A085461 -id:A085461 - cp's OEIS frontend

A006858 Expansion of g.f. x(1 + x)(1 + 6*x + x^2)/(1 - x)^7.

0, 1, 14, 84, 330, 1001, 2548, 5712, 11628, 21945, 38962, 65780, 106470, 166257, 251720, 371008, 534072, 752913, 1041846, 1417780, 1900514, 2513049, 3281916, 4237520, 5414500, 6852105, 8594586, 10691604, 13198654, 16177505, 19696656, 23831808, 28666352, 34291873

Offset: 0

Simon Plouffe and N. J. A. Sloane

Arises in enumerating paths in the plane.
a(n+1) is the determinant of the n X n Hankel matrix whose first row is the Catalan numbers C_n (A000108) beginning at C_4 = 14. Example (n=3): det[{{14, 42, 132}, {42, 132, 429}, {132, 429, 1430}}] = 330. - David Callan, Mar 30 2007
0 together with partial sums of A085461. - Arkadiusz Wesolowski, Aug 05 2012

			G.f. = x + 14*x^2 + 84*x^3 + 330*x^4 + 1001*x^5 + 2548*x^6 + 5712*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Richard P. Stanley, Enumerative Combinatorics, Volume 1, 1986, p. 221, Example 4.5.18.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paolo Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
G. Kreweras and H. Niederhausen, Solution of an enumerative problem connected with lattice paths, European J. Combin., 2 (1981), 55-60.
J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), 143-179. [Th. 7.2(ii), case a=1]
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 9, 24.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000108, A006332, A085461.

series((x+7*x^2+7*x^3+x^4)/(1-x)^7,x,50);
b:=binomial; t72b:= proc(a,k) ((a+k+1)/(a+1)) * b(k+2*a+1,k)*b(k+3*a/2+1,k)/(b(k+a/2,k)); end; [seq(t72b(1,k),k=0..40)];

a[n_]:= (n+1)*Binomial[2n+4, 5]/12;
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 17 2017, after Philippe Deléham *)

a(n) = (n+1)*binomial(2*n+4, 5)/12; \\ Michel Marcus, Oct 13 2016

[(n+1)*binomial(2*n+4, 5)/12 for n in (0..30)] # G. C. Greubel, Dec 14 2021

a(n) = (n+1)*binomial(2*n+4, 5)/12. - Philippe Deléham, Mar 06 2004
a(n) = a(-2-n) for all n in Z. - Michael Somos, Jun 27 2023
From Amiram Eldar, Jul 09 2023: (Start)
Sum_{n>=1} 1/a(n) = 30*Pi^2 - 295.
Sum_{n>=1} (-1)^(n+1)/a(n) = -15*Pi^2 + 240*Pi - 605. (End)
E.g.f.: exp(x)*x*(180 + 1080*x + 1350*x^2 + 555*x^3 + 84*x^4 + 4*x^5)/180. - Stefano Spezia, Dec 09 2023

Edited by N. J. A. Sloane, Oct 20 2007

A085465 Number of monotone n-weightings of complete bipartite digraph K(3,3).

Original entry on oeis.org

1, 15, 102, 442, 1443, 3885, 9100, 19188, 37269, 67771, 116754, 192270, 304759, 467481, 696984, 1013608, 1442025, 2011815, 2758078, 3722082, 4951947, 6503365, 8440356, 10836060, 13773565, 17346771, 21661290, 26835382, 33000927, 40304433

Offset: 1

Goran Kilibarda, Vladeta Jovovic, Jul 01 2003

A monotone n-(vertex) weighting of a digraph D=(V,E) is a function w: V -> {0,1,..,n-1} such that w(v1)<=w(v2) for every arc (v1,v2) from E.

a(n) = number of proper mergings of a 3-antichain and an (n-1)-chain. - Henri Mühle, Aug 17 2012

G. C. Greubel, Table of n, a(n) for n = 1..1000
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
H. Mühle, Counting Proper Mergings of Chains and Antichains, arXiv:1206.3922 [math.CO], 2012.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A006003, A006322, A006325, A079547, A085461-A085465.

