cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A049235 Sum of balls on the lawn for the s=3 tennis ball problem.

Original entry on oeis.org

0, 6, 75, 708, 5991, 47868, 369315, 2783448, 20631126, 151026498, 1094965524, 7878119760, 56330252412, 400703095284, 2838060684483, 20027058300144, 140874026880204, 988194254587242, 6915098239841331, 48285969880645908, 336521149274459979
Offset: 0

Views

Author

N. J. A. Sloane, Jan 19 2003

Keywords

Links

Crossrefs

The four sequences T_n, Y_n, A_n, S_n for s=2 are A000108, A000302, A000346, A031970, for s=3, A001764, A006256, A075045, this sequence, for s=4, A002293, A078995, A078999, A078516.
Cf. A079486.

Programs

  • Maple
    T := (n,s)->binomial(s*n,n)/((s-1)*n+1); Y := (n,s)->add(binomial(s*k,k)*binomial(s*(n-k),n-k),k=0..n); A := (n,s)->Y(n+1,s)/2-(1/2)*((2*s-3)*n+2*s-2)*T(n+1,s); S := (n,s)->(1/2)*(s*n^2+(3*s-1)*n+2*s)*T(n+1,s)-Y(n+1,s)/2;
F := 3*(2-3*t)*t*((t-1)*(3*t-1))^(-3);  G := t*(t-1)^2;   Ginv := RootOf(G-x,t);
ogf := series(eval(F,t=Ginv), x=0, 20);
  • Mathematica
    a[n_] := a[n] = Switch[n, 0, 0, 1, 6, 2, 75, 3, 708, 4, 5991, _, -((1/(8*(2*(n-5)^2 + 25*(n-5) + 78)))*(-(531441*(n-5)^2* a[n-5]) + 196830*(n-5)^2*a[n-4] - 24057*(n-5)^2*a[n-3] + 1809*(n-5)^2*a[n-2] - 232*(n-5)^2*a[n-1] - 1594323*(n-5)*a[n-5] + 747954*(n-5)*a[n-4] - 120285*(n-5)*a[n-3] + 16362*(n-5)*a[n-2] - 2798*(n-5)*a[n-1] - 1180980*a[n-5] + 656100*a[n-4] - 131220*a[n-3] + 36825*a[n-2] - 8352*a[n-1]))];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 02 2023, after Robert Israel *)

Formula

a(n) is asymptotic to c*sqrt(n)*(27/4)^n with c=2.4... - Benoit Cloitre, Jan 26 2003, c = 81*sqrt(3/Pi)/32 = 2.4735502165085321... - Vaclav Kotesovec, Feb 07 2019
G.f.: F(G^(-1)(x)) where F = 3*(2-3*t)*t*((t-1)*(3*t-1))^(-3) and G = t*(t-1)^2. - Mark van Hoeij, Oct 30 2011
D-finite with recurrence (531441*n^2 + 1594323*n + 1180980)*a(n) + (-196830*n^2 - 747954*n - 656100)*a(n + 1) + (24057*n^2 + 120285*n + 131220)*a(n + 2) + (-1809*n^2 - 16362*n - 36825)*a(n + 3) + (232*n^2 + 2798*n + 8352)*a(n + 4) + (-16*n^2 - 200*n - 624)*a(n + 5) = 0. - Robert Israel, Jun 20 2019

A171074 A115112 with initial term changed from 0 to 1.

Original entry on oeis.org

1, 4, 18, 68, 250, 922, 3430, 12868, 48618, 184754, 705430, 2704154, 10400598, 40116598, 155117518, 601080388, 2333606218, 9075135298, 35345263798, 137846528818, 538257874438, 2104098963718, 8233430727598, 32247603683098
Offset: 1

Views

Author

N. J. A. Sloane, Sep 06 2010

Keywords

Comments

From a variant of the tennis ball problem (cf. A031970, A049235). On turn n ball 2n-1 is introduced to the room, ball 2n is introduced to the garden, then one of the balls in the room is swapped with one of the balls in the garden. The present sequence gives the number of combinations, while A171075 gives the total on the lawn, A170076 gives the total in the room.

References

  • David Scambler, Just for fun, more tennis balls, Posting to the Sequence Fans Mailing List, Aug 25 2010.

Crossrefs

Cf. A031970, A049235, A171075, A171076.

A338403 Regular triangle read by rows: T(n,k) is the number of (n,k)-Duck words, for n>=1 and 0<=k<=n-1.

Original entry on oeis.org

1, 2, 3, 5, 23, 14, 14, 131, 233, 84, 42, 664, 2339, 2367, 594, 132, 3166, 18520, 36265, 24714, 4719, 429, 14545, 127511, 408311, 527757, 266219, 40898
Offset: 1

Views

Author

Michel Marcus, Oct 24 2020

Keywords

Comments

See link for the definition of Duck word.

Examples

			Triangle begins:
   1;
   2,   3;
   5,  23,   14;
  14, 131,  233,   84;
  42, 664, 2339, 2367, 594;
  ...

Links

Crossrefs

Cf. A000108 (column 0), A005700 (diagonal), A005789 (row sums), A031970 (column 1).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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