D n Verma - cp's OEIS frontend

A097552 Number of positive words of length n in the monoid Br_5 of positive braids on 6 strands.

1, 5, 20, 67, 209, 630, 1873, 5540, 16357, 48265, 142387, 420027, 1239006, 3654820, 10780958, 31801551, 93807834, 276713194, 816245143, 2407749755, 7102350204, 20950424039, 61799299470, 182294802589, 537730934397

Offset: 0

D n Verma, Aug 16 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,7,-4,1).

Cf. A097550, A097551, A097553, A097554, A097555, A097556.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x^2)^3/(1-5*x+8*x^2-7*x^3+4*x^4-x^5) )); // G. C. Greubel, Apr 19 2021

LinearRecurrence[{5,-8,7,-4,1}, {1,5,20,67,209,630,1873}, 40] (* G. C. Greubel, Apr 19 2021 *)

def A097552_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)^3/(1-5*x+8*x^2-7*x^3+4*x^4-x^5) ).list()
A097552_list(40) # G. C. Greubel, Apr 19 2021

G.f.: (1 + x^2)^3/(1 - 5*x + 8*x^2 - 7*x^3 + 4*x^4 - x^5). - T. D. Noe, Nov 02 2006

Corrected by T. D. Noe, Nov 02 2006

A097551 Number of positive words of length n in the monoid Br_4 of positive braids on 5 strands.

Original entry on oeis.org

1, 4, 13, 37, 101, 273, 737, 1990, 5374, 14513, 39194, 105848, 285855, 771985, 2084836, 5630344, 15205404, 41063976, 110898081, 299493268, 808816679, 2184304257, 5898969706, 15930859211, 43023152830, 116189067703

Offset: 0

D n Verma, Aug 16 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,5,-3,1).

Cf. A097550, A097552, A097553, A097554, A097555, A097556.

R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( (1+x^2)^2/(1-4*x+5*x^2-5*x^3+3*x^4-x^5) )); // G. C. Greubel, Apr 19 2021

<Ryan Propper, Sep 27 2005 *)
LinearRecurrence[{4,-5,5,-3,1}, {1,4,13,37,101}, 51] (* G. C. Greubel, Apr 19 2021 *)

def A097551_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)^2/(1-4*x+5*x^2-5*x^3+3*x^4-x^5) ).list()
A097551_list(50) # G. C. Greubel, Apr 19 2021

G.f.: (1 + x^2)^2/(1 - 4*x + 5*x^2 - 5*x^3 + 3*x^4 - x^5).

More terms from Ryan Propper, Sep 27 2005

A097553 Number of positive words of length n in the monoid Br_6 of positive braids on 7 strands.

Original entry on oeis.org

1, 6, 27, 101, 346, 1131, 3611, 11396, 35761, 111906, 349700, 1092039, 3409031, 10640179, 33206991, 103631414, 323402952, 1009233980, 3149469548, 9828376731, 30670834516, 95712596642, 298684343689, 932085486213, 2908700435744

Offset: 0

D n Verma, Aug 16 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,17,-17,11,-5,1).

Cf. A097550, A097551, A097552, A097554, A097555, A097556.

R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( (1+x^2)^4/(1-6*x+13*x^2-17*x^3+17*x^4-11*x^5+5*x^6-x^7) )); // G. C. Greubel, Apr 20 2021

CoefficientList[Series[(1+n^2)^4/(1-6n+13n^2-17n^3+17n^4-11n^5+5n^6-n^7),{n,0,30}],n] (* Harvey P. Dale, Sep 27 2019 *)
LinearRecurrence[{6,-13,17,-17,11,-5,1}, {1,6,27,101,346,1131,3611,11396,35761}, 40] (* G. C. Greubel, Apr 20 2021 *)

def A097553_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)^4/(1-6*x+13*x^2-17*x^3+17*x^4-11*x^5+5*x^6-x^7) ).list()
A097553_list(50) # G. C. Greubel, Apr 20 2021

G.f.: (1 +x^2)^4/(1 -6*x +13*x^2 -17*x^3 +17*x^4 -11*x^5 +5*x^6 -x^7).

Corrected and extended by Max Alekseyev, Jun 17 2011

A097554 Number of positive words of length n in the monoid Br_7 of positive braids on 8 strands.

