A000629 -id:A000629 - cp's OEIS frontend

A129062 T(n, k) = [x^k] Sum_{k=0..n} Stirling2(n, k)*RisingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

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, 0, 9366, 19334, 13650, 4270, 630, 42, 1, 0, 94586, 209580, 166376, 62160, 11900, 1176, 56, 1, 0, 1091670, 2562354, 2229444, 952728, 220500, 28476, 2016, 72, 1

Offset: 0

Wolfdieter Lang, May 04 2007

Matrix product of Stirling2 with unsigned Stirling1 triangle.
For the subtriangle without column no. m=0 and row no. n=0 see A079641.
The reversed matrix product |S1|. S2 is given in A111596.
As a product of lower triangular Jabotinsky matrices this is a lower triangular Jabotinsky matrix. See the D. E. Knuth references given in A039692 for Jabotinsky type matrices.
E.g.f. for row polynomials P(n,x):=sum(a(n,m)*x^m,m=0..n) is 1/(2-exp(z))^x. See the e.g.f. for the columns given below.
A048993*A132393 as infinite lower triangular matrices. - Philippe Deléham, Nov 01 2009
Triangle T(n,k), read by rows, given by (0,2,1,4,2,6,3,8,4,10,5,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 19 2011.
Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Triangle 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;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Olivier Bodini, Antoine Genitrini, Cécile Mailler, Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Wolfdieter Lang, First ten rows and more.

Columns k=0..3 give A000007, A000629(n-1), A129063, A129064.

Cf. A000629, A000670, A005649, A079641, A129062, A325872, A325873, A383140.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> polylog(-n,1/2), 9); # Peter Luschny, Jan 27 2016

rows = 9;
t = Table[PolyLog[-n, 1/2], {n, 0, rows}]; T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)
p[n_] := Sum[StirlingS2[n, k] Pochhammer[x, k], {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten (* Peter Luschny, Jun 27 2019 *)

def a_row(n):
    s = sum(stirling_number2(n,k)*rising_factorial(x,k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..9)] # Peter Luschny, Jun 28 2019

a(n,m) = Sum_{k=m..n} S2(n,k) * |S1(k,m)|, n>=0; S2=A048993, S1=A048994.
E.g.f. of column k (with leading zeros): (f(x)^k)/k! with f(x):= -log(1-(exp(x)-1)) = -log(2-exp(x)).
Sum_{0<=k<=n} T(n,k)*x^k = A153881(n+1), A000007(n), A000670(n), A005649(n) for x = -1,0,1,2 respectively. - Philippe Deléham, Nov 19 2011

New name by Peter Luschny, Jun 27 2019

A201339 Expansion of e.g.f. exp(x) / (3 - 2*exp(x)).

Original entry on oeis.org

1, 3, 15, 111, 1095, 13503, 199815, 3449631, 68062695, 1510769343, 37260156615, 1010843385951, 29916558512295, 959183053936383, 33118910817665415, 1225219266296167071, 48348200298184769895, 2027102674516399522623, 89990106205541777926215, 4216915299772659459872991

Offset: 0

Paul D. Hanna, Nov 30 2011

			E.g.f.: E(x) = 1 + 3*x + 15*x^2/2! + 111*x^3/3! + 1095*x^4/4! + 13503*x^5/5! + ...
O.g.f.: A(x) = 1 + 3*x + 15*x^2 + 111*x^3 + 1095*x^4 + 13503*x^5 + ...
where A(x) = 1 + 3*x/(1+x) + 2!*3^2*x^2/((1+x)*(1+2*x)) + 3!*3^3*x^3/((1+x)*(1+2*x)*(1+3*x)) + 4!*3^4*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000629, A004123, A050351, A123227.