[1/20*n*(n+1)*(n^2+1)*(n^2+2*n+2): n in [1..40]]; // Vincenzo Librandi, Oct 06 2017

Rest[CoefficientList[Series[x*(1 + 4*x + x^2)^2/(1 - x)^7, {x, 0, 50}], x]] (* G. C. Greubel, Oct 06 2017 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 15, 102, 442, 1443, 3885, 9100}, 40] (* Vincenzo Librandi, Oct 06 2017 *)

x='x+O('x^50); Vec(x*(1+4*x+x^2)^2/(1-x)^7) \\ G. C. Greubel, Oct 06 2017

a(n) = n + 13*binomial(n, 2) + 60*binomial(n, 3) + 120*binomial(n, 4) + 108*binomial(n, 5) + 36*binomial(n, 6) = 1/20*n*(n+1)*(n^2+1)*(n^2+2*n+2) = Sum_{i=1..n} ((n+1-i)^3-(n-i)^3)*i^3. More generally, number of monotone n-weightings of complete bipartite digraph K(s, t) is Sum_{i=1..n} ((n+1-i)^s-(n-i)^s)*i^t = Sum_{i=1..n} ((n+1-i)^t-(n-i)^t)*i^s.
G.f.: x*(1+4*x+x^2)^2/(1-x)^7. - Colin Barker, Apr 01 2012
a(n) = A006003(n)*A006003(n+1)/5 for n>0. - Bruno Berselli, Jun 26 2018

A110450 a(n) = n(n+1)(n^2+n+1)/2.

Original entry on oeis.org

0, 3, 21, 78, 210, 465, 903, 1596, 2628, 4095, 6105, 8778, 12246, 16653, 22155, 28920, 37128, 46971, 58653, 72390, 88410, 106953, 128271, 152628, 180300, 211575, 246753, 286146, 330078, 378885, 432915, 492528, 558096, 630003, 708645, 794430

Offset: 0

Reinhard Zumkeller, Jul 21 2005

This sequence is related to A085461 by 3*A085461(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n>0. - Bruno Berselli, Dec 27 2010

Subsequence of the triangular numbers A000217, see formulas below. - David James Sycamore, Jul 31 2018

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A002378, A061317, A085461, A110449, A014105.

List([0..40],n->n*(n+1)*(n^2+n+1)/2); # Muniru A Asiru, Aug 02 2018

[n*(n+1)*(n^2+n+1)/2: n in [0..40]]; // Vincenzo Librandi, Dec 26 2010

A110450:=n->n*(n+1)*(n^2+n+1)/2; seq(A110450(k), k=0..50); # Wesley Ivan Hurt, Sep 27 2013

Table[n (n + 1) (n^2 + n + 1)/2, {n, 0, 100}] (* Wesley Ivan Hurt, Sep 27 2013 *)
CoefficientList[Series[-3 x (x^2 + 2 x + 1)/(x - 1)^5, {x, 0, 36}], x] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 3, 21, 78, 210}, 36] (* Robert G. Wilson v, Jul 31 2018 *)