Original entry on oeis.org

1, 7, 36, 151, 570, 2019, 6893, 23034, 76020, 249077, 812614, 2644447, 8592693, 27895296, 90510106, 293576779, 952053411, 3087093728, 10009389358, 32452403488, 105214363653, 341111617862, 1105895184121, 3585328906357, 11623651559099

Offset: 0

D n Verma, Aug 16 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-18,25,-24,15,-6,1).

Cf. A097550, A097551, A097552, A097553, A097555, A097556.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x^2)^5/(1-7*x+18*x^2-25*x^3+24*x^4-15*x^5+6*x^6-x^7) )); // G. C. Greubel, Apr 20 2021

LinearRecurrence[{7,-18,25,-24,15,-6,1}, {1,7,36,151,570,2019,6893,23034,76020, 249077,812614}, 41] (* G. C. Greubel, Apr 20 2021 *)

def A097554_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)^5/(1-7*x+18*x^2-25*x^3+24*x^4-15*x^5+6*x^6-x^7) ).list()
A097554_list(40) # G. C. Greubel, Apr 20 2021

G.f.: (1 +x^2)^5/(1 -7*x +18*x^2 -25*x^3 +24*x^4 -15*x^5 +6*x^6 -x^7).

Corrected and extended by Max Alekseyev, Jun 17 2011

A097555 Number of positive words of length n in the monoid Br_8 of positive braids on 9 strands.

Original entry on oeis.org

1, 8, 45, 205, 831, 3133, 11294, 39585, 136302, 464026, 1568151, 5273999, 17681042, 59149925, 197598856, 659479754, 2199585548, 7333198205, 24441067317, 81444567492, 271360676916, 904051477063, 3011711782025, 10032660556567, 33420042561972

Offset: 0

D n Verma, Aug 16 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-25,45,-59,57,-41,21,-7,1).

Cf. A097550, A097551, A097552, A097553, A097554, A097556.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x^2)^6 /(1-8*x+25*x^2-45*x^3+59*x^4-57*x^5+41*x^6-21*x^7+7*x^8-x^9) )); // G. C. Greubel, Apr 20 2021

LinearRecurrence[{8,-25,45,-59,57,-41,21,-7,1}, {1,8,45,205,831,3133,11294,39585, 136302, 464026, 1568151, 5273999, 17681042}, 41] (* G. C. Greubel, Apr 20 2021 *)

def A097555_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)^6 /(1-8*x+25*x^2-45*x^3+59*x^4-57*x^5+41*x^6-21*x^7+7*x^8-x^9) ).list()
A097555_list(40) # G. C. Greubel, Apr 20 2021

G.f.: (1 +x^2)^6 /(1 -8*x +25*x^2 -45*x^3 +59*x^4 -57*x^5 +41*x^6 -21*x^7 +7*x^8 -x^9).

Edited and extended by Max Alekseyev, Jun 17 2011

A097556 Number of positive words of length n in the monoid Br_9 of positive braids on 10 strands.

Original entry on oeis.org

1, 9, 56, 279, 1223, 4932, 18833, 69345, 249166, 880525, 3076295, 10662459, 36749785, 126161246, 431880044, 1475412473, 5032964258, 17150277106, 58395929325, 198723871661, 675989712225, 2298799014859, 7815699898677, 26568450635871

Offset: 0

D n Verma, Aug 16 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-32,63,-84,81,-56,27,-8,1).

Cf. A097550, A097551, A097552, A097553, A097554, A097555.

R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( (1+x^2)^7/(1-9*x+32*x^2-63*x^3+84*x^4-81*x^5+56*x^6-27*x^7+8*x^8-x^9) )); // G. C. Greubel, Apr 20 2021

CoefficientList[Series[(1+x^2)^7/(1-9*x+32*x^2-63*x^3+84*x^4-81*x^5+56*x^6-27*x^7+8*x^8-x^9), {x,0,50}], x] (* G. C. Greubel, Apr 20 2021 *)

def A097556_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)^7/(1-9*x+32*x^2-63*x^3+84*x^4-81*x^5+56*x^6-27*x^7+8*x^8-x^9) ).list()
A097556_list(50) # G. C. Greubel, Apr 20 2021

G.f.: (1 +x^2)^7/(1 -9*x +32*x^2 -63*x^3 +84*x^4 -81*x^5 +56*x^6 -27*x^7 +8*x^8 -x^9).

