A022493 -id:A022493 - cp's OEIS frontend

A273714 Number of doublerises in all bargraphs having semiperimeter n (n>=2). A doublerise in a bargraph is any pair of adjacent up steps.

0, 1, 4, 14, 47, 155, 508, 1662, 5438, 17809, 58395, 191732, 630373, 2075221, 6840140, 22571800, 74564874, 246568051, 816099650, 2703492238, 8963064935, 29738123605, 98735734915, 328034119098, 1090509180192, 3627343273885, 12072071392105, 40197107361740, 133910579452363

Offset: 2

Emeric Deutsch, May 28 2016

nonn

a(n) appears to be the number of 021-avoiding ascent sequences (A022493) with exactly one repeated nonzero entry, where repeated means two consecutive equal entries. For example, a(4) = 4 counts 0011, 0110, 0112, 0122, and a(5) = 14 counts 00011, 00110, 00112, 00122, 01011, 01022, 01100, 0110 1, 01102, 01120, 01123, 0122 0, 01223, 01233. - David Callan, Nov 21 2021

			a(4) = 4 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and the corresponding drawings show that they have 0, 0, 1, 1, 2 doublerises.

M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
Emeric Deutsch and S. Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088 [math.CO], 2016.

Cf. A082582, A273713, A271943.

g := ((1-2*z-z^2-sqrt(1-4*z+2*z^2+z^4))*(1/2))/sqrt(1-4*z+2*z^2+z^4): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 2 .. 35);

F[k_] := DifferenceRoot[Function[{y, n}, {(2 + n) y[n] + (6 + 2 n) y[2 + n] + (-14 - 4 n) y[3 + n] + (4 + n) y[4 + n] == 0, y[0] == 1, y[1] == 2, y[2] == 5, y[3] == 14}]][k]; Table[1/2 (-F[n] - 2 F[n + 1] + F[n + 2]), {n, 0, 20}] (* Benedict W. J. Irwin, May 29 2016 *)

G.f.: g = (1 - 2z - z^2 - Q)/(2Q), where Q = sqrt(1 - 4z + 2z^2 + z^4).
a(n) = Sum_{k>0} k*A273713(n,k).
From Benedict W. J. Irwin, May 29 2016: (Start)
Let y(0)=1, y(1)=2, y(2)=5, y(3)=14,
Let (n+2)*y(n) + (2*n+6)*y(n+2) - (4*n+14)*y(n+3) + (n+4)*y(n+4)=0,
a(n) = (y(n+2)-2*y(n+1)-y(n))/2.
(End)
D-finite with recurrence n*a(n) +6*(-n+1)*a(n-1) +9*(n-2)*a(n-2) -6*a(n-3) +(-n+8) * a(n-4) +2*(-n+4)*a(n-5) +(-n+6)*a(n-6)=0. - R. J. Mathar, Jun 06 2016

A035378 Coefficients in expansion of Sum_{k>=0} Product_{j=1..k} (1-x^j) about x = -1.

Original entry on oeis.org

3, 11, 72, 635, 7085, 95911, 1528541, 28044762, 582314535, 13500314080, 345696545788, 9690223054222, 295132850278639, 9705001713289680, 342693270841135600, 12932930349605422101, 519485442041267214922

Offset: 0

Bill Gosper, Aug 19 2001

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..250
Hsien-Kuei Hwang, and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], p. 35, 2019.

Cf. A022493.

a(n)=polcoeff(sum(i=0,2*n+1,prod(j=1,i,1-(x-1)^j,1+x*O(x^n))),n)

a(n) ~ sqrt(12) * exp(Pi^2/48) * 24^(n+1) * n^(n+1) / (exp(n) * Pi^(2*n+2)). - Vaclav Kotesovec, May 04 2014

Conjectures: a(5*n+3) == 0 (mod 5), a(5*n+4) == 0 (mod 5) and a(5*n+2) - 2*a(5*n+1) == (0 mod 5) (all checked up to n = 49). - Peter Bala, Jun 19 2023

More terms from Vladeta Jovovic, Aug 20 2001

A189359 Number of homogeneous games for n players.

Original entry on oeis.org

0, 1, 3, 8, 23, 76, 293, 1307, 6642, 37882

Offset: 0

Fabián Riquelme, Apr 20 2011

I. Krohn and P. Sudhölter, Directed and weighted majority games, Mathematical Methods of Operation Research, 42, 2 (1995), 189-216. See Table 1, p. 213; also on ResearchGate.
P. Sudhölter, Homogeneous games as anti step functions, International Journal of Game Theory Vol. 18, (1989), 433-469.