a(n)=n*(n+1)*(n^2+n+1)/2 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = Sum_{k=0..n} A110449(n,k), sums of rows in triangle A110449.
From Bruno Berselli, Dec 27 2010: (Start)
G.f.: 3*x*(1 + x)^2/(1 - x)^5.
a(n) = A014105(A000217(n)). (End)
a(n) = Sum_{i=1..n*(n+1)} i. - Wesley Ivan Hurt, Sep 27 2013
a(n) = Sum_{i=0..n} i*(2*i^2+1), and these are the partial sums of A061317. - Bruno Berselli, Feb 09 2017
a(n) = t(n,t(n,A000217(n))), where t(n,k) = n*(n+1)/2 + k*n and k=0. - Bruno Berselli, Feb 28 2017
E.g.f.: (x/2)*(6 + 15*x + 8*x^2 + x^3)*exp(x). - G. C. Greubel, Aug 24 2017
a(n) = A000217(n*(n+1)). - David James Sycamore, Jul 31 2018
a(n) = A000217(2*A000217(n)) = A000217(A002378(n)). - Alois P. Heinz, Jul 31 2018
From R. J. Mathar, Mar 23 2021: (Start)
a(n) = A002378(n)+A062392(n).
a(n) = 3*A006325(n+1). (End)
Sum_{n>=1} 1/a(n) = 4 - 2*Pi*tanh(sqrt(3)*Pi/2)/sqrt(3). - Amiram Eldar, May 10 2025

A085462 Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1<=v4, v1<=v5, v2<=v4 and v3<=v4.

Original entry on oeis.org

1, 14, 77, 273, 748, 1729, 3542, 6630, 11571, 19096, 30107, 45695, 67158, 96019, 134044, 183260, 245973, 324786, 422617, 542717, 688688, 864501, 1074514, 1323490, 1616615, 1959516, 2358279, 2819467, 3350138, 3957863, 4650744

Offset: 1

Goran Kilibarda and Vladeta Jovovic, Jul 01 2003

Number of monotone n-weightings of a certain connected bipartite digraph. A monotone n-(vertex) weighting of a digraph D=(V,E) is a function w: V -> {0,1,..,n-1} such that w(v1)<=w(v2) for every arc (v1,v2) from E.

Dimensions of certain Lie algebra (see Landsberg-Manivel reference for precise definition). - N. J. A. Sloane, Oct 15 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), 143-179. [Th. 7.1, case a=-2/3]
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A006322, A006325, A079547, A085461, A085463, A085464, A085465.

[n*(n+1)*(2*n+1)*(3*n+1)*(3*n+2)/120: n in [0..50]]; // G. C. Greubel, Oct 07 2017

Rest[CoefficientList[Series[x*(1 + x)*(1 + 7*x + x^2)/(1 - x)^6, {x, 0, 50}], x]] (* or *) Table[n*(n+1)*(2*n+1)*(3*n+1)*(3*n+2)/120, {n,0,50}] (* G. C. Greubel, Oct 07 2017 *)

x='x+O('x^50); Vec(x*(1+x)*(1+7*x+x^2)/(1-x)^6) \\ G. C. Greubel, Oct 07 2017

a(n) = n + 12*binomial(n, 2) + 38*binomial(n, 3) + 45*binomial(n, 4) + 18*binomial(n, 5).
a(n) = n*(n+1)*(2*n+1)*(3*n+1)*(3*n+2)/120.
G.f.: x*(1+x)*(1+7*x+x^2)/(1-x)^6. - Colin Barker, Apr 01 2012
a(n) = sum(i=1..n, sum(j=1..n, i^2 * Min(i,j))). - Enrique Pérez Herrero, Jan 30 2013

A085463 Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1<=v4, v1<=v5, v2<=v4, v2<=v5 and v3<=v4.

Original entry on oeis.org

1, 12, 63, 219, 594, 1365, 2786, 5202, 9063, 14938, 23529, 35685, 52416, 74907, 104532, 142868, 191709, 253080, 329251, 422751, 536382, 673233, 836694, 1030470, 1258595, 1525446, 1835757, 2194633, 2607564, 3080439, 3619560, 4231656

Offset: 1

Goran Kilibarda, Vladeta Jovovic, Jul 01 2003

Number of monotone n-weightings of a certain connected bipartite digraph. A monotone n-(vertex) weighting of a digraph D=(V,E) is a function w: V -> {0,1,..,n-1} such that w(v1)<=w(v2) for every arc (v1,v2) from E.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A006322, A006325, A079547, A085461-A085465.