[&+[(-1)^(n-j)*3^j*Factorial(j)*StirlingSecond(n,j): j in [0..n]]: n in [0..20]]; // G. C. Greubel, Jun 08 2020

seq(coeff(series( 1/(3*exp(-x) -2) , x, n+1)*n!, x, n), n = 0..30); # G. C. Greubel, Jun 08 2020

Table[Sum[(-1)^(n-k)*3^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 13 2013 *)
With[{nn=20},CoefficientList[Series[Exp[x]/(3-2Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 16 2025 *)

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

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

{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, (-1)^(n-k)*3^k*Stirling2(n, k)*k!)}

[sum( (-1)^(n-j)*3^j*factorial(j)*stirling_number2(n,j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Jun 08 2020

O.g.f.: A(x) = Sum_{n>=0} n! * 3^n*x^n / Product_{k=0..n} (1+k*x).
O.g.f.: A(x) = 1/(1 - 3*x/(1-2*x/(1 - 6*x/(1-4*x/(1 - 9*x/(1-6*x/(1 - 12*x/(1-8*x/(1 - 15*x/(1-10*x/(1 - ...))))))))))), a continued fraction.
a(n) = Sum_{k=0..n} (-1)^(n-k) * 3^k * Stirling2(n,k) * k!.
a(n) = 3*A050351(n) for n>0.
a(n) = Sum_{k=0..n} A123125(n,k)*3^k*2^(n-k). - Philippe Deléham, Nov 30 2011
a(n) ~ n! / (2*log(3/2)^(n+1)). - Vaclav Kotesovec, Jun 13 2013
a(n) = log(3/2) * Integral_{x = 0..oo} (ceiling(x))^n * (3/2)^(-x) dx. - Peter Bala, Feb 06 2015
a(n) = 1 + 2 * Sum_{k=0..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jun 08 2020
From Seiichi Manyama, Nov 15 2023: (Start)
a(0) = 1; a(n) = -3*Sum_{k=1..n} (-1)^k * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3*a(n-1) + 2*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)
a(n) = (3/2)*A004123(n+1) - (1/2)*0^n. - Seiichi Manyama, Dec 21 2023

A201365 Expansion of e.g.f. exp(x) / (5 - 4*exp(x)).

Original entry on oeis.org

1, 5, 45, 605, 10845, 243005, 6534045, 204972605, 7348546845, 296387331005, 13282361478045, 654762261324605, 35211177242722845, 2051349014835939005, 128701394409842982045, 8651475271312083756605, 620334325261670875138845, 47259638324026516284867005

Offset: 0

Paul D. Hanna, Nov 30 2011

			E.g.f.: E(x) = 1 + 5*x + 45*x^2/2! + 605*x^3/3! + 10845*x^4/4! + 243005*x^5/5! + ...
O.g.f.: A(x) = 1 + 5*x + 45*x^2 + 605*x^3 + 10845*x^4 + 243005*x^5 + ...
where A(x) = 1 + 5*x/(1+x) + 2!*5^2*x^2/((1+x)*(1+2*x)) + 3!*5^3*x^3/((1+x)*(1+2*x)*(1+3*x)) + 4!*5^4*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...

G. C. Greubel, Table of n, a(n) for n = 0..358

Cf. A027882, A032183, A050353.
Cf. A201366, A201367, A201368.
Cf. A000629, A201339, A201354.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( 1/(5*Exp(-x) -4) ))); // G. C. Greubel, Jun 08 2020

seq(coeff(series(1/(5*exp(-x) - 4), x, n+1)*n!, x, n), n = 0..20); # G. C. Greubel, Jun 08 2020

Table[Sum[(-1)^(n-k)*5^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 13 2013 *)
With[{nn=20},CoefficientList[Series[Exp[x]/(5-4Exp[x]),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jul 09 2015 *)
a[n_]:= If[n<0, 0, PolyLog[ -n, 4/5]/4]; (* Michael Somos, Apr 27 2019 *)

{a(n)=n!*polcoeff(exp(x+x*O(x^n))/(5 - 4*exp(x+x*O(x^n))), n)}

{a(n)=polcoeff(sum(m=0, n, 5^m*m!*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}

{a(n)=sum(k=0, n, (-1)^(n-k)*5^k*stirling(n, k, 2)*k!)}

[sum( (-1)^(n-j)*5^j*factorial(j)*stirling_number2(n,j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Jun 08 2020