Subclass of A000617. Cf. A001532, A022493, A109456, A132183.

Conjecture: g.f.: Q(0)*x/(1-x), where Q(k) = 1 + (1-(1-x)^(2*k+2))/(1- (1-(1-x)^(2*k+3))/(1-(1-x)^(2*k+3) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 13 2013

Note that a(n) - a(n-1) = A022493(n) for 1 <= n <= 9. Does this equality hold for n > 9? If so, then we have the g.f. 1/(1 - x)*( Sum_{n >= 1} Product_{k = 1..n} (1 - (1 - x)^k) ). - Peter Bala, Dec 13 2021

A193387 Triangle read by rows: T(n,k) (n>=1, 1 <= k <= n) = number of n-element unlabeled interval posets of height k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 26, 10, 1, 1, 31, 100, 69, 15, 1, 1, 63, 366, 412, 150, 21, 1, 1, 127, 1317, 2305, 1270, 286, 28, 1, 1, 255, 4743, 12551, 9920, 3236, 497, 36, 1, 1, 511, 17275, 67933, 74525, 33301, 7210, 806, 45, 1, 1, 1023, 64029, 370168, 551232, 325860, 93926, 14540, 1239, 55, 1, 1, 2047, 242371, 2046980, 4072130, 3109628, 1151416, 232891, 27147, 1825, 66, 1

Offset: 1

N. J. A. Sloane, Aug 26 2011

			Triangle begins
1
1 1
1 3 1
1 7 6 1
1 15 26 10 1
1 31 100 69 15 1
1 63 366 412 150 21 1
1 127 1317 2305 1270 286 28 1
1 255 4743 12551 9920 3236 497 36 1
1 511 17275 67933 74525 33301 7210 806 45 1
...

Soheir M. Khamis, Height counting of unlabeled interval and N-free posets, Discrete Math. 275 (2004), no. 1-3, 165-175.
Soheir Mohamed Khamis, Exact Counting of Unlabeled Rigid Interval Posets Regarding or Disregarding Height, Order 29, pp. 443-461 (2012).

Row sums give A022493. Cf. A193357.

A193548 Decimal expansion of exp(Pi^2/12).

Original entry on oeis.org

2, 2, 7, 6, 1, 0, 8, 1, 5, 1, 6, 2, 5, 7, 3, 4, 0, 9, 4, 7, 9, 1, 0, 6, 1, 4, 1, 2, 0, 3, 1, 4, 9, 7, 4, 4, 6, 6, 9, 7, 9, 7, 4, 2, 6, 0, 3, 0, 0, 2, 3, 7, 7, 5, 6, 1, 5, 5, 1, 6, 1, 7, 0, 9, 8, 2, 7, 5, 0, 6, 3, 7, 2, 8, 6, 3, 0, 1, 4, 3, 1, 8, 6, 6, 8, 4, 6, 5, 7

Offset: 1

John M. Campbell, Jul 30 2011

Cf. A001113, A022493, A122214, A122215, A122216, A122217, A138265, A207651, A242153, A242154, A242155, A242156, A242157, A242158, A242159, A242160, A242161, A242162, A242163, A242164.

Product[Product[k^((1/(n+1))*(-1)^(k)*Binomial[n,k-1]*HarmonicNumber[n]),{k,1,n+1}],{n,1,Infinity}]
RealDigits[E^(Pi^2/12), 10, 100]

exp(Pi^2/12) \\ Charles R Greathouse IV, Jul 30 2011

exp(Pi^2/12) = Product_{n>=1} Product_{k=1..n+1} k^(1/(n+1)) * H(n) * (-1)^k * binomial(n, k-1) where H(n) is the n-th harmonic number.

exp(Pi^2/12) = lim_{n -> infinity} Product_{k=1..n} (1 + k/n)^(1/k). - Peter Bala, Feb 14 2015

A289317 The number of upper-triangular matrices whose nonzero entries are positive odd numbers summing to n and each row and each column contains a nonzero entry.