[n*(n+1)*(2*n+1)*(7*n^2 + 7*n + 6)/120: n in [1..25]]; // G. C. Greubel, Oct 07 2017

Table[n*(n+1)*(2*n+1)*(7*n^2 + 7*n + 6)/120, {n,1,25}] (* G. C. Greubel, Oct 07 2017 *)

for(n=1,25, print1(n*(n+1)*(2*n+1)*(7*n^2 + 7*n + 6)/120, ", ")) \\ G. C. Greubel, Oct 07 2017

a(n) = n + 10*binomial(n, 2) + 30*binomial(n, 3) + 35*binomial(n, 4) + 14*binomial(n, 5).
a(n) = n*(n+1)*(2*n+1)*(7*n^2 + 7*n + 6)/120.
G.f.: x*(1+6*x+6*x^2+x^3)/(1-x)^6. - Colin Barker, Apr 01 2012

A085464 Number of monotone n-weightings of complete bipartite digraph K(4,2).

Original entry on oeis.org

1, 19, 134, 586, 1919, 5173, 12124, 25572, 49677, 90343, 155650, 256334, 406315, 623273, 929272, 1351432, 1922649, 2682363, 3677374, 4962706, 6602519, 8671069, 11253716, 14447980, 18364645, 23128911, 28881594, 35780374, 44001091

Offset: 1

Goran Kilibarda and Vladeta Jovovic, Jul 01 2003

A monotone n-(vertex) weighting of a digraph D=(V,E) is a function w: V -> {0,1,..,n-1} such that w(v1)<=w(v2) for every arc (v1,v2) from E.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

Cf. A006322, A006325, A079547, A085461, A085462, A085463, A085465.

[(1/30)*n*(n+1)*(2*n^4+4*n^3+6*n^2+4*n-1): n in [1..25]]; // G. C. Greubel, Oct 07 2017

Table[(1/30)*n*(n+1)*(2*n^4+4*n^3+6*n^2+4*n-1), {n,1,50}] (* G. C. Greubel, Oct 07 2017 *)

a(n)=n*(n+1)*(2*n^4+4*n^3+6*n^2+4*n-1)/30 \\ Charles R Greathouse IV, Jan 16 2013

a(n) = n + 17*binomial(n, 2) + 80*binomial(n, 3) + 160*binomial(n, 4) + 144*binomial(n, 5) + 48*binomial(n, 6).
a(n) = (1/30)*n*(n+1)*(2*n^4+4*n^3+6*n^2+4*n-1).
a(n) = Sum_{i=1..n} ((n+1-i)^4-(n-i)^4)*i^2.
a(n) = Sum_{i=1..n} ((n+1-i)^2-(n-i)^2)*i^4.
More generally, number of monotone n-weightings of complete bipartite digraph K(s, t) is Sum_{i=1..n} ((n+1-i)^s-(n-i)^s)*i^t = Sum_{i=1..n} ((n+1-i)^t-(n-i)^t)*i^s.
G.f.: x*(1+x)^2*(1+10*x+x^2)/(1-x)^7. - Colin Barker, Apr 01 2012
a(n) = sum(i=1..n, sum (j=1..n, min(i,j)^4)). - Enrique Pérez Herrero, Jan 16 2013

A373424 Array read by ascending antidiagonals: T(n, k) = [x^k] cf(n) where cf(n) is the continued fraction (-1)^n/(~x - 1/(~x - ... 1/(~x - 1)))...) and where '~' is '-' if n is even, and '+' if n is odd, and x appears n times in the expression.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 5, 1, 0, 1, 5, 10, 14, 8, 1, 0, 1, 6, 15, 30, 31, 13, 1, 0, 1, 7, 21, 55, 85, 70, 21, 1, 0, 1, 8, 28, 91, 190, 246, 157, 34, 1, 0, 1, 9, 36, 140, 371, 671, 707, 353, 55, 1, 0, 1, 10, 45, 204, 658, 1547, 2353, 2037, 793, 89, 1, 0

Offset: 0

Peter Luschny, Jun 09 2024

A variant of both A050446 and A050447 which are the main entries. Differs in indexing and adds a first row to the array resp. a diagonal to the triangle.

