (x >= k) else 1
    return x*BB(n-1, k, x) + (n*k-x)*BB(n-1, k, x-k)
def EulerianPolynomial(n, k, x):
    if n == 0: return 1
    return add(BB(n+1, k, k*m+1)*x^m for m in (0..n))
def A123227(n) : return 3^n*EulerianPolynomial(n, 1, 1/3)
[A123227(n) for n in (0..18)]  # Peter Luschny, May 04 2013

a(n) = abs(A009362(n+1)).
a(n-1) = Sum_{k=1..n} 2^(n-k)*A028246(n,k), n>=1.
a(n) = Sum_{k=0..n} 3^k*A123125(n,k).
From Paul D. Hanna, Nov 30 2011: (Start)
a(n) = 3*A122704(n) for n>0.
a(n) = Sum_{k=0..n} (-2)^(n-k) * 3^k * Stirling2(n,k) * k!.
O.g.f.: Sum_{n>=0} 3^n * n!*x^n / Product_{k=0..n} (1+2*k*x).
O.g.f.: 1/(1 - 3*x/(1-x/(1 - 6*x/(1-2*x/(1 - 9*x/(1-3*x/(1 - 12*x/(1-4*x/(1 - 15*x/(1-5*x/(1 - ...))))))))))), a continued fraction.
(End)
a(n) ~ n! * (2/log(3))^(n+1). - Vaclav Kotesovec, Jun 24 2013
a(n) = 2^n*log(3)*Integral_{x = 0..oo} (ceiling(x))^n * 3^(-x) dx. - Peter Bala, Feb 06 2015
a(n) = (-1)^(n+1)*(LerchPhi(sqrt(3), -n, 0) + LerchPhi(-sqrt(3), -n, 0)) = (-1)^(n+1)*(Li_{-n}(sqrt(3)) + Li_{-n}(-sqrt(3))) - 2*0^n, where Li_n(x) is the polylogarithm. - Vladimir Reshetnikov, Oct 31 2015
a(n) = 2^(n+1)*Li_{-n}(1/3). - Peter Luschny, Nov 03 2015
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1). - Ilya Gutkovskiy, Jul 05 2020

Name changed and a(8) corrected by Paul D. Hanna, Nov 30 2011

A278075 Coefficients of the signed Fubini polynomials in ascending order, F_n(x) = Sum_{k=0..n} (-1)^nStirling2(n,k)k!*(-x)^k.

Original entry on oeis.org

1, 0, 1, 0, -1, 2, 0, 1, -6, 6, 0, -1, 14, -36, 24, 0, 1, -30, 150, -240, 120, 0, -1, 62, -540, 1560, -1800, 720, 0, 1, -126, 1806, -8400, 16800, -15120, 5040, 0, -1, 254, -5796, 40824, -126000, 191520, -141120, 40320, 0, 1, -510, 18150, -186480, 834120, -1905120, 2328480, -1451520, 362880

Offset: 0

Peter Luschny, Jan 09 2017

sign

Signed version of A131689.

Integral_{x=0..1} F_n(x) = B_n(1) where B_n(x) are the Bernoulli polynomials.

			Triangle of coefficients starts:
[1]
[0,  1]
[0, -1,    2]
[0,  1,   -6,    6]
[0, -1,   14,  -36,    24]
[0,  1,  -30,  150,  -240,   120]
[0, -1,   62, -540,  1560, -1800,    720]
[0,  1, -126, 1806, -8400, 16800, -15120, 5040]

Peter Luschny, Illustration of the polynomials.
Peter Luschny, The Bernoulli Manifesto.
Grzegorz Rządkowski, Bernoulli numbers and solitons - revisited, Journal of Nonlinear Mathematical Physics, (2010) 17:1, 121-126.
J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

Row sums are A000012, diagonal is A000142.
Cf. A131689 (unsigned), A019538 (n>0, k>0), A090582.
Let F(n, x) = Sum_{k=0..n} T(n,k)*x^k then, apart from possible differences in the sign or the offset, we have: F(n, -5) = A094418(n), F(n, -4) = A094417(n), F(n, -3) = A032033(n), F(n, -2) = A004123(n), F(n, -1) = A000670(n), F(n, 0) = A000007(n), F(n, 1) = A000012(n), F(n, 2) = A000629(n), F(n, 3) = A201339(n), F(n, 4) = A201354(n), F(n, 5) = A201365(n).
Cf. A163626, A173018.

function T(n, k)
    if k < 0 || k > n return 0 end
    if n == 0 && k == 0 return 1 end
    k*(T(n-1, k-1) - T(n-1, k))
end
for n in 0:7
    println([T(n,k) for k in 0:n])
end
# Peter Luschny, Mar 26 2020