Original entry on oeis.org

1, 1, 1, 3, 7, 23, 84, 364, 1792, 9953, 61455, 417720, 3098515, 24902930, 215538825, 1998518430, 19761943208, 207571259703, 2307812703419, 27075591512866, 334263981931669

Offset: 0

Peter Bala, Jul 25 2017

A Fishburn matrix of size n is defined to be an upper-triangular matrix with nonnegative integer entries which sum to n and each row and each column contains a nonzero entry. See A022493. Here we are considering Fishburn matrices where the nonzero entries are all odd.

The g.f. for primitive Fishburn matrices (i.e., Fishburn matrices with entries restricted to the set {0,1}), is F(x) = Sum_{n>=0} Product_{k=1..n} ( 1 - 1/(1 + x)^k ). See A138265. Let C(x) = x/(1 - x^2) = x + x^3 + x^5 + x^7 + .... Then applying Lemma 2.2.22 of Goulden and Jackson gives the g.f. for this sequence as the composition F(C(x)).

			a(4) = 7: The Fishburn matrices of size 4 with odd nonzero entries are
/3 0\ /1 0\
\0 1/ \0 3/
/1 1 0\ /1 0 1\ /1 0 0\
|0 1 0| |0 1 0| |0 1 1|
\0 0 1/ \0 0 1/ \0 0 1/
/1 1 0\
|0 0 1|
\0 0 1/
/1 0 0 0\
|0 1 0 0|
|0 0 1 0|
\0 0 0 1/

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 42.

Seiichi Manyama, Table of n, a(n) for n = 0..200
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A022493, A138265, A289312, A289313, A289314, A289315, A289316.

C:= x -> x/(1 - x^2):
G:= add(mul( 1 - 1/(1 + C(x))^k, k=1..n), n=0..20):
S:= series(G,x,21):
seq(coeff(S,x,j),j=0..20);

G.f.: A(x) = Sum_{n >= 0} Product_{k = 1..n} ( 1 - 1/(1 + x/(1 - x^2))^k ).

a(n) ~ 2^(n + 5/2) * 3^(n + 3/2) * n^(n+1) / (exp(n + Pi^2/12) * Pi^(2*n + 2)). - Vaclav Kotesovec, Aug 31 2023

A294219 Number T(n,k) of ascent sequences of length n where the maximum of 0 and all letter multiplicities equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 9, 4, 1, 0, 1, 26, 20, 5, 1, 0, 1, 82, 97, 30, 6, 1, 0, 1, 276, 496, 191, 42, 7, 1, 0, 1, 1014, 2686, 1259, 310, 56, 8, 1, 0, 1, 4006, 15481, 8784, 2416, 470, 72, 9, 1, 0, 1, 17046, 94843, 65012, 19787, 4141, 677, 90, 10, 1

Offset: 0

Alois P. Heinz, Oct 25 2017

			T(4,1) = 1: 0123.
T(4,2) = 9: 0011, 0012, 0101, 0102, 0110, 0112, 0120, 0121, 0122.
T(4,3) = 4: 0001, 0010, 0100, 0111.
T(4,4) = 1: 0000.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,     1;
  0, 1,     3,     1;
  0, 1,     9,     4,     1;
  0, 1,    26,    20,     5,     1;
  0, 1,    82,    97,    30,     6,    1;
  0, 1,   276,   496,   191,    42,    7,   1;
  0, 1,  1014,  2686,  1259,   310,   56,   8,  1;
  0, 1,  4006, 15481,  8784,  2416,  470,  72,  9,  1;
  0, 1, 17046, 94843, 65012, 19787, 4141, 677, 90, 10, 1;
  ...

Columns k=0-1 give: A000007, A057427.
Row sums give A022493.
Cf. A294220.

b:= proc(n, i, t, p, k) option remember; `if`(n=0, 1,
      add(`if`(coeff(p, x, j)=k, 0, b(n-1, j, t+
          `if`(j>i, 1, 0), p+x^j, k)), j=1..t+1))
    end:
A:= (n, k)-> b(n, 0$3, k):
T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, i_, t_, p_, k_] := b[n, i, t, p, k] = If[n == 0, 1, Sum[If[ Coefficient[p, x, j] == k, 0, b[n - 1, j, t + If[j > i, 1, 0], p + x^j, k]], {j, t + 1}]];
A[n_, k_] :=  b[n, 0, 0, 0, k];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2020, after Maple *)

T(n,k) = A294220(n,k) - A294220(n,k-1) for k>0, T(n,0) = A294220(n,k) = A000007(n).

