A000629 -id:A000629 - cp's OEIS frontend

A241168 Triangle read by rows: T(n,k) (1 <= k <= n) = Steffensen's bracket function [n,n-k].

1, 1, 2, 1, 5, 6, 1, 9, 25, 26, 1, 14, 67, 149, 150, 1, 20, 145, 525, 1081, 1082, 1, 27, 275, 1450, 4651, 9365, 9366, 1, 35, 476, 3430, 15421, 47229, 94585, 94586, 1, 44, 770, 7266, 43281, 180894, 545707, 1091669, 1091670, 1, 54, 1182, 14154, 107751, 581280, 2359225, 7087005, 14174521, 14174522

Offset: 1

N. J. A. Sloane, Apr 22 2014

Steffensen's bracket function [n,k] = Sum_{s=k..n-1} Stirling2(n,s+1)*s!/k!.

The numbers are used in numerical integration.

			Triangle begins:
1,
1, 2,
1, 5, 6,
1, 9, 25, 26,
1, 14, 67, 149, 150,
1, 20, 145, 525, 1081, 1082,
1, 27, 275, 1450, 4651, 9365, 9366,
1, 35, 476, 3430, 15421, 47229, 94585, 94586,
1, 44, 770, 7266, 43281, 180894, 545707, 1091669, 1091670,
...

J. F. Steffensen, On a class of polynomials and their application to actuarial problems, Skandinavisk Aktuarietidskrift, 11 (1928), 75-97.

J. F. Steffensen, On a class of polynomials and their application to actuarial problems, Skandinavisk Aktuarietidskrift, Vol. 11, pp. 75-97, 1928.

Diagonals include A000096, A000629, A002050, A002051, A241169, A241170.

with(combinat);
T:=proc(n,k) add(stirling2(n,s+1)*s!/k!,s=k..n-1); end;
for n from 1 to 12 do lprint([seq(T(n,n-k),k=1..n)]); od:

A242228 a(n) = Sum_{k=1..n} k^(2n-1) k! * Stirling2(n,k).

Original entry on oeis.org

1, 17, 1651, 473741, 300257371, 355743405917, 706872713310331, 2182548723605418941, 9894910566488309801851, 63052832687428562206049117, 545439670961897317869306191611, 6226501736967631584015448186252541, 91619831483112536750163352484302283131

Offset: 1

Vaclav Kotesovec, May 08 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..160

Cf. A000629, A220179, A229260, A244585.

Table[Sum[k^(2*n-1) * k! * StirlingS2[n,k], {k,1,n}], {n,1,20}]

a(n) ~ c * d^n * (n!)^3 / n^2, where d = r^3*(1+exp(2/r)) = 7.8512435106631367719817991716164612615296980032514..., r = 0.94520217245242431308104743874492469552738... is the root of the equation (1+exp(-2/r))*LambertW(-exp(-1/r)/r) = -1/r, and c = 0.15095210978787998524366903417512193343948127919...

E.g.f.: Sum_{k>=1} (exp(k^2*x) - 1)^k / k. - Seiichi Manyama, Jun 19 2024

A259533 Number of restricted barred preferential arrangements of an n-set having 3 bars in which 3 fixed sections are restricted sections and 1 section is a free section.

Original entry on oeis.org

1, 4, 18, 94, 582, 4294, 37398, 378214, 4366422, 56697574, 817979478, 12981058534, 224732536662, 4214866778854, 85130743747158, 1842265527790054, 42525237455785302, 1042966136232956134, 27084277306054500438, 742412698554626764774, 21421502369955072576342, 648998599988032588957414

Offset: 0

Sithembele Nkonkobe, Jul 02 2015

Also, number of preferential fuzzy subsets of length n+2 where the keychains are of length n+2.
Binomial transform of A007047.
Double binomial transform of A000629.
Conjecture: for fixed k = 1,2,..., the sequence obtained by reducing a(n) modulo k is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010. For example, modulo 24 the sequence becomes [1, 4, 18, 22, 6, 22, 6, 22, 6, ...] with an apparent period of 2 beginning at a(3). - Peter Bala, Jul 08 2022

Robert Israel, Table of n, a(n) for n = 0..387
V. Murali, Ordered partitions and finite fuzzy sets, Far East J. Math. Sci.(FJMS), 21(2006), 12-132.
S. Nkonkobe and V. Murali, A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements, arXiv:1503.06172 [math.CO], 2015.