F := (n,x) -> add((-1)^n*Stirling2(n,k)*k!*(-x)^k, k=0..n):
for n from 0 to 10 do PolynomialTools:-CoefficientList(F(n,x), x) od;

T[ n_, k_] := If[ n < 0 || k < 0, 0, (-1)^(n - k) k! StirlingS2[n, k]]; (* Michael Somos, Jul 08 2018 *)

{T(n, k) = if( n<0, 0, sum(i=0, k, (-1)^(n + i) * binomial(k, i) * i^n))};
/* Michael Somos, Jul 08 2018 */

T(n, k) = (-1)^(n-k) * Stirling2(n, k) * k!.
E.g.f.: 1/(1-x*(1-exp(-t))) = Sum_{n>=0} F_n(x) t^n/n!.
T(n, k) = k*(T(n-1, k-1) - T(n-1, k)) for 0 <= k <= n, T(0, 0) = 1, otherwise 0.
Bernoulli numbers are given by B(n) = Sum_{k = 0..n} T(n, k) / (k+1) with B(1) = 1/2. - Michael Somos, Jul 08 2018
Let F_n(x) be the row polynomials of this sequence and W_n(x) the row polynomials of A163626. Then F_n(1 - x) = W_n(x) and Integral_{x=0..1} U(n, x) = Bernoulli(n, 1) for U in {W, F}. - Peter Luschny, Aug 10 2021
T(n, k) = [z^k] Sum_{k=0..n} Eulerian(n, k)*z^(k+1)*(z-1)^(n-k-1) for n >= 1, where Eulerian(n, k) = A173018(n, k). - Peter Luschny, Aug 15 2022

A330353 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).

Original entry on oeis.org

1, 4, 18, 112, 810, 7144, 73458, 850672, 11069370, 161190904, 2575237698, 44571447232, 836188737930, 16970931765064, 368985732635538, 8524290269083792, 208874053200038490, 5428866923032585624, 149250273758730282978, 4318265042184721248352

Offset: 1

Ilya Gutkovskiy, Dec 11 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..420

Cf. A000041, A000203, A000629, A002745, A008277, A038048, A167137, A308555, A330351, A330352, A330354.

nmax = 20; CoefficientList[Series[Sum[(Exp[x] - 1)^k/(k (1 - (Exp[x] - 1)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] (k - 1)! DivisorSigma[1, k], {k, 1, n}], {n, 1, 20}]

E.g.f.: -Sum_{k>=1} log(1 - (exp(x) - 1)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A167137.
G.f.: Sum_{k>=1} (k - 1)! * sigma(k) * x^k / Product_{j=1..k} (1 - j*x), where sigma = A000203.
exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of the partition numbers (A000041).
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * sigma(k).
a(n) ~ n! * Pi^2 / (12 * (log(2))^(n+1)). - Vaclav Kotesovec, Dec 14 2019

A005462 Number of simplices in barycentric subdivision of n-simplex.

Original entry on oeis.org

1, 31, 602, 10206, 166824, 2739240, 46070640, 801496080, 14495120640, 273158645760, 5368729766400, 110055327782400, 2351983118284800, 52361635508582400, 1213240925049753600, 29227769646147072000, 731310069474496512000, 18984684514588176384000

Offset: 3

N. J. A. Sloane

R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.
R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Micah Manary, Table of n, a(n) for n = 3..138
R. Austin, R. K. Guy, and R. Nowakowski, Unpublished notes, 1987
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.

Cf. A001298, A005460, A005461, A028246.

[Factorial(n-3)*StirlingSecond(n+2,n-2): n in [3..30]]; // G. C. Greubel, Nov 22 2022

Table[(n-3)!*StirlingS2[n+2,n-2], {n,3,30}] (* G. C. Greubel, Nov 22 2022 *)

[factorial(n-3)*stirling_number2(n+2,n-2) for n in range(3,31)] # G. C. Greubel, Nov 22 2022

Essentially Stirling numbers of second kind - see A028246.

a(n) = Stirling2(n+2,n-2)*(n-3)!. - Alois P. Heinz, Aug 28 2022

A052862 Expansion of e.g.f. log(-1/(-2+exp(x)))*x.

Original entry on oeis.org

0, 0, 2, 6, 24, 130, 900, 7574, 74928, 851274, 10916700, 155919742, 2453941512, 42188446898, 786563892660, 15805750451430, 340522975054176, 7829628493247002, 191363568551328780, 4954089147107164238

Offset: 0

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

A simple grammar.

