A216794 -id:A216794 - cp's OEIS frontend

A080253 a(n) is the number of elements in the Coxeter complex of type B_n (or C_n).

1, 3, 17, 147, 1697, 24483, 423857, 8560947, 197613377, 5131725123, 148070287697, 4699645934547, 162723741209057, 6103779096411363, 246564971326084337, 10671541841672056947, 492664975795819140737, 24166020791610523843203

Offset: 0

Paul Boddington and Tim Honeywill, Feb 10 2003

There is a nice geometric interpretation. Let V be a Euclidean space containing a root system of type B_n. We can decompose V into a disjoint union of 'cells', a cell being simply a maximal connected subset C of V with the property that if C has nonempty intersection with the orthogonal complement of some root a, then C lies entirely within the orthogonal complement of a. a(n) is then the number of cells.
For example, if n=2 then we can take V=R^2 and the roots to be (1,0), (0,1), (1,1), (-1, -1) and their negatives. The 17 cells are as follows: the set containing the origin O; the eight "open" halflines radiating from O and containing a root (but not O); the eight connected components of V minus the union of the nine cells already described. The corresponding sequences for types A,D are A000670, A080254 respectively.
Also number of signed orders.

			a(2)=17 as follows. Let (W,S) be a Coxeter system of type B_2. By definition the elements of the associated complex are right cosets of "special parabolic subgroups". These are simply the subgroups generated by subsets of S. In our case they have orders 1,2,2,8 and hence have 8,4,4,1 cosets respectively, giving a total of 17.

Kenneth S. Brown, Buildings, Springer-Verlag, 1989.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Peter C. Fishburn, Signed Orders, Choice Probabilities and Linear Polytopes, Journal of Mathematical Psychology, Volume 45, Issue 1, (2001), pp. 53-80.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
Eric Weisstein's MathWorld, Polylogarithm.

Cf. A000670, A080254, A216794.

A080253 := proc(n) option remember; local k; if n <1 then 1 else 1 + add(2^r*binomial(n,r)*A080253(n-r),r=1..n); fi; end; seq(A080253(n),n=0..30); # Detlef Pauly

t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[c[n], {n, 0, 100}] (* Emanuele Munarini, Oct 04 2012 *)
CoefficientList[Series[E^x/(2-E^(2*x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 07 2015 *)
Round@Table[(-1)^(n + 1) (PolyLog[-n, Sqrt[2]] - PolyLog[-n, -Sqrt[2]])/(2 Sqrt[2]), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)

t(n):=sum(stirling2(n,k)*k!,k,0,n);
c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);
makelist(c(n),n,0,40); // Emanuele Munarini, Oct 04 2012

def A080253(n):
    return add(A060187(n, k) << (n-k) for k in (0..n))
[A080253(n) for n in (0..17)]  # Peter Luschny, Apr 26 2013

a(n) = 1 + Sum_{r=1..n} 2^r *binomial(n, r) *a(n-r).
E.g.f.: exp(x)/(2-exp(2*x)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 14 2003
a(n) = Sum_{t=0..n} binomial(n, t)*2^(n-t)*A000670(n-t). Fishburn 2001, p. 57.
a(n) = Sum_{k=0..n} Stirling2(n, k)*k!*A001333(k+1). - Vladeta Jovovic, Sep 28 2003
2*a(n) = Sum_{k>=0} (2*k+1)^n/2^k = 2^n*LerchPhi(1/2,-n,1/2). - Gerson Washiski Barbosa, May 11 2009, Dec 12 2010
An approximation formula can be derived from the latter, a(n) ~ (n!/(2*sqrt(2)))*(2/log(2))^(n+1), with relative errors approaching asymptotically zero as n increases. - Gerson Washiski Barbosa, Jun 26 2009
Half the row sums of triangle A154695. - Gerson Washiski Barbosa, Jun 26 2009
G.f.: 1 + x/G(0) where G(k) = 1 - x*3*(2*k+1) + x^2*(k+1)*(k+1)*(1-3^2)/G(k+1); (continued fraction due to Stieltjes). - Sergei N. Gladkovskii, Jan 11 2013
a(n) = Sum_{k = 0..n} A060187(n, k)*2^(n-k). - Peter Luschny, Apr 26 2013
G.f.: 1/Q(0), where Q(k) = 1 - 3*x*(2*k+1) - 8*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 28 2013
a(n) = log(2) * Integral_{x = 0..oo} (2*floor(x) + 1)^n * 2^(-x) dx. - Peter Bala, Feb 06 2015
From Vladimir Reshetnikov, Oct 31 2015: (Start)
a(n) = (-1)^(n+1)*(Li_{-n}(sqrt(2)) - Li_{-n}(-sqrt(2)))/(2*sqrt(2)), where Li_n(x) is the polylogarithm.
Li_{-n}(sqrt(2)) = (-1)^(n+1)*(2*A216794(n) + a(n)*sqrt(2)).
(End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 14 2003

A000556 Expansion of exp(-x) / (1 - exp(x) + exp(-x)).

