A055794 -id:A055794 - cp's OEIS frontend

A055796 T(2n+3,n), array T as in A055794.

1, 5, 16, 42, 98, 210, 420, 792, 1419, 2431, 4004, 6370, 9828, 14756, 21624, 31008, 43605, 60249, 81928, 109802, 145222, 189750, 245180, 313560, 397215, 498771, 621180, 767746, 942152, 1148488, 1391280, 1675520, 2006697, 2390829, 2834496, 3344874, 3929770

Offset: 0

Clark Kimberling, May 28 2000

If Y is a 2-subset of an n-set X then, for n>=6, a(n-6) is the number of 6-subsets of X which do not have exactly one element in common with Y. - Milan Janjic, Dec 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A051601.

[(n+1)*(n+2)*(n+3)*(n+4)*(n^2-n+30)/720: n in [0..40]]; // Vincenzo Librandi, Apr 30 2012

seq(binomial(n+4, 6)+binomial(n+4, 4), n=0..33) # Zerinvary Lajos, Jul 24 2006

a=1;b=2;c=3;d=4;s=5;lst={1,s};Do[a+=n;b+=a;c+=b;d+=c;s+=d;AppendTo[lst,s],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, May 24 2009 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,5,16,42,98, 210,420},50] (* Vincenzo Librandi, Apr 30 2012 *)
Table[(n+1)(n+2)(n+3)(n+4)(n^2-n+30)/720,{n,0,50}] (* Harvey P. Dale, Feb 12 2013 *)

a(n) = (n+1)(n+2)(n+3)(n+4)(n^2-n+30)/720.
a(n-4) = binomial(n,6) + binomial(n,4) for n>3. - Zerinvary Lajos, Jul 24 2006
G.f.: (1-2*x+2*x^2)/(1-x)^7. - Colin Barker, Feb 22 2012

More terms from Harvey P. Dale, Feb 12 2013

A055797 T(2n+4,n), array T as in A055794.

Original entry on oeis.org

1, 6, 22, 64, 162, 372, 792, 1584, 3003, 5434, 9438, 15808, 25636, 40392, 62016, 93024, 136629, 196878, 278806, 388608, 533830, 723580, 968760, 1282320, 1679535, 2178306, 2799486, 3567232, 4509384

Offset: 0

Clark Kimberling, May 28 2000

If Y is a 2-subset of an n-set X then, for n>=7, a(n-7) is the number of 7-subsets of X which do not have exactly one element in common with Y. - Milan Janjic, Dec 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A051601.

[Binomial(n,7) + Binomial(n,5): n in [5..40]]; // Vincenzo Librandi, May 01 2012

[seq(binomial(n,7)+binomial(n,5), n=5..34)]; # Zerinvary Lajos, Jul 24 2006

a=1;b=2;c=3;d=4;e=5;f=6;s=7;lst={s};Do[a+=n;b+=a;c+=b;d+=c;e+=d;f+=e;s+=f;AppendTo[lst,s],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, May 24 2009 *)
CoefficientList[Series[(1-2*x+2*x^2)/(1-x)^8,{x,0,30}],x] (* Vincenzo Librandi, May 01 2012 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,6,22,64,162,372,792,1584},30] (* Harvey P. Dale, Mar 11 2023 *)

a(n-5) = binomial(n,7) + binomial(n,5) for n>4. - Zerinvary Lajos, Jul 24 2006
G.f.: (1-2*x+2*x^2)/(1-x)^8. - Colin Barker, Feb 22 2012
a(n) = 8*(n-1) - 28*(n-2) + 56*(n-3) - 70*(n-4) + 56*(n-5) - 28*(n-6) + 8*(n-7) - (n-8). - Vincenzo Librandi, May 01 2012
a(n) = (n+5)*(n+4)*(n+3)*(n+2)*(n+1)*(n^2-n+42)/5040. - R. J. Mathar, Oct 01 2021

A055798 T(2n+5,n), array T as in A055794.

Original entry on oeis.org

1, 7, 29, 93, 255, 627, 1419, 3003, 6006, 11440, 20878, 36686, 62322, 102714, 164730, 257754, 394383, 591261, 870067, 1258675, 1792505, 2516085, 3484845, 4767165, 6446700, 8625006, 11424492, 14991724, 19501108, 25158980, 32208132, 40932804, 51664173

Offset: 0

Clark Kimberling, May 28 2000

If Y is a 2-subset of an n-set X then, for n>=8, a(n-8) is the number of 8-subsets of X which do not have exactly one element in common with Y. - Milan Janjic, Dec 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A051601.

[Binomial(n,8)-2*Binomial(n-2,7): n in [8..40]]; // Vincenzo Librandi, May 01 2012

CoefficientList[Series[(-2*(z - 1)*z - 1)/(z - 1)^9, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 16 2011 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,7,29,93,255,627,1419,3003,6006},50] (* Vincenzo Librandi, May 01 2012 *)

a(n-8) = binomial(n,8)-2*binomial(n-2,7), n=8,9,10,.... - Milan Janjic, Dec 28 2007
G.f.: (1-2*x+2*x^2)/(1-x)^9. [Colin Barker, Feb 22 2012]
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). - Vincenzo Librandi, May 01 2012

A055799 T(2n+6,n), array T as in A055794.