For n > 2, a(n) = 2 * n * A000670(n-2). - Gerald McGarvey, Nov 01 2007 [corrected by Seiichi Manyama, May 26 2022]

Seiichi Manyama, Table of n, a(n) for n = 0..425
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 830

Cf. A000629, A000670, A354413.

spec := [S,{B=Cycle(C),C=Set(Z,1 <= card),S=Prod(Z,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*((i+r)^(n-r)/(i!*(k-i-r)!)), {i, 0, k-r}], {k, r, n}];
Fubini[0, 1] = 1;
a[n_] := If[n == 2, 2, 2 n * Fubini[n-2, 1]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 11 2022 *)

my(x='x+O('x^25)); concat([0,0],Vec(serlaplace(log(-1/(-2+exp(x)))*x))) \\ Joerg Arndt, Oct 11 2022

a(n) ~ (n-1)! / log(2)^(n-1). - Vaclav Kotesovec, Aug 04 2014

A079641 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Stirling1-triangle |A008275(n,k)|.

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 26, 36, 12, 1, 150, 250, 120, 20, 1, 1082, 2040, 1230, 300, 30, 1, 9366, 19334, 13650, 4270, 630, 42, 1, 94586, 209580, 166376, 62160, 11900, 1176, 56, 1, 1091670, 2562354, 2229444, 952728, 220500, 28476, 2016, 72, 1, 14174522

Offset: 1

Vladeta Jovovic, Jan 30 2003

Triangle T(n,k), 1<=k<=n, read by rows, given by (0, 2, 1, 4, 2, 6, 3, 8, 4, 10, 5, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 22 2011
Subtriangle of triangle in A129062. - Philippe Deléham, Feb 17 2013
Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			Triangle begins:
  1;
  2,1;
  6,6,1;
  26,36,12,1;
  150,250,120,20,1;
  1082,2040,1230,300,30,1;
  ...
Triangle (0,2,1,4,2,6,3,8,4,...) DELTA (1,0,1,0,1,0,1,0,1,...) begins:
  1
  0, 1
  0, 2, 1
  0, 6, 6, 1
  0, 26, 36, 12, 1
  0, 150, 250, 120, 20, 1
  0, 1082, 2040, 1230, 300, 30, 1. - _Philippe Deléham_, Dec 22 2011

Nick Early, Canonical Bases for Permutohedral Plates, arXiv:1712.08520 [math.CO], 2017.
Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.
D. E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA], 1992; The Mathematica J., 2 (1992), 67-78.

Cf. A000670 (row sums), A000629 (first column), A195204, A195205. A209849, A129062

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> add((-1)^(n-k)*2^k*k!*combinat:-stirling2(n, k), k=0..n), 9); # Peter Luschny, Jan 26 2016

rows = 10;
t = Table[Sum[(-1)^(n-k)*2^k*k!*StirlingS2[n, k], {k,0,n}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

T(n, k) = Sum_{i=k..n} A008277(n, i) * |A008275(i, k)|.
E.g.f.: (2-exp(x))^(-y). - Vladeta Jovovic, Nov 22 2003
From Peter Bala, Sep 12 2011: (Start)
The row generating polynomials R(n,x) begin R(1,x) = x, R(2,x) = 2*x + x^2, R(3,x) = 6*x + 6*x^2 + x^3 and satisfy the recurrence R(n+1,x) = x*(2*R(n,x+1) - R(n,x)). They form a sequence of binomial type polynomials. In particular, denoting R(n,x) by x^[n] to emphasize the analogies with the monomial polynomials x^n, we have the binomial expansion (x + y)^[n] = Sum_{k = 0..n} binomial(n,k)*x^[n-k]*y^[k].
There is a Dobinski-type formula: exp(-x)*Sum_{k >= 0} (-k)^[n] * x^k/k! = Bell(n,-x). The alternating n-th row entries (-1)^k * T(n,k) are the connection coefficients expressing the polynomial Bell(n,-x) as a linear combination of Bell(k,x), 1 <= k <= n. For example, the list of coefficients of R(4,x) is [26, 36, 12, 1] and we have Bell(4,-x) = -26*Bell(1,x) + 36*Bell(2,x) - 12*Bell(3,x) + Bell(4,x).
The row polynomials also satisfy an analog of the Bernoulli's summation formula for powers of integers, namely, Sum_{k = 1..n} k^[p] = 1/(p+1) * Sum_{k = 0..p} binomial(p+1,k) * B_k * n^[p+1-k], where B_k denotes the Bernoulli numbers. Compare with A195204 and A195205. (End)
Let D be the forward difference operator D(f(x)) = f(x+1) - f(x). Then the n-th row polynomial R(n,x) = 1/f(x) * (x*D)^n(f(x)) with f(x) = 2^x. Cf. A209849. Also cf. A008277, where the row polynomials are given by 1/f(x) * (x*d/dx)^n(f(x)), where now f(x) = exp(x). - Peter Bala, Mar 16 2012
Conjecture: o.g.f. as a continued fraction of Stieltjes type: 1/(1 - x*z/(1 - 2*z/(1 - (x + 1)*z/(1 - 4*z/(1 - (x + 2)*z/(1 - 6*z/(1 - (x + 3)*z/(1 - 8*z/(1 - ... ))))))))) =  1 + x*z + (2*x + x^2)*z^2 + (6*x + 6*x^2 + x^3)*z^3 + .... - Peter Bala, Dec 12 2024

A302922 Raw moments of a Fibonacci-geometric probability distribution.