O.g.f.: A(x) = Sum_{n>=0} n! * 5^n*x^n / Product_{k=0..n} (1+k*x).
O.g.f.: A(x) = 1/(1 - 5*x/(1-4*x/(1 - 10*x/(1-8*x/(1 - 15*x/(1-12*x/(1 - 20*x/(1-16*x/(1 - 25*x/(1-20*x/(1 - ...))))))))))), a continued fraction.
a(n) = Sum_{k=0..n} (-1)^(n-k) * 5^k * Stirling2(n,k) * k!.
a(n) = Sum_{k=0..n} A123125(n,k)*5^k*4^(n-k). - Philippe Deléham, Nov 30 2011
a(n) ~ n! / (4*(log(5/4))^(n+1)) . - Vaclav Kotesovec, Jun 13 2013
a(n) = log(5/4) * Integral_{x = 0..oo} (ceiling(x))^n * (5/4)^(-x) dx. - Peter Bala, Feb 14 2015
a(n) = (1/4) Sum_{k>=1} (4/5)^k * n^k. - Michael Somos, Apr 27 2019
a(n) = 1 + 4 * Sum_{k=0..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jun 08 2020
From Seiichi Manyama, Nov 15 2023: (Start)
a(0) = 1; a(n) = -5*Sum_{k=1..n} (-1)^k * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 5*a(n-1) + 4*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)
a(n) = (5/4)*A094417(n) - (1/4)*0^n. - Seiichi Manyama, Dec 21 2023

A005461 Number of simplices in barycentric subdivision of n-simplex.

Original entry on oeis.org

1, 15, 180, 2100, 25200, 317520, 4233600, 59875200, 898128000, 14270256000, 239740300800, 4249941696000, 79332244992000, 1556132497920000, 32011868528640000, 689322235650048000, 15509750302126080000, 364022962973429760000, 8898339094906060800000

Offset: 1

N. J. A. Sloane

			G.f. = x + 15*x^2 + 180*x^3 + 2100*x^4 + 25200*x^5 + 317520*x^6 + ...

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).

G. C. Greubel, Table of n, a(n) for n = 1..440
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, Vol. 10, No. 7 (2022), 1161.

Cf. A001113, A001620, A028246, A091725, A099285.
Cf. A005460, A005462, A005463, A005464, A005465.
Cf. A001297.

[Factorial(n-1)*StirlingSecond(n+3,n): n in [1..35]]; // G. C. Greubel, Nov 23 2022

a:=n->sum((n-j)*n!/4!, j=3..n): seq(a(n), n=4..17); # Zerinvary Lajos, Apr 29 2007

Table[(n(n+1)(n+3)!)/48,{n,20}] (* Harvey P. Dale, Mar 14 2012 *)
a[ n_] := If[ n < 0, 0, n (n + 1) (n + 3)! / 48]; (* Michael Somos, May 27 2014 *)

[factorial(m+1)*binomial(m-1,2)/24 for m in range(3, 19)] # Zerinvary Lajos, Jul 05 2008

[binomial(n,4)*factorial (n-2)/2 for n in range(4, 18)] #  Zerinvary Lajos, Jul 07 2009

a(n) = n*(n + 1)*(n + 3)!/48.
Essentially Stirling numbers of second kind - see A028246.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j) then a(n-3) = (-1)^n*f(n,4,-3), (n>=4). - Milan Janjic, Mar 01 2009
E.g.f.: t*(3*t + 2)/(2*(t - 1)^6). - Ran Pan, Jul 10 2016
a(n) ~ sqrt(Pi/2)*exp(-n)*n^(n+1/2)*(n^5/24 + 85*n^4/288 + 5065*n^3/6912 + 955841*n^2/1244160 + 3710929*n/11943936). - Ilya Gutkovskiy, Jul 10 2016
From Amiram Eldar, May 06 2022: (Start)
Sum_{n>=1} 1/a(n) = 16*(e + gamma - Ei(1)) - 64/3, where e = A001113, gamma = A001620, and Ei(1) = A091725.
Sum_{n>=1} (-1)^(n+1)/a(n) = 32*(gamma - Ei(-1)) - 16/e - 56/3, where Ei(-1) = -A099285. (End)
a(n) = (n-1)! * Stirling2(n+3, n). - G. C. Greubel, Nov 23 2022

More terms from Harvey P. Dale, Mar 14 2012

A084785 Diagonal of the triangle (A084783) and the self-convolution of the first column (A084784).

Original entry on oeis.org

1, 2, 5, 16, 66, 348, 2298, 18504, 176841, 1958746, 24661493, 347548376, 5415830272, 92410046544, 1712819553864, 34258146124320, 735267392077962, 16852848083339700, 410809882438699346, 10611174406149372736, 289493459925589039804, 8317946739043065421640

Offset: 0

Paul D. Hanna, Jun 13 2003

In the triangle (A084783), the diagonal (this sequence) is the self-convolution of the first column (A084784) and the row sums (A084786) gives the differences of the diagonal and the first column.

