A079547 -id:A079547 - cp's OEIS frontend

A083200 Polynexus numbers of order 7.

0, 1, 17, 118, 514, 1681, 4529, 10612, 22380, 43473, 79057, 136202, 224302, 355537, 545377, 813128, 1182520, 1682337, 2347089, 3217726, 4342394, 5777233, 7587217, 9847036, 12642020, 16069105, 20237841, 25271442, 31307878, 38501009, 47021761, 57059344, 68822512

Offset: 1

Xavier Acloque, Jun 01 2003

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A079547, A088889-A088894 (similar sequences).

Table[((n^7 - (n - 1)^7) - (n^3 - (n - 1)^3))/120, {n, 31}] (* Bruno Berselli, Feb 13 2012 *)

a(n) = ((n^7-(n-1)^7)-(n^3-(n-1)^3))/120.

G.f.: x^2*(1+10*x+20*x^2+10*x^3+x^4)/(1-x)^7. - Bruno Berselli, Feb 13 2012

More terms from David Wasserman, Oct 26 2004

Offset changed (according to the formula) from Bruno Berselli, Feb 13 2012

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)

A088889 Polynexus numbers of order 8.

Original entry on oeis.org

0, 1, 26, 245, 1353, 5368, 17017, 45878, 109446, 237291, 476476, 898403, 1607255, 2750202, 4529539, 7216924, 11169884, 16850757, 24848238, 35901697, 50928437, 71054060, 97646109, 132351154, 177135490, 234329615, 306676656, 397384911, 510184675, 649389518

Offset: 1

Xavier Acloque, Oct 21 2003

Bruno Berselli, Table of n, a(n) for n = 1..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [dead link, domain owned by a domain grabber].
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A079547, A083200, A088890, A088891, A088892, A088893, A088894.

Table[((n^8 - (n - 1)^8) - (n^4 - (n - 1)^4))/240, {n, 26}] (* Bruno Berselli, Feb 10 2012 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,1,26,245,1353,5368,17017,45878},30] (* Harvey P. Dale, Oct 31 2024 *)

a(n) = ((n^8-(n-1)^8)-(n^4-(n-1)^4))/240 = ((n^8-(n-1)^8)-(n^4-(n-1)^4))/(2^8-2^4).

G.f.: x^2*(1+x)*(1+17*x+48*x^2+17*x^3+x^4)/(1-x)^8. - Bruno Berselli, Feb 10 2012

First term added according to the formula from Bruno Berselli, Feb 10 2012

A088894 Polynexus numbers of order 16.

Original entry on oeis.org

0, 1, 656, 64895, 2263323, 40728358, 464160877, 3788798018, 23985732786, 124343403711, 548683300726, 2120490369833, 7334132330405, 23085114905772, 67009386720139, 181293280189204, 461148470725844, 1110780513139917, 2548868780082828, 5600100525524707

Offset: 1

Xavier Acloque, Oct 21 2003

Bruno Berselli, Table of n, a(n) for n = 1..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Cf. A079547, A083200, A088889, A088890, A088891, A088892, A088893.

Table[((n^16 - (n - 1)^16) - (n^4 - (n - 1)^4))/65520, {n, 20}] (* Bruno Berselli, Feb 08 2012 *)

a(n) = ((n^16-(n-1)^16)-(n^4-(n-1)^4))/65520 = ((n^16-(n-1)^16)-(n^4-(n-1)^4))/(2^16-2^4).

G.f.: x^2*(1+x)*(1+639*x+53880*x^2+1249283*x^3+10687767*x^4+38266494*x^5+59151072*x^6+38266494*x^7+10687767*x^8+1249283*x^9+53880*x^10+639*x^11+x^12)/(1-x)^16. - Bruno Berselli, Feb 08 2012

First term added according to the formula from Bruno Berselli, Feb 08 2012

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

A088890 Polynexus numbers of order 8.

Original entry on oeis.org

0, 1, 25, 234, 1290, 5115, 16211, 43700, 104244, 226005, 453805, 855646, 1530750, 2619279, 4313895, 6873320, 10638056, 16048425, 23665089, 34192210, 48503410, 67670691, 92996475, 126048924, 168700700, 223171325, 292073301, 378462150, 485890534, 618466615

Offset: 1

Xavier Acloque, Oct 21 2003

Bruno Berselli, Table of n, a(n) for n = 1..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A079547, A083200, A088889, A088891, A088892, A088893, A088894.

Table[((n^8 - (n - 1)^8) - (n^2 - (n - 1)^2))/252, {n, 26}] (* Bruno Berselli, Feb 10 2012 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,1,25,234,1290,5115,16211,43700},30] (* Harvey P. Dale, Nov 24 2019 *)

a(n) = ((n^8-(n-1)^8)-(n^2-(n-1)^2))/252 = ((n^8-(n-1)^8)-(n^2-(n-1)^2))/(2^8-2^2).