			Generating functions of the rows:
   gf0 =  1;
   gf1 = -1/( x-1);
   gf2 =  1/(-x-1/(-x-1));
   gf3 = -1/( x-1/( x-1/( x-1)));
   gf4 =  1/(-x-1/(-x-1/(-x-1/(-x-1))));
   gf5 = -1/( x-1/( x-1/( x-1/( x-1/( x-1)))));
   gf6 =  1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1))))));
   ...
Array A(n, k) starts:
  [0] 1, 0,  0,  0,   0,    0,    0,     0,      0,      0, ...  A000007
  [1] 1, 1,  1,  1,   1,    1,    1,     1,      1,      1, ...  A000012
  [2] 1, 2,  3,  5,   8,   13,   21,    34,     55,     89, ...  A000045
  [3] 1, 3,  6, 14,  31,   70,  157,   353,    793,   1782, ...  A006356
  [4] 1, 4, 10, 30,  85,  246,  707,  2037,   5864,  16886, ...  A006357
  [5] 1, 5, 15, 55, 190,  671, 2353,  8272,  29056, 102091, ...  A006358
  [6] 1, 6, 21, 91, 371, 1547, 6405, 26585, 110254, 457379, ...  A006359
   A000027,A000330,   A085461,     A244881, ...
       A000217, A006322,    A108675, ...
.
Triangle T(n, k) = A(n - k, k) starts:
  [0] 1;
  [1] 1,  0;
  [2] 1,  1,  0;
  [3] 1,  2,  1,  0;
  [4] 1,  3,  3,  1,  0;
  [5] 1,  4,  6,  5,  1,  0;
  [6] 1,  5, 10, 14,  8,  1, 0;

T. Kyle Petersen and Yan Zhuang, Zig-zag Eulerian polynomials, arXiv:2403.07181 [math.CO], 2024. (Table 3)

Cf. A050446, A050447, A276313 (main diagonal), A373353 (row sums of triangle).

Cf. A373423.

row := proc(n, len) local x, a, j, ser; if irem(n, 2) = 1 then
a :=  x - 1; for j from 1 to n do a :=  x - 1 / a od: a :=  a - x; else
a := -x - 1; for j from 1 to n do a := -x - 1 / a od: a := -a - x;
fi; ser := series(a, x, len + 2); seq(coeff(ser, x, j), j = 0..len) end:
A := (n, k) -> row(n, 12)[k+1]:      # array form
T := (n, k) -> row(n - k, k+1)[k+1]: # triangular form

def Arow(n, len):
    R. = PowerSeriesRing(ZZ, len)
    if n == 0: return [1] + [0]*(len - 1)
    x = -x if n % 2 else x
    a = x + 1
    for _ in range(n):
        a = x - 1 / a
    a = x - a if n % 2 else a - x
    return a.list()
for n in range(7): print(Arow(n, 10))

A373423 Array read by ascending antidiagonals: T(n, k) = [x^k] cf(n) where cf(0) = 1, cf(1) = -1/(x - 1), and for n > 1 is cf(n) = ~( ~x - 1/(~x - 1/(~x - 1/(~x - 1/(~x - ... 1/(~x + 1))))...) ) where '~' is '-' if n is even, and '+' if n is odd, and x appears n times in the expression.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 1, 1, 0, 1, 4, 3, 1, 1, 0, 1, 5, 6, 5, 1, 1, 0, 1, 6, 10, 14, 8, 1, 1, 0, 1, 7, 15, 30, 31, 13, 1, 1, 0, 1, 8, 21, 55, 85, 70, 21, 1, 1, 0, 1, 9, 28, 91, 190, 246, 157, 34, 1, 1, 0, 1, 10, 36, 140, 371, 671, 707, 353, 55, 1, 1, 0