Original entry on oeis.org

1, 8, 37, 130, 385, 1012, 2431, 5434, 11440, 22880, 43758, 80444, 142766, 245480, 410210, 667964, 1062347, 1653608, 2523675, 3782350, 5574855, 8090940, 11575785, 16342950, 22789650, 31414656, 42839148, 57830872

Offset: 0

Clark Kimberling, May 28 2000

If Y is a 2-subset of an n-set X then, for n>=9, a(n-9) is the number of 9-subsets of X which do not have exactly one element in common with Y. - Milan Janjic, Dec 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A051601.

[Binomial(n,9)-2*Binomial(n-2,8):n in [9..40]]; // Vincenzo Librandi, May 01 2012

a=1;b=2;c=3;d=4;e=5;f=6;g=7;s=8;lst={1,s};Do[a+=n;b+=a;c+=b;d+=c;e+=d;f+=e;g+=f;s+=g;AppendTo[lst,s],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, May 24 2009 *)
CoefficientList[Series[(1-2*x+2*x^2)/(1-x)^10,{x,0,30}],x] (* Vincenzo Librandi, May 01 2012 *)

a(n-9) = binomial(n,9) - 2*binomial(n-2,8), n=9, 10, ... . - Milan Janjic, Dec 28 2007
G.f.: (1-2*x+2*x^2)/(1-x)^10. - Colin Barker, Feb 21 2012
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4)+ 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10). - Vincenzo Librandi, May 01 2012

A055795 a(n) = binomial(n,4) + binomial(n,2).

Original entry on oeis.org

0, 1, 3, 7, 15, 30, 56, 98, 162, 255, 385, 561, 793, 1092, 1470, 1940, 2516, 3213, 4047, 5035, 6195, 7546, 9108, 10902, 12950, 15275, 17901, 20853, 24157, 27840, 31930, 36456, 41448, 46937, 52955, 59535, 66711, 74518, 82992, 92170, 102090, 112791, 124313, 136697

Offset: 1

Clark Kimberling, May 28 2000

Answer to the question: if you have a tall building and 4 plates and you need to find the highest story from which a plate thrown does not break, what is the number of stories you can handle given n tries?
If Y is a 2-subset of an n-set X then, for n >= 4, a(n-3) is the number of 4-subsets of X which do not have exactly one element in common with Y. - Milan Janjic, Dec 28 2007
Antidiagonal sums of A139600. - Johannes W. Meijer, Apr 29 2011
Also the number of maximal cliques in the n-tetrahedral graph for n > 5. - Eric W. Weisstein, Jun 12 2017
Mark each point on an 8^(n-2) grid with the number of points that are visible from the point; for n > 3, a(n) is the number of distinct values in the grid. - Torlach Rush, Mar 25 2021
Antidiagonal sums of both A057145 and also A134394 yield this sequence without the initial term 0. - Michael Somos, Nov 23 2021

James Spahlinger, Table of n, a(n) for n = 1..1000
Michael Boardman, The Egg-Drop Numbers, Mathematics Magazine, 77 (2004), 368-372.
Milan Janjic, Two Enumerative Functions
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
Eric Weisstein's World of Mathematics, Johnson Graph
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Tetrahedral Graph
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

T(2n+1, n), array T as in A055794. Cf. A004006, A000127.

Cf. A134394, A051601, A057145.

[n*(n^3-6*n^2+23*n-18)/24: n in [1..100]]; // Wesley Ivan Hurt, Sep 29 2013

A055795:=n->binomial(n,4)+binomial(n,2); # Zerinvary Lajos, Jul 24 2006

Table[Binomial[n, 4] + Binomial[n, 2], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, May 24 2009 *)
Table[n (n^3 - 6 n^2 + 23 n - 18)/24, {n, 100}] (* Wesley Ivan Hurt, Sep 29 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 3, 7, 15}, 50] (* Harvey P. Dale, Dec 07 2015 *)
Total[Binomial[Range[20], #] & /@ {2, 4}] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[x (-1 + 2 x - 2 x^2)/(-1 + x)^5, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017~ *)

A055795(n):=n*(n^3-6*n^2+23*n-18)/24$ makelist(A055795(n), n, 1, 100); /* Wesley Ivan Hurt, Sep 29 2013 */

a(n)= n*(n^3-6*n^2+23*n-18)/24 \\ Wesley Ivan Hurt, Sep 29 2013

a(n) = A000127(n)-1. Differences give A000127.
a(1) = 1; a(n) = a(n-1) + 1 + A004006(n-1).
a(n+1) = C(n, 1) + C(n, 2) + C(n, 3) + C(n, 4). - James Sellers, Mar 16 2002
Row sums of triangle A134394. Also, binomial transform of [1, 2, 2, 2, 1, 0, 0, 0, ...]. - Gary W. Adamson, Oct 23 2007
O.g.f.: -x^2(1-2x+2x^2)/(x-1)^5. a(n) = A000332(n) + A000217(n-1). - R. J. Mathar, Apr 13 2008
a(n) = n*(n^3 - 6*n^2 + 23*n - 18)/24. - Gary Detlefs, Dec 08 2011
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(1)=0, a(2)=1, a(3)=3, a(4)=7, a(5)=15. - Harvey P. Dale, Dec 07 2015

Better description from Leonid Broukhis, Oct 24 2000
Edited by Zerinvary Lajos, Jul 24 2006
Offset corrected and Sellers formula adjusted by Gary Detlefs, Nov 28 2011

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A055796 T(2n+3,n), array T as in A055794.

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A055797 T(2n+4,n), array T as in A055794.

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A055798 T(2n+5,n), array T as in A055794.

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A055799 T(2n+6,n), array T as in A055794.

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A055795 a(n) = binomial(n,4) + binomial(n,2).

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