A154695 -id:A154695 - cp's OEIS frontend

A174674 Sequence A154695 adjusted to leading one:t(n,m)=A154695(n,m)-A154695(n,0)+1.

1, 1, 1, 1, 20, 1, 1, 130, 130, 1, 1, 744, 1824, 744, 1, 1, 4234, 20152, 20152, 4234, 1, 1, 24484, 210796, 376704, 210796, 24484, 1, 1, 143686, 2183524, 6233224, 6233224, 2183524, 143686, 1, 1, 851504, 22549360, 99411264, 149600192, 99411264

Offset: 0

Roger L. Bagula, Mar 26 2010

Row sums are:
1, 2, 22, 262, 3314, 48774, 847266, 17120870, 395224450, 10263445126,
296140564130,...

			{1},
{1, 1},
{1, 20, 1},
{1, 130, 130, 1},
{1, 744, 1824, 744, 1},
{1, 4234, 20152, 20152, 4234, 1},
{1, 24484, 210796, 376704, 210796, 24484, 1},
{1, 143686, 2183524, 6233224, 6233224, 2183524, 143686, 1},
{1, 851504, 22549360, 99411264, 149600192, 99411264, 22549360, 851504, 1},
{1, 5075122, 231836368, 1562973472, 3331837600, 3331837600, 1562973472, 231836368, 5075122, 1},
{1, 30344508, 2370195636, 24248921920, 72553861536, 97733916928, 72553861536, 24248921920, 2370195636, 30344508, 1}

A154690, A154692, A154693, A154694, A154695

Clear[t, p, q, n, m, a];
p[x_, n_] = 2^n*(1 - x)^(n + 1)*LerchPhi[x, -n, 1/2];
a = Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
p = 2; q = 1;
t[n_, m_] := (p^(n - m)*q^m + p^m*q^(n - m))*a[[n + 1]][[m + 1]];
Table[Table[t[n, m] - t[n, 0] + 1, {m, 0, n}], {n, 0, 10}];
Flatten[%]

t(n,m)=A154695(n,m)-A154695(n,0)+1

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

Original entry on oeis.org

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

A174675 Sequence A154696 adjusted to leading one:t(n,m)=A154696(n,m)-A154696(n,0)+1.

Original entry on oeis.org

1, 1, 1, 1, 60, 1, 1, 656, 656, 1, 1, 5832, 16464, 5832, 1, 1, 49496, 302486, 302486, 49496, 1, 1, 419412, 4933332, 10171944, 4933332, 419412, 1, 1, 3593036, 76425506, 280498526, 280498526, 76425506, 3593036, 1, 1, 31167600, 1157982288

Offset: 0

Roger L. Bagula, Mar 26 2010

Row sums are:
1, 2, 62, 1314, 28130, 703966, 20877434, 721034138, 28453293026,
1263142713270, 62305874244266,...

			{1},
{1, 1},
{1, 60, 1},
{1, 656, 656, 1},
{1, 5832, 16464, 5832, 1},
{1, 49496, 302486, 302486, 49496, 1},
{1, 419412, 4933332, 10171944, 4933332, 419412, 1},
{1, 3593036, 76425506, 280498526, 280498526, 76425506, 3593036, 1},
{1, 31167600, 1157982288, 6978681888, 12117629472, 6978681888, 1157982288, 31167600, 1},
{1, 273237776, 17387745806, 164112248126, 449798124926, 449798124926, 164112248126, 17387745806, 273237776, 1},
{1, 2414712204, 260247533196, 3735760480536, 15279843395064, 23749342002264, 15279843395064, 3735760480536, 260247533196, 2414712204, 1}

A154690, A154692, A154693, A154694, A154695, A154696

Clear[t, p, q, n, m, a];
p[x_, n_] = 2^n*(1 - x)^(n + 1)*LerchPhi[x, -n, 1/2];
a = Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
p = 2; q = 3;
t[n_, m_] := (p^(n - m)*q^m + p^m*q^(n - m))*a[[n + 1]][[m + 1]];
Table[Table[t[n, m] - t[n, 0] + 1, {m, 0, n}], {n, 0, 10}];
Flatten[%]

t(n,m)=A154696(n,m)-A154696(n,0)+1

cp's OEIS Frontend

A174674 Sequence A154695 adjusted to leading one:t(n,m)=A154695(n,m)-A154695(n,0)+1.

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

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A174675 Sequence A154696 adjusted to leading one:t(n,m)=A154696(n,m)-A154696(n,0)+1.

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Programs

Mathematica

Formula