Offset: 0

Peter Luschny, Jun 09 2024

			Generating functions of row n:
   gf0 = 1;
   gf1 =   - 1/( x-1);
   gf2 = x + 1/(-x+1);
   gf3 = x - 1/( x-1/( x+1));
   gf4 = x + 1/(-x-1/(-x-1/(-x+1)));
   gf5 = x - 1/( x-1/( x-1/( x-1/( x+1))));
   gf6 = x + 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x+1)))));
.
Array begins:
  [0] 1, 0,  0,   0,   0,    0,     0,     0,      0, ...
  [1] 1, 1,  1,   1,   1,    1,     1,     1,      1, ...
  [2] 1, 2,  1,   1,   1,    1,     1,     1,      1, ...  A373565
  [3] 1, 3,  3,   5,   8,   13,    21,    34,     55, ...  A373566
  [4] 1, 4,  6,  14,  31,   70,   157,   353,    793, ...  A373567
  [5] 1, 5, 10,  30,  85,  246,   707,  2037,   5864, ...  A373568
  [6] 1, 6, 15,  55, 190,  671,  2353,  8272,  29056, ...  A373569
       A000217,  A006322,     A108675, ...
            A000330,   A085461,      A244881, ...
.
Triangle starts:
  [0] 1;
  [1] 1, 0;
  [2] 1, 1,  0;
  [3] 1, 2,  1,  0;
  [4] 1, 3,  1,  1,  0;
  [5] 1, 4,  3,  1,  1,  0;
  [6] 1, 5,  6,  5,  1,  1, 0;

Cf. A373424, A276312 (main diagonal).
Rows include: A373565, A373566, A373567, A373568, A373569.
Columns include: A000217, A000330, A006322, A085461, A108675, A244881.

row := proc(n, len) local x, a, j, ser;
if n = 0 then a := -1 elif n = 1 then a := -1/(x - 1) elif irem(n, 2) = 1 then
  a :=  x + 1; for j from 1 to n-1 do a :=  x - 1 / a od: else
  a := -x + 1; for j from 1 to n-1 do a := -x - 1 / a od: fi;
ser := series((-1)^(n-1)*a, x, len + 2); seq(coeff(ser, x, j), j = 0..len) end:
A := (n, k) -> row(n, 12)[k+1]:      # array form
T := (n, k) -> row(n - k, k+1)[k+1]: # triangular form
seq(lprint([n], row(n, 9)), n = 0..9);

def Arow(n, len):
    R. = PowerSeriesRing(ZZ, len)
    if n == 0: return [1] + [0]*(len - 1)
    if n == 1: return [1]*(len - 1)
    x = x if n % 2 == 1 else -x
    a = x + 1
    for _ in range(n - 1):
        a = x - 1 / a
    if n % 2 == 0: a = -a
    return a.list()
for n in range(8): print(Arow(n, 9))

cp's OEIS Frontend

A006858 Expansion of g.f. x*(1 + x)*(1 + 6*x + x^2)/(1 - x)^7.

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A085465 Number of monotone n-weightings of complete bipartite digraph K(3,3).

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A110450 a(n) = n*(n+1)*(n^2+n+1)/2.

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A085462 Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1<=v4, v1<=v5, v2<=v4 and v3<=v4.

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A085463 Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1<=v4, v1<=v5, v2<=v4, v2<=v5 and v3<=v4.

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A085464 Number of monotone n-weightings of complete bipartite digraph K(4,2).

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A373424 Array read by ascending antidiagonals: T(n, k) = [x^k] cf(n) where cf(n) is the continued fraction (-1)^n/(~x - 1/(~x - ... 1/(~x - 1)))...) and where '~' is '-' if n is even, and '+' if n is odd, and x appears n times in the expression.

A006858 Expansion of g.f. x(1 + x)(1 + 6*x + x^2)/(1 - x)^7.

A110450 a(n) = n(n+1)(n^2+n+1)/2.