			G.f.: A(x) = (1-x)^(-1/2)*(1-2*x)^(-1/4)*(1-3*x)^(-1/8)*(1-4*x)^(-1/16)*... - _Paul D. Hanna_, Jun 16 2010

Vaclav Kotesovec, Table of n, a(n) for n = 0..350
Chao-Ping Chen, Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant, Journal of Number Theory, 2016, Volume 172, March 2017, Pages 145-159.
Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 17.

Cf. A000629, A000670, A019538, A084783, A084784, A084786, A090352.

m:=40;
f:= func< n,x | Exp((&+[(&+[(-2)^j*Factorial(j)*StirlingSecond(k,j)*(-x)^k/k: j in [1..k]]): k in [1..n+2]])) >;
R:=PowerSeriesRing(Rationals(), m+1);  // A084785
Coefficients(R!( f(m,x) )); // G. C. Greubel, Jun 08 2023

nmax = 19; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[(1+x)^2 * A[x] - A[x/(1+x)]^2 + O[x]^(n+1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)
With[{m=40}, CoefficientList[Series[Exp[Sum[Sum[(-2)^j*j!*StirlingS2[k, j], {j,k}]*(-x)^k /k, {k,m+1}]], {x,0,m}], x]] (* G. C. Greubel, Jun 08 2023 *)

A = matrix(25, 25); A[1, 1] = 1; rs = 1; print(1); for (n=2, 25, sc = sum(i=2, n-1, A[i, 1]*A[n+1-i, 1]); A[n, 1] = rs - sc; rs = A[n, 1]; for (k=2, n, A[n, k] = A[n, k-1] + A[n-1, k-1]; rs += A[n, k]); print(A[n, n])); \\ David Wasserman, Jan 06 2005

{a(n)=local(A); if(n<0, 0, A=1; for(k=1,n, A=truncate(A+O(x^k))+x*O(x^k); A+=A-(subst(1/A,x,x/(1+x))*(1+x))^-2;); polcoeff(A,n))} /* Michael Somos, Feb 18 2006 */

def f(n, x): return exp(sum(sum( (-2)^j*factorial(j)* stirling_number2(k,j)*(-x)^k/k for j in range(1,k+1)) for k in range(1,n+2)))
m=50
def A084785_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(m,x) ).list()
A084785_list(m-9) # G. C. Greubel, Jun 08 2023

G.f. A(x) satisfies (1+x)^2 = A(x/(1+x))^2/A(x). - Michael Somos, Feb 16 2006
G.f.: A(x) = Product_{n>=1} 1/(1 - n*x)^(1/2^n). - Paul D. Hanna, Jun 16 2010
a(n) ~ (n-1)! / (log(2))^(n+1). - Vaclav Kotesovec, Nov 19 2014
From Peter Bala, May 26 2001: (Start)
O.g.f.: A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ), where b(n) = (-1)^n*Sum_{k = 1..n} k!*Stirling2(n,k)*(-2)^k = A000629(n) = 2*A000670(n) for n >= 1. Cf. A090352.
sqrt(A(x)) = 1/(1 + x)*A(x/(1 + x)) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 137*x^5 + ... is the o.g.f. for A084784. See also A019538. (End)

More terms from David Wasserman, Jan 06 2005

A334282 Number of properly colored labeled graphs on n nodes so that the color function is surjective onto {c_1,c_2,...,c_k} for some k, 1<=k<=n.

Original entry on oeis.org

1, 1, 5, 73, 2849, 277921, 65067905, 35545840513, 44384640206849, 124697899490480641, 778525887500557625345, 10693248499002776513697793, 320453350845793018626300755969, 20807125028666778079876193487790081, 2909872870574162514727072641529432735745

Offset: 0

Geoffrey Critzer, Apr 21 2020

nonn

Also 1 together with the row sums of A046860.

