A085465 -id:A085465 - cp's OEIS frontend

A079547 a(n) = ((n^6 - (n-1)^6) - (n^2 - (n-1)^2))/60.

0, 1, 11, 56, 192, 517, 1183, 2408, 4488, 7809, 12859, 20240, 30680, 45045, 64351, 89776, 122672, 164577, 217227, 282568, 362768, 460229, 577599, 717784, 883960, 1079585, 1308411

Offset: 1

Xavier Acloque, Jan 22 2003

Polynexus numbers of order 6.
A polynexus (subtractive) function is composed of two or more subtracted nexus numbers divided by an integer x. The general form of the formula is a(n)=((n^p-(n-1)^p)-(n^q-(n-1)^q))/x, where n, p, q and x are integers.
Already known: ((n^5-(n-1)^5) - (n^3-(n-1)^3))/24, giving A006322 for n>1; ((n^4-(n-1)^4) - (n^2-(n-1)^2))/12, giving A000330; ((n^3-(n-1)^3) - (n^1-(n-1)^1))/6, giving A000217; ((n^2-(n-1)^2) - (n^1-(n-1)^1))/2, giving n; ((n^2-(n-1)^2) - (n^0-(n-1)^0))/1, giving 2*n-1. In those examples, x is equal to 1,2,6,12,24, and 3 is also possible.
Also number of monotone n-weightings of complete bipartite digraph K(3,2) if offset were 0; cf. A085464-A085465. - Goran Kilibarda, Vladeta Jovovic, Jul 01 2003
Partial sums of A037270. - J. M. Bergot, Jun 07 2012

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A006322, A000330, A000217, A047969, A003215, A083200, A088889-A088894.

List([1..30], n-> n*(6*n^4-15*n^3+20*n^2-15*n+4)/60) # G. C. Greubel, Jun 19 2019

[n*(6*n^4-15*n^3+20*n^2-15*n+4)/60: n in [1..30]]; // G. C. Greubel, Jun 19 2019

Table[((n^6 -(n-1)^6) - (n^2 -(n-1)^2))/60, {n, 30}] (* Bruno Berselli, Feb 13 2012 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,11,56,192,517},30] (* Harvey P. Dale, Feb 21 2023 *)

a(n) = n*(6*n^4-15*n^3+20*n^2-15*n+4)/60 \\ Charles R Greathouse IV, Jan 16 2013

[n*(6*n^4-15*n^3+20*n^2-15*n+4)/60 for n in (1..30)] # G. C. Greubel, Jun 19 2019

a(n+1) = Sum_{i=1..n} (i^2 + i^4)/2 = n*(2*n+1)*(n+1)*(3*n^2+3*n+4)/60. - Vladeta Jovovic, Mar 17 2006
G.f.: x^2*(x+1)*(1+4*x+x^2)/(1-x)^6. - Bruno Berselli, Feb 13 2012
a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} min(i,j)^3. - Enrique Pérez Herrero, Jan 16 2013
E.g.f.: x^2*(30 + 80*x + 45*x^2 + 6*x^3)*exp(x)/60. - G. C. Greubel, Jun 19 2019

A085461 Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1 <= v5, v2 <= v5, v2 <= v4 and v3 <= v4.

Original entry on oeis.org

1, 13, 70, 246, 671, 1547, 3164, 5916, 10317, 17017, 26818, 40690, 59787, 85463, 119288, 163064, 218841, 288933, 375934, 482734, 612535, 768867, 955604, 1176980, 1437605, 1742481, 2097018, 2507050, 2978851, 3519151, 4135152, 4834544

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.
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
Can be constructed by taking the product of the three members of a Pythagorean triples and dividing by 60. Formula: n*(n^2-1)*(n^2+1)/240 where n runs through the odd numbers >= 3. - Pierre Gayet, Apr 04 2009
Number of composable morphisms in a height-n tower of retractions. A retraction between objects X and Y is a pair of maps s:X->Y and r:Y->X such that r(s(x))=x for all x in X. Given objects X_0,X_1,X_2,...,X_n, we can ask for retractions s_i:X_i->X_{i+1},r_i:X_{i+1}->X_i, for each 0 <= i < n. The total number of morphisms in that category is 0^2 + 1^2 + 2^2 + ... + n^2 (cf. A000330). The total number of composable pairs of morphisms in that category is the sequence given here. - David Spivak, Feb 26 2014

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 168).

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.
Daeseok Lee and H.-K. Ju, An Extension of Hibi's palindromic theorem, arXiv preprint arXiv:1503.05658 [math.CO], 2015.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
See p. 31
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A006322, A006325, A079547, A085462, A085463, A085464, A085465.

Rest[CoefficientList[Series[x*(1 + x)*(1 + 6*x + x^2)/(1 - x)^6, {x, 0, 50}], x]] (* G. C. Greubel, Oct 06 2017 *)

x='x+O('x^50); Vec(x*(1+x)*(1+6*x+x^2)/(1-x)^6) \\ G. C. Greubel, Oct 06 2017

a(n) = n + 11*binomial(n, 2) + 34*binomial(n, 3) + 40*binomial(n, 4) + 16*binomial(n, 5) = 1/30*n*(n+1)*(2*n+1)*(2*n^2 + 2*n + 1).
From Bruno Berselli, Dec 27 2010: (Start)
G.f.: x*(1+x)*(1+6*x+x^2)/(1-x)^6.
a(n) = ( n*A110450(n) - Sum_{i=0..n-1} A110450(i) )/3. (End)

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

A274778 Number of proper mergings of an n-antichain and an n-chain.

Original entry on oeis.org

0, 3, 26, 442, 12899, 582381, 37700452, 3315996468, 380835212037, 55380159334315, 9950025870043126, 2165134468142294430, 561245519520167902471, 170913803045738754172185, 60421582956702701927410120, 24543570079301728283314502248, 11353373604627607560431407875081

Offset: 0

Henri Mühle, Nov 11 2016

nonn

a(n) is also the number of monotone (n+1)-colorings of a complete bipartite digraph K(n,n), where a monotone (n+1)-coloring is a labeling w of the vertices of K(n,n) with integers in {1,2,...,n+1} such that for every arc (e1, e2) we have w(e1) <= w(e2).

			For n=1, the three proper mergings of a 1-chain {x} and a 1-antichain {y} are x<y, y<x, and x,y.

H. Mühle, Counting Proper Mergings of Chains and Antichains, Discrete Math., Vol. 327(C), 2014, 118-129. Also arXiv:1206.3922 [math.CO], 2012.

Cf. A085465.

a := n -> add(((n-i+1)^n-(n-i)^n)*(i+1)^n, i=0..n):
seq(a(n), n=0..16); # Peter Luschny, Nov 11 2016

a[0] = 0; a[n_] := Sum[((n-i+1)^n - (n-i)^n)*(i+1)^n, {i, 0, n}];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jul 14 2018, after Peter Luschny *)

a(n) = sum(i=1, n+1, ((n+2-i)^n - (n+1-i)^n)*i^n); \\ Michel Marcus, Jul 14 2018

a(n) = Sum_{i=1..n+1} ((n+2-i)^n - (n+1-i)^n)*i^n.

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A079547 a(n) = ((n^6 - (n-1)^6) - (n^2 - (n-1)^2))/60.

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A085461 Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1 <= v5, v2 <= v5, v2 <= v4 and v3 <= v4.

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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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A274778 Number of proper mergings of an n-antichain and an n-chain.

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