OFFSET
0,8
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
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
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