A binary relation R on [n] is periodic iff there is a d>=2 such that R^d = R. Let A be the class of non-arcless strongly connected periodic relations (A000629). Then a(n) is the number of binary relations on [n] whose strongly connected components are in A. - Geoffrey Critzer, Dec 12 2023

Alois P. Heinz, Table of n, a(n) for n = 0..77

Cf. A046860, A322280, A011266, A361560, A003024, A365534, A365590.

b:= proc(n, k) option remember; `if`([n, k]=[0$2], 1,
      add(binomial(n, r)*2^(r*(n-r))*b(r, k-1), r=0..n-1))
    end:
a:= n-> add(b(n,k), k=0..n):
seq(a(n), n=0..15);  # Alois P. Heinz, Apr 21 2020

nn = 15; e2[x_] := Sum[x^n/(n! 2^Binomial[n, 2]), {n, 0, nn}];
Table[n! 2^Binomial[n, 2], {n, 0, nn}] CoefficientList[Series[1/(1 - (e2[x] - 1)), {x, 0, nn}], x]

Sum_{n>=0} a_n*x^n/(n!*2^C(n,2)) = 1/(2-Sum_{n>=0} x^n/(n!*2^C(n,2))).

A052856 E.g.f.: (1-3exp(x)+exp(2x))/(exp(x)-2).

Original entry on oeis.org

1, 2, 4, 14, 76, 542, 4684, 47294, 545836, 7087262, 102247564, 1622632574, 28091567596, 526858348382, 10641342970444, 230283190977854, 5315654681981356, 130370767029135902, 3385534663256845324

Offset: 0

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

Previous name was: A simple grammar.
Stirling transform of A005212(n-1)=[1,1,0,6,0,120,0,...] is a(n-1)=[1,2,4,14,76,...]. - Michael Somos, Mar 04 2004
Stirling transform of (-1)^n*A052612(n-1)=[0,2,-2,12,-24,...] is a(n-1)=[0,2,4,14,76,...]. - Michael Somos, Mar 04 2004
Stirling transform of A000142(n)=[2,2,6,24,120,...] is a(n)=[2,2,4,14,76,...]. - Michael Somos, Mar 04 2004

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 824

A000670(n)=a(n)-1, if n>0. A032109(n)=a(n)/2, if n>0.

A000629, A000670, A002050, A052856, A076726 are all more-or-less the same sequence. - N. J. A. Sloane, Jul 04 2012

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

With[{nn=20},CoefficientList[Series[(1-3Exp[x]+Exp[x]^2)/(-2+Exp[x]),{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Nov 24 2012 *)

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

E.g.f.: (1-3*exp(x)+exp(x)^2)/(-2+exp(x))

a(n) ~ n!/(2*(log(2))^(n+1)). - Vaclav Kotesovec, Oct 05 2013

New name using e.g.f., Vaclav Kotesovec, Oct 05 2013

A076726 a(n) = Sum_{k>=0} k^n/2^k.

Original entry on oeis.org

2, 2, 6, 26, 150, 1082, 9366, 94586, 1091670, 14174522, 204495126, 3245265146, 56183135190, 1053716696762, 21282685940886, 460566381955706, 10631309363962710, 260741534058271802, 6771069326513690646

Offset: 0

Charles G. Waldman (cgw(AT)alum.mit.edu), Oct 27 2002

nonn

			a(0) = 2 because 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 2; a(1) = 2 because 0 + 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + ... = 2.
G.f. = 2 + 2*x + 6*x^2 + 26*x^3 + 150*x^4 + 1082*x^5 + 9366*x^6 + 94586*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 17.
Ramesh L. Srigiriraju, Recurrences for A076726

Same as A000629 except for a(0).

A000629, A000670, A002050, A052856, A076726 are all more-or-less the same sequence. - N. J. A. Sloane, Jul 04 2012

a[n_] := Sum[(k^n)/(2^k), {k, 0, Infinity}]; Table[ a[n], {n, 0, 18}]
a[n_] := (-1)^(n+1) PolyLog[-n, 2] (* Vladimir Reshetnikov, Jan 23 2011 *)

a(n)=abs(polylog(-n, 2)) \\ Charles R Greathouse IV, Jul 15 2014

a(n) = 2*A000670(n). - Philippe Deléham, Mar 06 2004
a(n) ~ n! / (log(2))^(n+1). - Vaclav Kotesovec, Nov 28 2013
From Jianing Song, May 04 2022: (Start)
a(0) = 2, a(n) = Sum_{k=0..n-1} binomial(n,k)*a(k) for n >= 1.
G.f.: Sum_{k>=0} 1/(2^k*(1-k*x)).
E.g.f.: 1/(1-exp(x)/2). (End)

More terms from Robert G. Wilson v, Oct 29 2002

A180875 Sum_{j>=1} j^n2^j/binomial(2j,j) = r_n*Pi/2 + s_n with integer r_n and s_n; sequence gives s_n.