Cf. A000670, A000629, A007047.

S:= series(exp(3*x)/(2-exp(x)),x,31):
seq(coeff(S,x,j)*j!, j=0..30); # Robert Israel, Aug 11 2015

Range[0, 25]! CoefficientList[Series[E^(3 x)/(2 - E^(x)), {x, 0, 25}], x] (* Vincenzo Librandi, Jul 06 2015 *)

{ my(x = xx + O(xx^40)); Vec(serlaplace(exp(3*x)/(2-exp(x)))) } \\ Michel Marcus, Jul 06 2015

E.g.f.: exp(3*x)/(2-exp(x)).
a(n) = 3^n + Sum_{k = 0..n-1} binomial(n,k)*a(k). - Robert Israel, Aug 11 2015
a(n) ~ 4*n! / (log(2))^(n+1). - Vaclav Kotesovec, Sep 27 2017
a(n) = Sum_{k>=0} (k + 3)^n / 2^(k+1). - Ilya Gutkovskiy, Jun 27 2020
a(n) = 8*A000670(n) - (2^n + 2 + 4*0^n). - Seiichi Manyama, Dec 21 2023

More terms from Michel Marcus, Jul 06 2015

A344489 a(n) = 1 + Sum_{k=0..n-2} binomial(n-1,k) * a(k).

Original entry on oeis.org

1, 1, 2, 4, 11, 34, 122, 487, 2144, 10276, 53165, 294760, 1740950, 10899841, 72033470, 500664496, 3648211139, 27792215302, 220802394110, 1825428024367, 15672798590804, 139499676115312, 1285109772354941, 12235037442987028, 120220980122266010, 1217655627762149857

Offset: 0

Ilya Gutkovskiy, May 21 2021

nonn

Cf. A000296, A000629, A007476, A186021, A337186, A344490, A344491, A344492, A344493.

a[n_] := a[n] = 1 + Sum[Binomial[n - 1, k] a[k] , {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
nmax = 25; A[] = 0; Do[A[x] = (1 + x A[x/(1 - x)])/(1 - x^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = (1 + x * A(x/(1 - x))) / (1 - x^2).

A354413 Expansion of e.g.f. 1/(2 - exp(x))^x.

Original entry on oeis.org

1, 0, 2, 6, 36, 250, 2100, 20594, 231168, 2923722, 41149140, 637972522, 10804678632, 198480649250, 3930963078588, 83500876635570, 1893745346613216, 45672635292831322, 1167233799092342148, 31510575263852229242, 896028017040096045720

Offset: 0

Seiichi Manyama, May 25 2022

nonn

Cf. A000629, A000670, A052862, A351739, A354412.

With[{nn=20},CoefficientList[Series[1/(2-Exp[x])^x,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 03 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x))^x))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*sum(k=1, j-1, (k-1)!*stirling(j-1, k, 2))*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A052862(k) * binomial(n-1,k-1) * a(n-k).

a(n) ~ n! / (Gamma(log(2)) * 2^log(2) * n^(1 - log(2)) * log(2)^(n + log(2))). - Vaclav Kotesovec, Jun 08 2022

A027882 a(n) = Sum_{k>=1} k^n (2/3)^k.

