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A213808 Triangle of numbers C^(7)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 7 appearances allowed. 2
1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 10, 20, 35, 1, 5, 15, 35, 70, 126, 1, 6, 21, 56, 126, 252, 462, 1, 7, 28, 84, 210, 462, 924, 1716, 1, 8, 36, 120, 330, 792, 1716, 3432, 6427, 1, 9, 45, 165, 495, 1287, 3003, 6435, 12861, 24229, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24300, 48520, 91828 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

For k <= 6, the triangle coincides with triangle A213745.

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

FORMULA

T(n,k) = Sum_{r=0..floor(k/8)} (-1)^r*C(n,r)*C(n-8*r+k-1, n-1).

T(n,0)=1, T(n,1)=n, T(n,2)=A000217(n) for n > 1, T(n,3)=A000292(n) for n >= 3, T(n,4)=A000332(n) for n >= 7, T(n,5)=A000389(n) for n >= 9, T(n,6)=A000579(n) for n >= 11, T(n,7)=A000580(n) for n >= 13.

EXAMPLE

Triangle begins

n/k |  0     1     2     3     4     5     6     7     8

----+---------------------------------------------------

  0 |  1

  1 |  1     1

  2 |  1     2     3

  3 |  1     3     6    10

  4 |  1     4    10    20    35

  5 |  1     5    15    35    70   126

  6 |  1     6    21    56   126   252   462

  7 |  1     7    28    84   210   462   924  1716

  8 |  1     8    36   120   330   792  1716  3432  6427

MATHEMATICA

Table[Sum[(-1)^r*Binomial[n, r]*Binomial[n - 8*r + k - 1, n - 1], {r, 0, Floor[k/8]}], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Nov 25 2017 *)

PROG

(PARI) for(n=0, 10, for(k=0, n, print1(if(n==0 && k==0, 1, sum(r=0, floor(k/8), (-1)^r*binomial(n, r)*binomial(n-8*r + k-1, n-1))), ", "))) \\ G. C. Greubel, Nov 25 2017

CROSSREFS

Cf. A007318, A005725, A059481, A111808, A187925, A213742, A213743, A213744, A000217, A000292, A000332, A000389, A000579, A000580.

Sequence in context: A213743 A213744 A213745 * A027555 A059481 A113592

Adjacent sequences:  A213805 A213806 A213807 * A213809 A213810 A213811

KEYWORD

nonn,tabl

AUTHOR

Vladimir Shevelev and Peter J. C. Moses, Jun 20 2012

STATUS

approved

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Last modified June 17 17:12 EDT 2019. Contains 324196 sequences. (Running on oeis4.)