A005340 -id:A005340 - cp's OEIS frontend

A005337 Number of ways in which n identical balls can be distributed among 4 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

15, 40, 76, 124, 185, 260, 350, 456, 579, 720, 880, 1060, 1261, 1484, 1730, 2000, 2295, 2616, 2964, 3340, 3745, 4180, 4646, 5144, 5675, 6240, 6840, 7476, 8149, 8860, 9610, 10400, 11231, 12104, 13020, 13980, 14985

Offset: 8

N. J. A. Sloane, Simon Plouffe

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 8..1000
D. R. Breach, Letter to N. J. A. Sloane, Jun 1980
Philippe Flajolet, Balls and Urns, etc., A problem in submarine detection (solution to problem 68-16).
M. Hayes (proposer) and D. R. Breach (solver), A combinatorial problem, Problem 68-16, SIAM Rev. 12 (1970), 294-297.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005338, A005339, A005340. A column of A259975.

A005337:=(15-20*z+6*z**2)/(z-1)**4; # Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(15 - 20 x + 6 x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 14 2013 *)
LinearRecurrence[{4,-6,4,-1},{15,40,76,124},50] (* Harvey P. Dale, May 11 2014 *)

G.f.: x^8*(15 - 20*x + 6*x^2)/(1 - x)^4.

a(n) = (546 - 169*n + 6*n^2 + n^3)/6. [Colin Barker, Jul 08 2012]

G.f. corrected by Colin Barker, Jul 08 2012

Name clarified by Alois P. Heinz, Oct 02 2017

A005338 Number of ways in which n identical balls can be distributed among 5 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

Original entry on oeis.org

1, 8, 31, 85, 190, 360, 610, 956, 1415, 2005, 2745, 3655, 4756, 6070, 7620, 9430, 11525, 13931, 16675, 19785, 23290, 27220, 31606, 36480, 41875, 47825, 54365, 61531, 69360, 77890, 87160, 97210, 108081, 119815, 132455, 146045, 160630

Offset: 8

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 8..1000
D. R. Breach, Letter to N. J. A. Sloane, Jun 1980
Philippe Flajolet, Balls and Urns, etc., A problem in submarine detection (solution to problem 68-16).
M. Hayes (proposer) and D. R. Breach (solver), A combinatorial problem, Problem 68-16, SIAM Rev. 12 (1970), 294-297.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005337, A005339, A005340.

I:=[1, 8, 31, 85, 190, 360, 610]; [n le 7 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, May 11 2012

f[x_] := x^8*(1 + 3*x + x^2 - 11*x^5 + 7*x^6)/(1 - x)^5; Drop[ CoefficientList[ Series[f[x], {x, 0, 44}], x], 8] (* Jean-François Alcover, Oct 05 2011, after Vladeta Jovovic *)
LinearRecurrence[{5,-10,10,-5,1},{1,8,31,85,190,360,610},40] (* Harvey P. Dale, Aug 26 2019 *)

G.f.: x^8*(1 + 3*x + x^2 - 11*x^5 + 7*x^6)/(1 - x)^5. - Vladeta Jovovic, Apr 13 2008

a(n) = (n^4 + 10*n^3 - 445*n^2 + 2690*n - 1656)/24 for n > 9. - Colin Barker, May 10 2012

Corrected and extended by Vladeta Jovovic, Apr 13 2008

Name clarified by Alois P. Heinz, Oct 02 2017

A005339 Number of ways in which n identical balls can be distributed among 6 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

Original entry on oeis.org

35, 154, 424, 930, 1775, 3080, 4985, 7650, 11256, 16006, 22126, 29866, 39501, 51332, 65687, 82922, 103422, 127602, 155908, 188818, 226843, 270528, 320453, 377234, 441524, 514014, 595434, 686554, 788185, 901180, 1026435, 1164890, 1317530

Offset: 12

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. R. Breach, Letter to N. J. A. Sloane, Jun 1980
Philippe Flajolet, Balls and Urns, etc., A problem in submarine detection (solution to problem 68-16).
M. Hayes (proposer) and D. R. Breach (solver), A combinatorial problem, Problem 68-16, SIAM Rev. 12 (1970), 294-297.
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Cf. A005337, A005338, A005340.

Drop[CoefficientList[Series[x^12(35-56x+25x^2-4x^3+x^6)/(1-x)^6, {x,0, 60}], x],12] (* or *) Join[{35},LinearRecurrence[{6,-15,20,-15,6,-1},{154,424,930,1775,3080,4985},48]] (* Harvey P. Dale, Aug 12 2011 *)

G.f.: x^12*(35 - 56*x + 25*x^2 - 4*x^3 + x^6)/(1-x)^6. - Vladeta Jovovic, Apr 13 2008

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6), a(12)=35, a(13)=154, a(14)=424, a(15)=930, a(16)=1775, a(17)=3080, a(18)=4985. - Harvey P. Dale, Aug 12 2011

More terms from Vladeta Jovovic, Apr 13 2008

Name clarified by Alois P. Heinz, Oct 02 2017

A259975 Irregular triangle read by rows: T(n,k) = number of ways of placing n balls into k boxes in such a way that any two adjacent boxes contain at least 4 balls.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 6, 4, 1, 7, 9, 1, 8, 16, 1, 9, 25, 15, 1, 1, 10, 35, 40, 8, 1, 11, 46, 76, 31, 1, 12, 58, 124, 85, 1, 13, 71, 185, 190, 35, 1, 1, 14, 85, 260, 360, 154, 13, 1, 15, 100, 350, 610, 424, 76, 1, 16, 116, 456, 956, 930, 295

Offset: 0

N. J. A. Sloane, Jul 12 2015

			Triangle begins:
  1;
  1;
  1;
  1;
  1,  5,   1;
  1,  6,   4;
  1,  7,   9;
  1,  8,  16;
  1,  9,  25,  15,   1;
  1, 10,  35,  40,   8;
  1, 11,  46,  76,  31;
  1, 12,  58, 124,  85;
  1, 13,  71, 185, 190,  35,  1;
  1, 14,  85, 260, 360, 154, 13;
  1, 15, 100, 350, 610, 424, 76;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
D. R. Breach, Letter to N. J. A. Sloane, Jun 1980
M. Hayes (proposer) and D. R. Breach (solver), A combinatorial problem, Problem 68-16, SIAM Rev. 12 (1970), 294-297.

Columns: A004120, A005337, A005338, A005339, A005340.

Row sums give A257666.

b:= proc(n, v) option remember; expand(`if`(n=0,
      `if`(v=0, 1+x, 1), add(x*b(n-j, max(0, 4-j)), j=v..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, 0)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 12 2015

b[n_, v_] := b[n, v] = Expand[If[n == 0, If[v == 0, 1+x, 1], Sum[x*b[n-j, Max[0, 4-j]], {j, v, n}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 13 2016, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jul 12 2015

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A005337 Number of ways in which n identical balls can be distributed among 4 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

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A005338 Number of ways in which n identical balls can be distributed among 5 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

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A005339 Number of ways in which n identical balls can be distributed among 6 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

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Extensions

A259975 Irregular triangle read by rows: T(n,k) = number of ways of placing n balls into k boxes in such a way that any two adjacent boxes contain at least 4 balls.

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Maple

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