Original entry on oeis.org

1, 1, 5, 31, 257, 2671, 33305, 484471, 8054177, 150635551, 3130337705, 71556251911, 1784401334897, 48205833997231, 1402462784186105, 43716593539939351, 1453550100421124417, 51350258701767067711, 1920785418183176050505, 75839622064482770570791

Offset: 0

N. J. A. Sloane

Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin's summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..401
Gregory Dresden, On the Brousseau sums Sum_{i=1..n} i^p*Fibonacci(i), arxiv.org:2206.00115 [math.NT], 2022.
Paul Kinlaw, Michael Morris, and Samanthak Thiagarajan, Sums related to the Fibonacci sequence, Husson University (2021).
G. Ledin, Jr., On a certain kind of Fibonacci sums, Fib. Quart., 5 (1967), 45-58.
R. L. Ollerton and A. G. Shannon, A Note on Brousseau's Summation Problem, Fibonacci Quart. 58 (2020), no. 5, 190-199.
Eric Weisstein's MathWorld, Polylogarithm.
Eric Weisstein's MathWorld, Golden Ratio.
Eric Weisstein's MathWorld, Lucas Number.

Cf. A005923, A216794.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*(2^j-1), j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 05 2019

CoefficientList[Series[E^(-x)/(1-E^x+E^(-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, May 04 2015 *)
Round@Table[(-1)^(n+1) (PolyLog[-n, 1-GoldenRatio] GoldenRatio + PolyLog[-n, GoldenRatio]/GoldenRatio)/Sqrt[5], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 30 2015 *)

a(n) = sum(k=0, n, k!*fibonacci(k+1)*stirling(n, k, 2)); \\ Michel Marcus, Oct 30 2015

a(n) = Sum_{k=0..n} k!*Fibonacci(k+1)*Stirling2(n,k).
E.g.f.: 1/(1 + U(0)) where U(k) = 1 - 2^k/(1 - x/(x - (k+1)*2^k/U(k+1) )); (continued fraction 3rd kind, 3-step ). - Sergei N. Gladkovskii, Dec 05 2012
a(n) ~ 2*n! / ((5+sqrt(5)) * log((1+sqrt(5))/2)^(n+1)). - Vaclav Kotesovec, May 04 2015
a(n) = (-1)^(n+1)*(Li_{-n}(1-phi)*phi + Li_{-n}(phi)/phi)/sqrt(5), where Li_n(x) is the polylogarithm, phi=(1+sqrt(5))/2 is the golden ratio. - Vladimir Reshetnikov, Oct 30 2015
John W. Layman observes that this is also Sum (-2)^k*binomial(n, k)*b(n-k), where b() = A005923.
From Greg Dresden, May 13 2022 (Start):
For n > 0, a(n) = 1 + 2*Sum_{k=0..floor(n/2-1)} binomial(n,2*k+1) * a(n-2*k-1).
For n > 0, a(n) = Sum_{k=0..n-1} binomial(n,k)*A000557(k).
(End)

A336950 E.g.f.: 1 / (1 - x * exp(2*x)).

Original entry on oeis.org

1, 1, 6, 42, 392, 4600, 64752, 1063216, 19952256, 421227648, 9880951040, 254960721664, 7176891675648, 218857588139008, 7187394935347200, 252897556424140800, 9491754142468702208, 378509920569294684160, 15982018774576565649408, 712306819507400060502016

Offset: 0

Ilya Gutkovskiy, Aug 08 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..394

Column k=2 of A351790.

Cf. A001787, A006153, A122704, A216794, A235328, A336947, A336951, A336952.

nmax = 19; CoefficientList[Series[1/(1 - x Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(2 (n - k))^k/k!, {k, 0, n}], {n, 1, 19}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

seq(n)={ Vec(serlaplace(1 / (1 - x*exp(2*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020

a(n) = n! * Sum_{k=0..n} (2 * (n-k))^k / k!.
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 2^(k-1) * a(n-k).
a(n) ~ n! * (2/LambertW(2))^n / (1 + LambertW(2)). - Vaclav Kotesovec, Aug 09 2021

A367977 Expansion of e.g.f. exp(-x) / (2 - exp(2*x)).