Original entry on oeis.org

1, 3, 11, 55, 355, 2807, 26259, 283623, 3473315, 47552791, 719718067, 11932268231, 215053088835, 4186305575415, 87534887434835, 1956680617267879, 46561960552921315, 1175204650272267479, 31357650670190565363, 881958890078887314567, 26078499305918584929155, 808742391638178302137783

Offset: 0

Jonathan Vos Post, Sep 23 2010

In the references, the infinite series is S_n(2) = A014307(n+1)*Pi/2 + A180875(n) for n >= 1 (and S_0(2) is not defined). - Petros Hadjicostas, May 14 2020

Seiichi Manyama, Table of n, a(n) for n = 0..423
F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011; see Table IV on p. 14.
F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's interesting series, Amer. Math. Monthly, 120 (2013), 116-130; see Table 2.
D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(7) (1985), 449-457.
Feng Qi and Mark Daniel Ward, Closed-form formulas and properties of coefficients in Maclaurin's series expansion of Wilf's function, arXiv:2110.08576 [math.CO], 2021.

The values of r_n give A014307.

Cf. A000629, A129063.

f := n -> sum(j^n*(j!)^2*2^j/(2*j)!, j = 1..infinity):
seq(f(n), n = 0..5); # gives
# [1+(1/2)*Pi, 3+Pi, 11+(7/2)*Pi, 55+(35/2)*Pi, 355+113*Pi, 2807+(1787/2)*Pi].

Table[Expand[FunctionExpand[FullSimplify[Sum[j^n*2^j/Binomial[2*j, j], {j, 1, Infinity}]]]][[1]], {n, 0, 20}] (* Vaclav Kotesovec, May 14 2020 *)

N=20; x='x+O('x^N); f=sqrt(exp(x)/(2-exp(x))); Vec(serlaplace(deriv(f*intformal(f)))) \\ Seiichi Manyama, Oct 22 2019

# An alternative version of the sequence starts (for n >= 0):
# 0, 1, 3, 11, ..., or in terms of the approximation: [(1/2)*Pi, 1+(1/2)*Pi,
# 3+Pi, 11+(7/2)*Pi, ...]. Similar to the formula of Detlef Meya above, the
# sequence then can be computed (without a special initial case) as:
from functools import cache
from math import comb as binomial
@cache
def a(n): return n + sum((binomial(n, j) - 1) * a(n - j) for j in range(1, n))
print([a(n) for n in range(23)])  # Peter Luschny, Jun 09 2023

a(0)=1; if n>=1, then a(n) = a(n-1) + 1 + Sum_{m=1..n} binomial(n,m)*a(n-m). - Detlef Meya, Jan 22 2018
E.g.f.: 2*(arcsin(exp(x/2)/sqrt(2)) - Pi/4) * sqrt(exp(x)/(2-exp(x))^3) + exp(x)/(2-exp(x)). - Seiichi Manyama, Oct 21 2019
a(n) ~ Pi * n^(n+1) / (sqrt(2) * exp(n) * (log(2))^(n + 3/2)). - Vaclav Kotesovec, Oct 22 2019
E.g.f.: d/dx (f(x) * Integral f(x) dx), where f(x) = sqrt(exp(x)/(2-exp(x))), cf. A014307. - Seiichi Manyama, Oct 22 2019

Attribution corrected by M. Lawrence Glasser, Sep 25 2010
Provided a better definition following a suggestion from Herb Conn. - N. J. A. Sloane, Feb 08 2011
Missing a(15) inserted by Seiichi Manyama, Oct 20 2019

A002051 Steffensen's bracket function [n,2].