Original entry on oeis.org

2, 6, 30, 222, 2190, 27006, 399630, 6899262, 136125390, 3021538686, 74520313230, 2021686771902, 59833117024590, 1918366107872766, 66237821635330830, 2450438532592334142, 96696400596369539790

Offset: 0

Stephen J. Greenfield (greenfie(AT)math.rutgers.edu)

nonn

C. G. Bower, Transforms
Steffen Greenfield, Source
Index entries for sequences related to necklaces

Cf. A000629, A032183.

Table[ PolyLog[n, 2/3], {n, 0, -18, -1}] (* Robert G. Wilson v, Aug 05 2010 *)
Table[Sum[StirlingS2[n, k] * (k-1)! * 2^k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jul 12 2018 *)

a(n)=polylog(-n,2/3) \\ Charles R Greathouse IV, Aug 27 2014

Also "CIJ" (necklace, indistinct, labeled) transform of 2, 2, 2, 2...
E.g.f. (for offset 1): -log(3-2*exp(x)). Sum_{k=1..n} 2^k*(k-1)!*Stirling2(n, k). - Vladeta Jovovic, Sep 14 2003
a(n) ~ n! / (log(3/2))^(n+1). - Vaclav Kotesovec, Oct 07 2013

More terms from Christian G. Bower

A091804 a(n) = 2^(n(n+1)/2)A055209(n).

Original entry on oeis.org

1, 2, 32, 9216, 84934656, 39137889484800, 1298501242170900480000, 4221953171600428240994304000000, 1757091687362157576297527910177177600000000

Offset: 0

Philippe Deléham, Mar 07 2004

nonn

This sequence is the Hankel transform (see A001906 for definition) of A000629, A000670 (Fubini numbers) and A052841.

G. C. Greubel, Table of n, a(n) for n = 0..30
P. Barry, Eulerian Polynomials as Moments, via Exponential Riordan Arrays, J. Int. Seq. 14 (2011) # 11.9.5.
P. Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2. [_N. J. A. Sloane_, Dec 27 2012]

Cf. A000629, A000670, A001906, A052841, A055209.

[(&*[(2*(k+1)*Floor((2*k+3)/2))^(n-k): k in [0..n]]): n in [0..10]]; // G. C. Greubel, Oct 14 2018

Table[Product[(2(k+1)Floor[(2k+3)/2])^(n-k),{k,0,n}],{n,0,10}] (* Harvey P. Dale, Aug 10 2011 *)

for(n=0,10, print1(prod(k=0,n, (2*(k+1)*floor((2*k+3)/2))^(n-k)), ", ")) \\ G. C. Greubel, Oct 14 2018

a(n) = Product{k=0..n} (2(k+1)*floor((2k+3)/2))^(n-k). - Paul Barry, Mar 30 2010

A129063 Third column (m=2) of triangle A129062 and second column of triangle A079641.

Original entry on oeis.org

1, 6, 36, 250, 2040, 19334, 209580, 2562354, 34915680, 524986462, 8636859924, 154357103498, 2978418173640, 61718786864310, 1367098836863868, 32236969655283682, 806313056758966320, 21322699350055313678, 594440128269066768612, 17424632249851351374906

Offset: 0

Wolfdieter Lang, May 04 2007

Seiichi Manyama, Table of n, a(n) for n = 0..422

A000629 gives second, resp. first column of A129062, resp. A079641.

Cf. A180875.

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

a(n) = A129062(n+2,2), n>=0.
a(n) = A079641(n+1,1), n>=0.
E.g.f.: (d^2/dx^2)((-log(2-exp(x)))^2)/2.
E.g.f.: d/dx (f(x) * Integral f(x) dx), where f(x) = exp(x)/(2-exp(x)), cf. A000629. - Seiichi Manyama, Oct 22 2019
a(n) ~ n! * n * log(n) / (log(2))^(n+2) * (1 + (gamma - log(2) - log(log(2))) / log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 22 2019

A162312 Triangular array, inverse of 2*P - I, where P is Pascal's triangle and I is the identity matrix.