Original entry on oeis.org

1, 1, 9, 73, 849, 12241, 211929, 4280473, 98806689, 2565862561, 74035143849, 2349822967273, 81361870604529, 3051889548205681, 123282485663042169, 5335770920836028473, 246332487897909570369, 12083010395805261921601, 627555570373369525058889, 34404109751876393769480073

Offset: 0

Ilya Gutkovskiy, Dec 07 2023

nonn

G. C. Greubel, Table of n, a(n) for n = 0..380

Cf. A000670, A052841, A080253, A216794, A344037, A367979, A367980, A367981, A367982, A367983.

R:=PowerSeriesRing(Rationals(), 50);
Coefficients(R!(Laplace( Exp(-x)/(2-Exp(2*x)) ))) // G. C. Greubel, Jun 10 2024

nmax = 19; CoefficientList[Series[Exp[-x]/(2 - Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-1)^n + Sum[Binomial[n, k] 2^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

def A367977_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-x)/(2-exp(2*x)) ).egf_to_ogf().list()
A367977_list(50) # G. C. Greubel, Jun 10 2024

a(n) = Sum_{k>=0} (2*k-1)^n / 2^(k+1).
a(n) = (-1)^n + Sum_{k=1..n} binomial(n,k) * 2^k * a(n-k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 2^k * A000670(k).

A328182 Expansion of e.g.f. 1 / (2 - exp(3*x)).

Original entry on oeis.org

1, 3, 27, 351, 6075, 131463, 3413907, 103429791, 3581223435, 139498558263, 6037616347587, 287444492409231, 14929010774254395, 839982382565841063, 50897213545996785267, 3304312091004451756671, 228821504027595115886955, 16836102104577636004291863, 1311625494765417347634022947

Offset: 0

Ilya Gutkovskiy, Oct 06 2019

nonn

Cf. A000670, A216794, A247452, A328183.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*3^j, j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 06 2019

nmax = 18; CoefficientList[Series[1/(2 - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[3^k Binomial[n, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
Table[3^n HurwitzLerchPhi[1/2, -n, 0]/2, {n, 0, 18}]

a(0) = 1; a(n) = Sum_{k=1..n} 3^k * binomial(n,k) * a(n-k).
a(n) = Sum_{k>=0} (3*k)^n / 2^(k + 1).
a(n) = 3^n * A000670(n).
a(n) ~ n! * 3^n / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Aug 09 2021

A352117 Expansion of e.g.f. 1/sqrt(2 - exp(2*x)).

Original entry on oeis.org

1, 1, 5, 37, 377, 4921, 78365, 1473277, 31938737, 784384561, 21523937525, 652667322517, 21672312694697, 782133969325801, 30481907097849485, 1275870745561131757, 57083444567425884257, 2718602143583362124641, 137315150097164841942245

Offset: 0

Seiichi Manyama, Mar 05 2022

nonn

Cf. A000670, A097397, A216794, A352118, A352119, A305404, A346982 - A346985.

m = 18; Range[0, m]! * CoefficientList[Series[(2 - Exp[2*x])^(-1/2), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)

a[n]:=if n=0 then 1 else sum(a[n-k]*(1-k/n/2)*binomial(n,k)*2^k,k,1,n);
makelist(a[n],n,0,50); /* Tani Akinari, Sep 06 2023 */

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(2-exp(2*x))))

a(n) = sum(k=0, n, 2^(n-k)*prod(j=0, k-1, 2*j+1)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} 2^(n-k) * (Product_{j=0..k-1} (2*j+1)) * Stirling2(n,k).
a(n) ~ 2^n * n^n / (log(2)^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Mar 05 2022
Conjectural o.g.f. as a continued fraction of Stieltjes type: 1/(1 - x/(1 - 4*x/(1 - 3*x/(1 - 8*x/(1 - ... - (2*n-1)*x/(1 - 4*n*x/(1 - ... ))))))). Cf. A346982.  - Peter Bala, Aug 22 2023
For n > 0, a(n) = Sum_{k=1..n} a(n-k)*(1-k/n/2)*binomial(n,k)*2^k. - Tani Akinari, Sep 06 2023
a(0) = 1; a(n) = a(n-1) - 2*Sum_{k=1..n-1} (-2)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 18 2023

A328183 Expansion of e.g.f. 1 / (2 - exp(4*x)).