Original entry on oeis.org

1, 6, 58, 822, 15514, 366006, 10361818, 342239862, 12918651034, 548600581686, 25885279045978, 1343513774912502, 76071145660848154, 4666162902628259766, 308236822886732856538, 21815861409181135034742, 1646982315540717414270874, 132109620398598537723816246

Offset: 0

Albert Gordon Smith, Apr 15 2018

nonn

If F(k) is the k-th Fibonacci number, where F(0)=0, F(1)=1, and F(n)=F(n-1)+F(n-2), then p(k)=F(k-1)/2^k is a normalized probability distribution on the positive integers.
For example, it is the probability that k coin tosses are required to get two heads in a row, or the probability that a random series of k bits has its first two consecutive 1's at the end.
The g.f. for this distribution is g(x) = x^2/(4-2x-x^2) = (1/4)x^2 + (1/8)x^3 + (1/8)x^4 + (3/32)x^5 + ....
The n-th moments about zero of this distribution, known as raw moments, are defined by a(n) = Sum_{k>=1} (k^n)p(k). They appear to be integers and form this sequence.
The e.g.f. for the raw moments is g(e^x) = 1 + 6x + 58x^2/2! + 822x^3/3! + ....
For n >= 1, a(n) appears to be even.
Dividing these terms by 2 gives sequence A302923.
The central moments (i.e., the moments about the mean) also appear to be integers. They form sequence A302924.
The central moments also appear to be even for n >= 1. Dividing them by 2 gives sequence A302925.
The cumulants of this distribution, defined by the cumulant e.g.f. log(g(e^x)), also appear to be integers. They form sequence A302926.
The cumulants also appear to be even for n >= 0. Dividing them by 2 gives sequence A302927.
Note: Another probability distribution on the positive integers that has integral moments and cumulants is the geometric distribution p(k)=1/2^k. The sequences related to these moments are A000629, A000670, A052841, and A091346.

			a(0)=1 is the 0th raw moment of the distribution, which is the total probability.
a(1)=6 is the 1st raw moment, known as the mean of the distribution. It is the arithmetic average of integers following the distribution.
a(2)=58 is the 2nd raw moment. It is the arithmetic average of the squares of integers following the distribution.

Albert Gordon Smith, Table of n, a(n) for n = 0..300
Ignas Gasparavičius, Andrius Grigutis, and Juozas Petkelis, Picturesque convolution-like recurrences and partial sums' generation, arXiv:2507.23619 [math.NT], 2025. See p. 23.
Christopher Genovese, Double Heads

Raw half-moments: A302923.
Central moments: A302924.
Central half-moments: A302925.
Cumulants: A302926.
Half-cumulants: A302927.
Cf. A000629, A000670, A052841, A091346.

Module[{max, r, g},
  max = 17;
  r = Range[0, max];
  g[x_] := x^2/(4 - 2 x - x^2);
  r! CoefficientList[Normal[Series[g[Exp[x]], {x, 0, max}]], x]
]

Vec(serlaplace(exp(2*x)/(4-2*exp(x)-exp(2*x)))) \\ Michel Marcus, Apr 17 2018

In the following,
F(k) is the k-th Fibonacci number, as defined in the Comments.
phi=(1+sqrt(5))/2 is the golden ratio, and psi=(1-sqrt(5))/2.
Li(s,z) is the polylogarithm of order s and argument z.
When s is a negative integer as it is here, Li(s,z) is a rational function of z: Li(-n,z) = (z(d/dz))^n(z/(1-z)).
For n>=0:
a(n) = Sum_{k>=1} ((k^n)(F(k-1)/2^k));
a(n) = Sum_{k>=1} ((k^n)(((phi^(k-1)-psi^(k-1))/sqrt(5))/2^k));
a(n) = (Li(-n,phi/2)/phi-Li(-n,psi/2)/psi)/sqrt(5).
E.g.f.: g(e^x) where g(x) = x^2/(4-2x-x^2) is the g.f. for the probability distribution.
a(n) ~ n! * (5 - sqrt(5)) / (10 * (log(sqrt(5) - 1))^(n+1)). - Vaclav Kotesovec, Apr 13 2022

cp's OEIS Frontend

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A005462 Number of simplices in barycentric subdivision of n-simplex.

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