Original entry on oeis.org

1, -2, 1, 6, -4, 1, -26, 18, -6, 1, 150, -104, 36, -8, 1, -1082, 750, -260, 60, -10, 1, 9366, -6492, 2250, -520, 90, -12, 1, -94586, 65562, -22722, 5250, -910, 126, -14, 1, 1091670, -756688, 262248, -60592, 10500, -1456, 168, -16, 1, -14174522, 9825030

Offset: 0

Peter Bala, Jul 01 2009

We make a few remarks about the general array M(a) := (I - a*P)^-1, where a <> 1, and its connection with weighted sums of powers of positive integers. The present case corresponds to -M(2).
The array M(a) begins
/
| 1/(1-a)
| a/(1-a)^2............... 1/(1-a)
| (a+a^2)/(1-a)^3......... 2*a/(1-a)^2........ 1/(1-a)
| (a+4*a^2+a^3)/(1-a)^4... 3*(a+a^2)/(1-a)^3.. 3*a/(1-a)^2... 1/(1-a)
| ...
\
In the first column the numerator polynomials are the Eulerian polynomials A_n(a). See A008292.
The e.g.f. for this array is
(1)... exp(x*t)/(1-a*exp(t)) = 1/(1-a) + [a/(1-a)^2 + x/(1-a)]*t
+ [(a+a^2)/(1-a)^3 + 2*a*x/(1-a)^2 + x^2/(1-a)]*t^2/2! + ....
The row generating polynomials P_m(x) of the array M(a), which, of course, depend on a, have properties similar to those of the Bernoulli polynomials. They form an Appell sequence and may be expressed in terms of the Eulerian polynomials as
(2)... P_m(x) = sum {k=0..m} binomial(m,k) * A_k(a) / (1-a)^(k+1) * x^(m-k).
As a Newton series we have
(3)... P_m(x) = 1/(1-a)*sum {j = 0..m} sum {k = j..m}(a/(1-a))^j * k! * Stirling2(m,k) * binomial(x,k-j).
The proof of this result in the particular case a = -1 given in [Roman, p. 100] can be easily generalized to a proof of (3).
A result equivalent to (3) is
(4)... P_m(x) = 1/(1-a)*sum {j = 0..m} sum {k = 0..j} (a/(1-a))^j * (-1)^(j-k) * comb(j,k) * (x + k)^m,
which in turn leads to the infinite series expansion
(5)... P_m(x) = sum {k = 0..inf} a^k * (x + k)^m,
provided |a| < 1. See [Nelsen].
The polynomials P_m(x) satisfy the difference equation
(6)... P_m(x) - a*P_m(x + 1) = x^m (recall a <> 1),
which leads easily to the evaluation of the weighted sums of powers of integers
(7)... sum {k = 0..n-1} a^k * k^m = P_m(0) - a^n * P_m(n).
for m = 0,1,2,... and a <> 1.
More generally we have
(8)... sum {k = 0..n-1} a^k * (x + k)^m = P_m(x) - a^n * P_m(x + n).
for m = 0,1,2,... and a <> 1.
In the remaining case a = 1 the sums are evaluated in terms of the Bernoulli polynomials.
The most well-studied case is when a = -1. The row polynomials of the array M(-1) are then one half of the Euler polynomials E_m(x), which may be used to evaluate the alternating sums of powers of integers
(9)... 2*sum {k = 1..n-1} (-1)^k * k^m = E_m(0) - (-1)^n * E_m(n).

			Triangle begins
====================================================
n\k|.....0......1......2......3......4......5......6
====================================================
0..|.....1
1..|....-2......1
2..|.....6.....-4......1
3..|...-26.....18.....-6......1
4..|...150...-104.....36.....-8......1
5..|.-1082....750...-260.....60....-10......1
6..|..9366..-6492...2250...-520.....90....-12......1
...

