OFFSET
0,10
LINKS
G. C. Greubel, Rows n = 0..25 of the irregular triangle, 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) = Sum_{j=0..3} binomial(k-1, j) * T(n-1, k-j-1), with T(n, n) = 1, T(n, k) = 0 if n < 1 or n > k.
Sum_{k=0..4*n} T(n, k) = A144508(n).
EXAMPLE
Irregular triangle begins:
1;
0, 1, 1, 1, 1;
0, 0, 1, 3, 7, 15, 25, 35, 35;
0, 0, 0, 1, 6, 25, 90, 280, 770, 1855, 3675, 5775, 5775;
...
MAPLE
T := proc(n, k) option remember;
if n = k then 1;
elif k < n then 0;
elif n < 1 then 0;
else T(n - 1, k - 1) + (k - 1)*T(n - 1, k - 2) + 1/2*(k - 1)*(k - 2)*T(n - 1, k - 3) + 1/6*(k - 1)*(k - 2)*(k - 3)*T(n - 1, k - 4);
end if;
end proc;
MATHEMATICA
T[n_, k_]:= T[n, k]= Which[n==k, 1, k<n, 0, n<1, 0, True, T[n-1, k-1] + (k-1)*T[n-1, k-2] + 1/2*(k-1)*(k-2)*T[n-1, k-3] + 1/6*(k-1)*(k-2)*(k-3)*T[n-1, k-4]]; Table[T[n, k], {n, 0, 5}, {k, 0, 4n}]//Flatten (* Jean-François Alcover, Mar 20 2014, after Maple *)
Table[BellY[k, n, {1, 1, 1, 1}], {n, 0, 12}, {k, 0, 4*n}]]//Flatten (* G. C. Greubel, Oct 11 2023 *)
PROG
(Magma)
function t(n, k)
if k eq n then return 1;
elif k le n-1 or n le 0 then return 0;
else return (&+[Binomial(k-1, j)*t(n-1, k-j-1): j in [0..3]]);
end if;
end function;
A144643:= func< n, k | t(n, k) >;
[A144643(n, k): k in [0..4*n], n in [0..8]]; // G. C. Greubel, Oct 11 2023
(SageMath)
@CachedFunction
def t(n, k):
if (k==n): return 1
elif (k<n or n<1): return 0
else: return sum(binomial(k-1, j)*t(n-1, k-j-1) for j in range(4))
def A144643(n, k): return t(n, k)
flatten([[A144643(n, k) for k in range(4*n+1)] for n in range(13)]) # G. C. Greubel, Oct 11 2023
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
David Applegate and N. J. A. Sloane, Jan 25 2009
STATUS
approved