Original entry on oeis.org

0, 0, 1, 9, 67, 525, 4651, 47229, 545707, 7087005, 102247051, 1622631549, 28091565547, 526858344285, 10641342962251, 230283190961469, 5315654681948587, 130370767029070365, 3385534663256714251, 92801587319328148989, 2677687796244383678827, 81124824998504072833245, 2574844419803190382447051

Offset: 1

N. J. A. Sloane

a(n) is the number of ways to arrange the blocks of the partitions of {1,2,...,n} in an undirected cycle of length 3 or more, see A000629. - Geoffrey Critzer, Nov 23 2012
From Gus Wiseman, Jun 24 2020: (Start)
Also the number of (1,2)-matching length-n sequences covering an initial interval of positive integers. For example, the a(2) = 1 and a(3) = 9 sequences are:
  (1,2)  (1,1,2)
         (1,2,1)
         (1,2,2)
         (1,2,3)
         (1,3,2)
         (2,1,2)
         (2,1,3)
         (2,3,1)
         (3,1,2)
Missing from this list are:
  (1,1)  (1,1,1)
  (2,1)  (2,1,1)
         (2,2,1)
         (3,2,1)
(End)

			a(4) = 9. There are 6 partitions of {1,2,3,4} into exactly three blocks and one way to put them in an undirected cycle of length three. There is one partition of {1,2,3,4} into four blocks and 3 ways to make an undirected cycle of length four. 6 + 3 = 9. - _Geoffrey Critzer_, Nov 23 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Steffensen, J. F. Interpolation. 2d ed. Chelsea Publishing Co., New York, N. Y., 1950. ix+248 pp. MR0036799 (12,164d)

T. D. Noe, Table of n, a(n) for n = 1..100
G. J. Simmons, Letter to N. J. Sloane, May 29 1974
J. F. Steffensen, On a class of polynomials and their application to actuarial problems, Skandinavisk Aktuarietidskrift, Vol. 11, pp. 75-97, 1928.
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

A diagonal of the triangular array in A241168.
(1,2)-avoiding patterns are counted by A011782.
(1,1)-matching patterns are counted by A019472.
(1,2)-matching permutations are counted by A033312.
(1,2)-matching compositions are counted by A056823.
(1,2)-matching permutations of prime indices are counted by A335447.
(1,2)-matching compositions are ranked by A335485.
Patterns are counted by A000670 and ranked by A333217.
Patterns matched by compositions are counted by A335456.
Cf. A056986, A335454, A335465, A335486, A335515.

a[n_] := Sum[ k!*StirlingS2[n-1, k], {k, 0, n-1}] - 2^(n-2); Table[a[n], {n, 3, 17}] (* Jean-François Alcover, Nov 18 2011, after Manfred Goebel *)
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],!GreaterEqual@@#&]],{n,0,5}] (* Gus Wiseman, Jun 24 2020 *)

a(n) = sum(s=2, n-1, stirling(n,s+1,2)*s!/2); \\ Michel Marcus, Jun 24 2020

[n,2] = Sum_{s=2..n-1} Stirling2(n,s+1)*s!/2 (cf. A241168).
a(1)=0; for n >= 2, a(n) = A000670(n-1) - 2^(n-2). - Manfred Goebel (mkgoebel(AT)essex.ac.uk), Feb 20 2000; formula adjusted by N. J. A. Sloane, Apr 22 2014. For example, a(5) = 67 = A000670(4)-2^3 = 75-8 = 67.
E.g.f.: (1 - exp(2*x) - 2*log(2 - exp(x)))/4 = B(A(x)) where A(x) = exp(x)-1 and B(x) = (log(1/(1-x))- x - x^2/2)/2. - Geoffrey Critzer, Nov 23 2012

Entry revised by N. J. A. Sloane, Apr 22 2014

cp's OEIS Frontend

A129062 T(n, k) = [x^k] Sum_{k=0..n} Stirling2(n, k)*RisingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

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A201339 Expansion of e.g.f. exp(x) / (3 - 2*exp(x)).

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A201365 Expansion of e.g.f. exp(x) / (5 - 4*exp(x)).

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A005461 Number of simplices in barycentric subdivision of n-simplex.

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A084785 Diagonal of the triangle (A084783) and the self-convolution of the first column (A084784).

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A334282 Number of properly colored labeled graphs on n nodes so that the color function is surjective onto {c_1,c_2,...,c_k} for some k, 1<=k<=n.

A052856 E.g.f.: (1-3exp(x)+exp(2x))/(exp(x)-2).

A180875 Sum_{j>=1} j^n2^j/binomial(2j,j) = r_n*Pi/2 + s_n with integer r_n and s_n; sequence gives s_n.