G.f.: x^2*(1+x)*(1+16*x+46*x^2+16*x^3+x^4)/(1-x)^8. - Bruno Berselli, Feb 10 2012

First term added according to the formula from Bruno Berselli, Feb 10 2012

A088891 Polynexus numbers of order 9.

Original entry on oeis.org

0, 1, 38, 481, 3355, 16120, 60071, 186238, 502386, 1215435, 2694340, 5559191, 10803013, 19953466, 35282365, 60071660, 98945236, 158276613, 246683346, 375619645, 560079455, 819422956, 1178340163, 1667966026, 2327162150, 3203980975, 4357328976, 5858846163

Offset: 1

Xavier Acloque, Oct 21 2003

Bruno Berselli, Table of n, a(n) for n = 1..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A079547, A083200, A088889, A088890, A088892, A088893, A088894.

Table[((n^9-(n-1)^9)-(n^3-(n-1)^3))/504,{n,30}] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{0,1,38,481,3355,16120,60071,186238,502386},30] (* Harvey P. Dale, Jan 18 2012 *)

a(n) = ((n^9-(n-1)^9)-(n^3-(n-1)^3))/504 = ((n^9-(n-1)^9)-(n^3-(n-1)^3))/(2^9-2^3).
a(1)=1, a(2)=38, a(3)=481, a(4)=3355, a(5)=16120, a(6)=60071, a(7)=186238, a(8)=502386, a(9)=1215435, a(n)=9*a(n-1)-36*a(n-2)+ 84*a(n-3)- 126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - Harvey P. Dale, Jan 18 2012
G.f.: x^2*(1+29*x+175*x^2+310*x^3+175*x^4+29*x^5+x^6)/(1-x)^9. - Bruno Berselli, Feb 10 2012

Offset changed and first term added (according to the formula) from Bruno Berselli, Feb 08 2012

A088892 Polynexus numbers of order 14.

Original entry on oeis.org

0, 1, 291, 16096, 356232, 4411517, 36621423, 227095448, 1128128568, 4708376529, 17078744419, 55199550120, 161993768080, 438011626365, 1103841220991, 2616890599056, 5880356075792, 12602902382337, 25897027973187, 51245013077968, 98017089897528, 181801058663389

Offset: 1

Xavier Acloque, Oct 21 2003

Bruno Berselli, Table of n, a(n) for n = 1..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Cf. A079547, A083200, A088889, A088890, A088891, A088893, A088894.

Table[((n^14 - (n - 1)^14) - (n^2 - (n - 1)^2))/16380, {n, 20}] (* Bruno Berselli, Feb 08 2012 *)
LinearRecurrence[{14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1},{0,1,291,16096,356232,4411517,36621423,227095448,1128128568,4708376529,17078744419,55199550120,161993768080,438011626365},20] (* Harvey P. Dale, Mar 04 2024 *)

a(n) = ((n^14-(n-1)^14)-(n^2-(n-1)^2))/16380 = ((n^14-(n-1)^14)-(n^2-(n-1)^2))/(2^14-2^2).

G.f.: x^2*(1+x)*(1+276*x+11837*x^2+145168*x^3+638914*x^4+1068728*x^5+638914*x^6+145168*x^7+11837*x^8+276*x^9+x^10)/(1-x)^14. - Bruno Berselli, Feb 08 2012

First term added according to the formula from Bruno Berselli, Feb 08 2012

A088893 Polynexus numbers of order 15.

Original entry on oeis.org

0, 1, 437, 32338, 898774, 13420861, 130567049, 929084572, 5210829060, 24240197433, 96985597357, 342789175982, 1092151142842, 3186269918917, 8618003504977, 21826239750488, 52182586901800, 118565859736497, 257462955231909, 536839834252906, 1079191653936254

Offset: 1

Xavier Acloque, Oct 21 2003

Bruno Berselli, Table of n, a(n) for n = 1..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Cf. A079547, A083200, A088889, A088890, A088891, A088892, A088894.

Table[((n^15 - (n - 1)^15) - (n^3 - (n - 1)^3))/32760, {n, 20}] (* Bruno Berselli, Feb 08 2012 *)

a(n) = ((n^15-(n-1)^15)-(n^3-(n-1)^3))/32760 = ((n^15-(n-1)^15)-(n^3-(n-1)^3))/(2^15-2^3).

G.f.: x^2*(1+422*x+25888*x^2+459134*x^3+3137271*x^4+9505116*x^5+13661136*x^6+9505116*x^7+3137271*x^8+459134*x^9+25888*x^10+422*x^11+x^12)/(1-x)^15. - Bruno Berselli, Feb 08 2012

First term added according to the formula from Bruno Berselli, Feb 08 2012

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

cp's OEIS Frontend

A083200 Polynexus numbers of order 7.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A088889 Polynexus numbers of order 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A088894 Polynexus numbers of order 16.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A088890 Polynexus numbers of order 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A088891 Polynexus numbers of order 9.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A088892 Polynexus numbers of order 14.

Views

Author

Keywords

Links