S. Roman, The Umbral Calculus, Dover Publications.

G. F. C. De Bruyn, Formulas for a + a^2*2^p + a^3*3^p + ... + a^n*n^p, Fibonacci Quart. 33 (1995), no. 2, 98-103.
R. B. Nelsen, Problem E3062, Amer. Math. Monthly, Vol. 94, No. 4 (Apr., 1987), 376-377.

Cf. A000629, A000670, A007047, A008292, A080253, A154921.

#A162312
with(combinat):
T := (n,k) -> (-1)^(n+k)*binomial(n,k)
*add( j!*stirling2(n-k+1,j+1),j = 0..n):
for n from 0 to 9 do
seq(T(n,k), k = 0..n);
end do;

Table[(-1)^(n+k) Binomial[n, k] PolyLog[k-n, 1/2], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 14 2019 *)

matrix(10, 10, n, k, 2*binomial(n-1,k-1) - (n==k))^(-1) \\ Michel Marcus, Jul 12 2018

TABLE ENTRIES
(1)... T(n,k) = (-1)^(n+k) * binomial(n,k) * A000629(n-k).
(2)... T(n,k) = (-1)^(n+k) * binomial(n,k) * sum {j = 0..n} j! * Stirling2(n-k+1,j+1).
GENERATING FUNCTION
(3)... exp(x*t)/(2*exp(t)-1) = 1 + (-2+x)*t + (6-4*x+x^2)*t^2/2!
+ ....
PROPERTIES OF ROW POLYNOMIALS
The row generating polynomials R_n(x) form an Appell sequence. The first few values are R_0(x) = 1, R_1(x) = x-2, R_2(x) = x^2-4*x+6 and R_3(x) = x^3-6*x^2+18*x-26.
They may be recursively computed by means of
(4)... R_n(x) = x^n - 2*sum {k = 0..n-1} binomial(n,k) * R_k(x).
Explicit formulas are
(5)... R_n(x) = sum {j = 0..n} sum {k = j..n} (-2)^j * k! * Stirling2(n,k) * binomial(x,k-j),
(6)... R_n(x) = (-1)^n * sum {j = 0..n} sum {k = j..n} k! * Stirling2(n,k) * binomial(-x+1,k-j),
and
(7)... R_n(x) = sum {j = 0..n} sum {k = 0..j} 2^j * (-1)^k * comb(j,k) * (x + k)^n.
Other expansions include
(8)... R_n(x) = sum {k = 0..n} binomial(n,k) * (-1)^k * A000670(k) * (x-1)^(n-k),
(9)... R_n(x) = sum {k = 0..n} binomial(n,k) * (-1/2)^k * A080253(k) * (x-1/2)^(n-k)
and
(10)... R_n(x) = sum {k = 0..n} binomial(n,k) * (-1)^k * A007047(k) * (x+1)^(n-k).
SUMS OF POWERS OF INTEGERS
The row polynomials satisfy the difference equation
(11)... 2*R_n(x+1) - R_n(x) = x^n,
and so may be used to evaluate the weighted sum of powers of integers
(12)... sum {k = 0..n-1} 2^k * k^m = 2^n*R_m(n) - R_m(0).
For example, m = 3 gives
(13)... sum {k = 0..n-1} 2^k * k^3 = 2^n*(n^3-6*n^2+18*n-26) + 26.
More generally we have
(14)... sum {k = 0..n-1} 2^k * (x + k)^m = 2^n * R_m(x + n) - R_m(x).
RELATIONS WITH OTHER SEQUENCES
(15)... Row sums [1,-1,3,-13,75,...] = (-1)^n*A000670(n).
(16)... Alt. row sums [1,-3,11,-51,299,...] = (-1)^n * A007047(n).
(17)... Column 0: (-1)^n * A000629(n).
(18)... (-2)^n * R_n(1/2) = A080253(n).
(19)... R_n(1-x) = (-1)^n * P_n(x),
where P_n(x) are the row generating polynomials of A154921.
This provides the connection between (12) and the result
(20)... sum {k = 0..n-1} (1/2)^k * k^m = 2*P_m(0) - (1/2)^(n-1) * P_m(n).

