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A189804 Triangle read by rows: T(n,k) is the number of compositions of set {1, 2, ..., k} into exactly n blocks, each of size 1, 2 or 3 (n >= 0, 0 <= k <= 3*n). 1
1, 0, 1, 1, 1, 0, 0, 2, 6, 14, 20, 20, 0, 0, 0, 6, 36, 150, 450, 1050, 1680, 1680, 0, 0, 0, 0, 24, 240, 1560, 7560, 29400, 90720, 218400, 369600, 369600, 0, 0, 0, 0, 0, 120, 1800, 16800, 117600, 667800, 3137400, 12243000, 38808000, 96096000, 168168000, 168168000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
Row n has 3*n+1 entries.
Column sums give A188896, row sums give A144422. - Adi Dani, May 14 2011
LINKS
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
FORMULA
T(n, k) = k*T(n-1, k-1) + (1/2)*k*(k-1)*T(n-1, k-2) + (1/6)*k*(k-1)*(k-2)*T(n-1, k-3).
E.g.f.: sum(n>=0, T(n, k)*x^k/k!) = (x+x^2/2+x^3/6)^k.
EXAMPLE
Triangle begins:
[1]
[0, 1, 1, 1]
[0, 0, 2, 6, 14, 20, 20]
[0, 0, 0, 6, 36, 150, 450, 1050, 1680, 1680]
[0, 0, 0, 0, 24, 240, 1560, 7560, 29400, 90720, 218400, 369600, 369600]
[0, 0, 0, 0, 0, 120, 1800, 16800, 117600, 667880, 3137400, 12243000, 3880800, 96096000, 168168000, 168168000]
MAPLE
T := proc(n, k)
option remember;
if n = k then 1;
elif k < n then 0;
elif n < 1 then 0;
else =k *T(n - 1, k - 1) + (1/2)*k*(k - 1)*T(n - 1, k - 2)+ (1/6)*k* (k - 1)*(k - 2)*T(n - 1, k - 3);
end if;
end proc; for n from 0 to 12 do lprint([seq(T(n, k), k=0..3*n)]); od:
MATHEMATICA
Table[Sum[ n!/(2^(n + j - 2m)3^(m - j))Binomial[m, j]Binomial[j, n + 2j - 3m], {j, 0, 3m - n}], {m, 0, 5}, {n, 0, 3m}]//Flatten
PROG
(PARI) for(m=0, 7, for(n=0, 3*m, print1(sum(j=0, 3*m-n, (n!/(2^(n+j-2*m)*3^(m-j)))*binomial(m, j)*binomial(j, n+2*j-3*m)), ", "))) \\ G. C. Greubel, Jan 16 2018
CROSSREFS
Sequence in context: A215807 A373437 A140525 * A308394 A246068 A032643
KEYWORD
nonn,tabf
AUTHOR
Adi Dani, Apr 27 2011
EXTENSIONS
Terms a(44) and a(47) corrected by G. C. Greubel, Jan 16 2018
STATUS
approved

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Last modified July 18 15:13 EDT 2024. Contains 374388 sequences. (Running on oeis4.)