Original entry on oeis.org

1, 4, 48, 832, 19200, 553984, 19181568, 774848512, 35771842560, 1857882947584, 107214340620288, 6805814291464192, 471298297319915520, 35356865248765149184, 2856513752723261227008, 247264693517100223823872, 22830563015939200206766080, 2239752722978295095737974784

Offset: 0

Ilya Gutkovskiy, Oct 06 2019

nonn

Cf. A000670, A216794, A328182.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*4^j, j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 06 2019

nmax = 17; CoefficientList[Series[1/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[4^k Binomial[n, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
Table[2^(2 n - 1) HurwitzLerchPhi[1/2, -n, 0], {n, 0, 17}]

a(0) = 1; a(n) = Sum_{k=1..n} 4^k * binomial(n,k) * a(n-k).
a(n) = Sum_{k>=0} (4*k)^n / 2^(k + 1).
a(n) = 4^n * A000670(n).

A367835 Expansion of e.g.f. 1/(2 - x - exp(2*x)).

Original entry on oeis.org

1, 3, 22, 242, 3544, 64872, 1424976, 36517840, 1069533824, 35240047232, 1290137297152, 51955085596416, 2282489348834304, 108630445541684224, 5567741266098944000, 305752314499878569984, 17909736027185859100672, 1114647522476340562132992

Offset: 0

Seiichi Manyama, Dec 02 2023

nonn

Cf. A006155, A367836, A367837.

Cf. A126390, A216794, A343672, A355110, A367838.

A367835 := proc(n)
    option remember ;
    if n = 0 then
        1 ;
    else
        n*procname(n-1)+add(2^k*binomial(n,k)*procname(n-k),k=1..n) ;
    end if;
end proc:
seq(A367835(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

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

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

A336947 E.g.f.: 1 / (exp(-2*x) - x).

Original entry on oeis.org

1, 3, 14, 98, 920, 10792, 151888, 2494032, 46803072, 988095104, 23178247424, 598074306304, 16835199087616, 513385352524800, 16859837094942720, 593234633904293888, 22265289445252628480, 887889931920920313856, 37489832605652634763264, 1670894259596134872711168

Offset: 0

Ilya Gutkovskiy, Aug 08 2020

nonn

Cf. A072597, A216794, A336948, A336949, A336950.

nmax = 19; CoefficientList[Series[1/(Exp[-2 x] - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(2 (n - k + 1))^k/k!, {k, 0, n}], {n, 0, 19}]
a[0] = 1; a[n_] := a[n] = 3 n a[n - 1] - Sum[Binomial[n, k] (-2)^k a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 19}]

seq(n)={ Vec(serlaplace(1 / (exp(-2*x + O(x*x^n)) - x))) } \\ Andrew Howroyd, Aug 08 2020

a(n) = n! * Sum_{k=0..n} (2 * (n-k+1))^k / k!.
a(0) = 1; a(n) = 3 * n * a(n-1) - Sum_{k=2..n} binomial(n,k) * (-2)^k * a(n-k).
a(n) ~ n! / ((1 + LambertW(2)) * (LambertW(2)/2)^(n+1)). - Vaclav Kotesovec, Aug 09 2021

A354313 Expansion of e.g.f. 1/(1 - x/2 * (exp(2 * x) - 1)).

Original entry on oeis.org

1, 0, 2, 6, 40, 280, 2496, 25424, 297984, 3920256, 57349120, 922611712, 16193375232, 307896882176, 6304666798080, 138318662000640, 3236895083167744, 80483201605795840, 2118875812456366080, 58882581280649117696, 1722441885524719042560

Offset: 0

Seiichi Manyama, May 23 2022

nonn

Cf. A052848, A354314.

Cf. A216794, A353998, A354311.

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

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

a(n) = n!*sum(k=0, n\2, 2^(n-2*k)*k!*stirling(n-k, k, 2)/(n-k)!);

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

a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-2*k) * k! * Stirling2(n-k,k)/(n-k)!.

cp's OEIS Frontend

A080253 a(n) is the number of elements in the Coxeter complex of type B_n (or C_n).

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A000556 Expansion of exp(-x) / (1 - exp(x) + exp(-x)).

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A336950 E.g.f.: 1 / (1 - x * exp(2*x)).

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A367977 Expansion of e.g.f. exp(-x) / (2 - exp(2*x)).

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A328182 Expansion of e.g.f. 1 / (2 - exp(3*x)).

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A352117 Expansion of e.g.f. 1/sqrt(2 - exp(2*x)).

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PARI

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A328183 Expansion of e.g.f. 1 / (2 - exp(4*x)).

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A367835 Expansion of e.g.f. 1/(2 - x - exp(2*x)).