Typo corrected by Peter Bala, Nov 05 2010

A179929 a(n) = 2^n*A(n, -1/2), A(n, x) the Eulerian polynomials.

Original entry on oeis.org

1, 2, 2, -6, -30, 42, 882, 954, -39870, -203958, 2300562, 29677914, -120958110, -4657703958, -7059175758, 807984602874, 6667870853250, -145556787011958, -2827006784652078, 21703953751815834, 1108558810703202210

Offset: 0

Peter Luschny, Aug 03 2010

sign

Stanislav Sykora, Table of n, a(n) for n = 0..199
OEIS Wiki, Eulerian polynomials

Cf. A000629 = 2^n*A(n, 1/2).

Cf. A212846, A123125, A173018, A195205.

a[n_] := Sum[3^(n-k) (-1)^k k! StirlingS2[n+1, k+1], {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 17 2019 *)

A179929(n) = {my(s,k,term);
  term = 3^n;s = term*stirling(n+1,1,2);
  for (k=1,n,term *= -k/3;s += term*stirling(n+1,k+1,2););
return(s);} \\ Stanislav Sykora, May 15 2014

From Philippe Deléham, Dec 22 2011: (Start)
a(n) = Sum_{k, 0<=k<=n} A123125(n,k)*(-1)^(n-k)*2^k
a(n) = Sum_{k, 0<=k<=n} A173018(n,k)*2^(n-k)*(-1)^k. (End)
From Peter Bala, Mar 12 2013: (Start)
E.g.f.: 3/(1 + 2exp(-3x)) = 1 + 2x + 2x^2/2! - 6x^3/3! - 30x^4/4! + ....
Recurrence equation: a(n+1) = 3a(n) - Sum_{k=0..n} binomial(n,k) a(k)a(n-k).
(-1)^n*a(n) are the coefficients of a delta operator associated with a sequence of polynomials of binomial type - see A195205. (End)
a(n) ~ n! * 2*3^(n+1)/(Pi^2+(log(2))^2)^(n/2+1) * (Pi*sin(n*arctan(Pi/log(2))) - log(2)*cos(n*arctan(Pi/log(2)))). - Vaclav Kotesovec, Oct 09 2013
From Stanislav Sykora, May 15 2014: (Start)
a(n) = -2*A212846(n) for n > 0.
a(n) = -3^(n+1)*Li(-n, -1/2), with Li(-n, x) = Sum_{k>=0} ((k^n)*(x^k)) the polylogarithm.
a(n) = Sum_{k=0..n} 3^(n-k)*(-1)^k*k!*S(n+1, k+1), S(m, l) the Stirling number of second kind. (End)

cp's OEIS Frontend

A241168 Triangle read by rows: T(n,k) (1 <= k <= n) = Steffensen's bracket function [n,n-k].

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A242228 a(n) = Sum_{k=1..n} k^(2*n-1) * k! * Stirling2(n,k).

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A259533 Number of restricted barred preferential arrangements of an n-set having 3 bars in which 3 fixed sections are restricted sections and 1 section is a free section.

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A344489 a(n) = 1 + Sum_{k=0..n-2} binomial(n-1,k) * a(k).

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A354413 Expansion of e.g.f. 1/(2 - exp(x))^x.

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A027882 a(n) = Sum_{k>=1} k^n (2/3)^k.

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A091804 a(n) = 2^(n*(n+1)/2)*A055209(n).

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Magma

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A129063 Third column (m=2) of triangle A129062 and second column of triangle A079641.

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A242228 a(n) = Sum_{k=1..n} k^(2n-1) k! * Stirling2(n,k).

A091804 a(n) = 2^(n(n+1)/2)A055209(n).