Edited and extended by Max Alekseyev, Jun 17 2011

A097550 Number of positive words of length n in the monoid Br_3 of positive braids on 4 strands.

Original entry on oeis.org

1, 3, 8, 19, 44, 102, 237, 551, 1281, 2978, 6923, 16094, 37414, 86977, 202197, 470051, 1092736, 2540303, 5905488, 13728594, 31915109, 74193627, 172479257, 400965626, 932131991, 2166943978, 5037533578, 11710844769, 27224411129, 63289077427

Offset: 0

D n Verma, Aug 16 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

Cf. A097551, A097552, A097553, A097554, A097555, A097556.

Cf. A095263, A135364, A136302, A136303, A136304, A136305, A137229, A137234, A137249.

[n le 3 select Fibonacci(2*n) else 3*Self(n-1) -2*Self(n-2) +Self(n-3): n in [1..31]]; // G. C. Greubel, Apr 19 2021

a:= n-> (<<1|1|2>>. <<3|1|0>, <-2|0|1>, <1|0|0>>^n)[1$2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 24 2008

LinearRecurrence[{3,-2,1},{1,3,8},30] (* Harvey P. Dale, Jul 10 2019 *)

@CachedFunction
def A095263(n): return sum( binomial(n+j+2, 3*j+2) for j in (0..n//2) )
def A097550(n): return A095263(n) +A095263(n-2)
[A097550(n) for n in (0..30)] # G. C. Greubel, Apr 19 2021

G.f.: (1+x^2)/(1 - 3*x+ 2*x^2 - x^3).
a(n) = term (1,1) in the 1 X 3 matrix [1,1,2].[3,1,0; -2,0,1; 1,0,0]^n. - Alois P. Heinz, Jul 24 2008
a(n) = A095263(n) + A095263(n-2). - G. C. Greubel, Apr 19 2021

More terms from Ryan Propper, Sep 27 2005

A012250 a(n) = A012249(2n) / 2^(2n-1).

Original entry on oeis.org

1, 3, 40, 1225, 67956, 5986134, 769550496, 136151219061, 31753157473180, 9445432588519642, 3491687484842443536, 1570713950508131878618, 845034544811095556274280, 535857105694970626486925100, 395590680969537758258609408640, 336386798400777928783348084420365

Offset: 1

D n Verma

nonn

G. C. Greubel, Table of n, a(n) for n = 1..225
M. Hering and B. Howard, The ring of evenly weighted points on the line, arXiv:1211.3941 [math.AG], 2012-2014, see p. 8.
B. Howard, J. Millson, A. Snowden, and R. Vakil, The moduli space of n points on the line is cut out by simple quadrics when n is not six, p. 12.
Richard Stanley, Access to a preprint by D. N. Verma.
R. P. Stanley and F. Zanello, Unimodality of partitions with distinct parts inside Ferrers shapes, see Theorem 4.6.
R. P. Stanley and F. Zanello, Unimodality of partitions with distinct parts inside Ferrers shapes, arXiv:1305.6083 [math.CO], 2013, see Theorem 4.6 and Remark 4.7.
R. P. Stanley and F. Zanello, Unimodality of partitions with distinct parts inside Ferrers shapes, European Journal of Combinatorics, Volume 49, October 2015, Pages 194-202.
D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished. [Scanned copy of annotated version of preprint given to me by the author in 1997. - _N. J. A. Sloane_, Oct 04 2021]

Cf. A012249.

A012250:= func< n | (&+[(-1)^(j+1)*Binomial(2*n+2,j)*(n-j+1)^(2*n-1) : j in [0..n]])/2 >;
[A012250(n): n in [1..20]]; // G. C. Greubel, Feb 27 2024

A012250 := n -> 1/2*add((-1)^(j+1)*binomial(2*n+2,j)*(n-j+1)^(2*n-1)*(2*j-2*n-1),j=0..n); seq(A012250(i),i=1..9); # Peter Luschny, Mar 03 2013