A294281 Number of ascent sequences of length n with alternating ascents and descents (unaffected by level steps).

Original entry on oeis.org

1, 1, 2, 4, 9, 22, 59, 172, 547, 1886, 7047, 28360, 122675, 567210, 2796999, 14641044, 81191947, 475148678, 2929442263, 18965690560, 128754649699, 914056305794, 6777666961735, 52367331911180, 421188392986843, 3519168714308702, 30519733808467031

Offset: 0

Alois P. Heinz, Oct 26 2017

nonn

			a(3) = 4: 000, 001, 010, 011.
a(4) = 9: 0000, 0001, 0010, 0011, 0100, 0101, 0102, 0110, 0111.
a(5) = 22: 00000, 00001, 00010, 00011, 00100, 00101, 00102, 00110, 00111, 01000, 01001, 01002, 01010, 01011, 01020, 01021, 01022, 01100, 01101, 01102, 01110, 01111.

Alois P. Heinz, Table of n, a(n) for n = 0..550

Cf. A022493, A099960.

b:= proc(n, i, t, u) option remember; `if`(n<1, 1, add(
       b(n-1, j, t+`if`(j>i, 1, 0), `if`(i=j, u, 1-u)),
       j=`if`(u=0, i..t+1, 0..i)))
    end:
a:= n-> b(n-1, 0$3):
seq(a(n), n=0..30);

a(n) = Sum_{j=0..n} binomial(n-1,j) * A099960(n-j).

A317784 Number of ascent sequences of length n avoiding the pattern 0000.

Original entry on oeis.org

1, 1, 2, 5, 14, 47, 180, 773, 3701, 19488, 111890, 695786, 4656185, 33356828, 254675642, 2063984616, 17694054723, 159958176316, 1520689121858, 15165205111010

Offset: 0

Alois P. Heinz, Aug 06 2018

Paul Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv:1109.3641 [math.CO], 2011.

Column k=3 of A294220.

Cf. A022493.

b:= proc(n, i, t, p) option remember; `if`(n=0, 1, add(
      `if`(coeff(p, x, j)=3, 0, b(n-1, j, t+
      `if`(j>i, 1, 0), p+x^j)), j=1..t+1))
    end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..12);

b[n_,i_,t_,p_,k_]:=b[n,i,t,p,k]=If[n==0,1,Sum[If[Coefficient[p,x,j]==k,0,b[n-1,j,t+If[j>i,1,0],p+x^j,k]],{j,1,t+1}]]; a[n_]:=b[n,0,0,0,Min[n,3]];
Table[Print["a(",n,") = ",a[n]];a[n],{n, 0, 15}] (* Vincenzo Librandi, Feb 12 2020 *)

a(n) <= A022493(n) with equality only for n < 4.

a(18) from Vaclav Kotesovec, Aug 20 2018

a(19) from Vaclav Kotesovec, Aug 23 2018

A336071 Number of inversion sequences avoiding the vincular pattern 1-01 (or 1-10).

Original entry on oeis.org

1, 2, 6, 23, 107, 584, 3655, 25790, 202495, 1750763

Offset: 1

Michael De Vlieger, Jul 07 2020

Juan S. Auli, Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020. See p. 5, Table 1. Gives terms 1-10.

Cf. A000079, A000110, A022493, A091768, A102038, A113227, A263777, A328441, A336070, A336072.

cp's OEIS Frontend

A273714 Number of doublerises in all bargraphs having semiperimeter n (n>=2). A doublerise in a bargraph is any pair of adjacent up steps.

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A035378 Coefficients in expansion of Sum_{k>=0} Product_{j=1..k} (1-x^j) about x = -1.

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A189359 Number of homogeneous games for n players.

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A193387 Triangle read by rows: T(n,k) (n>=1, 1 <= k <= n) = number of n-element unlabeled interval posets of height k.

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A193548 Decimal expansion of exp(Pi^2/12).

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A289317 The number of upper-triangular matrices whose nonzero entries are positive odd numbers summing to n and each row and each column contains a nonzero entry.

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A294219 Number T(n,k) of ascent sequences of length n where the maximum of 0 and all letter multiplicities equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A294281 Number of ascent sequences of length n with alternating ascents and descents (unaffected by level steps).

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A317784 Number of ascent sequences of length n avoiding the pattern 0000.

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