Table[Sum[(-1)^(j + 1)*Binomial[2*n + 2, j]*(n - j + 1)^(2*n - 1)/2, {j, 0, n}], {n, 15}] (* Wesley Ivan Hurt, Nov 11 2014 *)

def A012250(n): return sum( (-1)^(j+1)*binomial(2*n+2,j)*(n-j+1)^(2*n-1) for j in range(n+1))/2
[A012250(n) for n in range(1,21)] # G. C. Greubel, Feb 27 2024

a(n) = (1/2)*Sum_{j=0..n} (-1)^(j+1)*binomial(2*n+2,j)*(n-j+1)^(2*n-1). - Richard Stanley, Mar 31 2013

a(n) ~ 3^(3/2) * 2^(2*n) * n^(2*n-2) / exp(2*n). - Vaclav Kotesovec, Oct 07 2021

Edited and extended using Richard Stanley's formula. - N. J. A. Sloane, Jun 10 2013

A012249 Volume of a certain rational polytope whose points with given denominator count certain sets of Standard Tableaux.

Original entry on oeis.org

1, 2, 5, 24, 154, 1280, 13005, 156800, 2189726, 34793472, 620169186, 12259602432, 266267950740, 6304157663232, 161624247752253, 4461403146190848, 131936409635518774, 4161949856324648960, 139508340802911502422, 4952126960969786064896, 185585825504872433198636

Offset: 1

D n Verma

nonn

It should be noticed that Richard Stanley's formula (cf. A012250) gives a(9) = 2189726 instead of 2189725 as given in Verma (1997). - Jean-François Alcover, Nov 28 2013

G. C. Greubel, Table of n, a(n) for n = 1..350
MathOverflow, Access to a preprint by D. N. Verma, Feb 2013.
D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished. [Scanned copy of annotated version of preprint given to me by the author in 1997. - _N. J. A. Sloane_, Oct 03 2021]

Cf. A012250.

Row sums of A348211.

A012249:= func< n | 2^(n-2)*(&+[(-1)^(j+1)*Binomial(n+2,j)*(n/2-j+1)^(n-1) : j in [0..1+Floor(n/2)]] ) >;
[A012249(n): n in [1..30]]; // G. C. Greubel, Feb 28 2024

A012249 := proc(n)
     add( (-1)^(j+1)*(n/2-j+1)^(n-1)*binomial(n+2,j),j=0..ceil(n/2)) ;
     %*2^(n-2) ;
end proc:
seq(A012249(n),n=1..20) ; # R. J. Mathar, Oct 07 2021

a[n_] := 2^(n-2)*Sum[(-1)^(j+1)*(n/2-j+1)^(n-1)*Binomial[n+2, j], {j, 0, Ceiling[n/2]}]; Table[a[n], {n, 1, 16}] (* Jean-François Alcover, Nov 25 2013, after Richard Stanley's formula in A012250. *)

def A012249(n): return 2^(n-2)*sum( (-1)^(j+1)*binomial(n+2,j)*(n/2-j+1)^(n-1) for j in range(n//2+2))
[A012249(n) for n in range(1,31)] # G. C. Greubel, Feb 28 2024

a(n) ~ 3^(3/2) * 2^(n+1) * n^(n-2) / exp(n). - Vaclav Kotesovec, Oct 07 2021

a(n) = 2^(n-2)*Sum_{j=0..ceiling(n/2)} (-1)^(j+1)*(n/2-j+1)^(n-1) * binomial(n+2, j) (based on Richard Stanley's formula in A012250). - Jean-François Alcover, Nov 25 2013

Corrected and extended by R. J. Mathar, Oct 07 2021

Edited by N. J. A. Sloane, Oct 07 2021

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User: D n Verma

A097552 Number of positive words of length n in the monoid Br_5 of positive braids on 6 strands.

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A097551 Number of positive words of length n in the monoid Br_4 of positive braids on 5 strands.

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A097553 Number of positive words of length n in the monoid Br_6 of positive braids on 7 strands.

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A097554 Number of positive words of length n in the monoid Br_7 of positive braids on 8 strands.

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A097555 Number of positive words of length n in the monoid Br_8 of positive braids on 9 strands.

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A097556 Number of positive words of length n in the monoid Br_9 of positive braids on 10 strands.

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A097550 Number of positive words of length n in the monoid Br_3 of positive braids on 4 strands.

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A012250 a(n) = A012249(2n) / 2^(2n-1).