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%I #18 Oct 13 2023 07:36:49
%S 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,
%T 5775,5775,0,0,0,0,1,10,65,350,1645,6930,26425,90475,275275,725725,
%U 1576575,2627625,2627625,0,0,0,0,0,1,15,140,1050,6825,39795,211750,1033725,4629625
%N Triangle read by rows: T(n,k) = number of partitions of [1..k] into n nonempty clumps of sizes 1, 2, 3 or 4 (n >= 0, 0 <= k <= 4n).
%H G. C. Greubel, <a href="/A144643/b144643.txt">Rows n = 0..25 of the irregular triangle, flattened</a>
%H Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1701.08394">Analysis of the Gift Exchange Problem</a>, arXiv:1701.08394 [math.CO], 2017.
%H David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">The Gift Exchange Problem</a>, arXiv:0907.0513 [math.CO], 2009.
%F 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.
%F Sum_{k=0..4*n} T(n, k) = A144508(n).
%e Irregular triangle begins:
%e 1;
%e 0, 1, 1, 1, 1;
%e 0, 0, 1, 3, 7, 15, 25, 35, 35;
%e 0, 0, 0, 1, 6, 25, 90, 280, 770, 1855, 3675, 5775, 5775;
%e ...
%p T := proc(n, k) option remember;
%p if n = k then 1;
%p elif k < n then 0;
%p elif n < 1 then 0;
%p 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);
%p end if;
%p end proc;
%t 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 *)
%t Table[BellY[k, n, {1,1,1,1}], {n,0,12}, {k,0,4*n}]]//Flatten (* _G. C. Greubel_, Oct 11 2023 *)
%o (Magma)
%o function t(n,k)
%o if k eq n then return 1;
%o elif k le n-1 or n le 0 then return 0;
%o else return (&+[Binomial(k-1,j)*t(n-1,k-j-1): j in [0..3]]);
%o end if;
%o end function;
%o A144643:= func< n,k | t(n,k) >;
%o [A144643(n,k): k in [0..4*n], n in [0..8]]; // _G. C. Greubel_, Oct 11 2023
%o (SageMath)
%o @CachedFunction
%o def t(n,k):
%o if (k==n): return 1
%o elif (k<n or n<1): return 0
%o else: return sum(binomial(k-1,j)*t(n-1,k-j-1) for j in range(4))
%o def A144643(n,k): return t(n,k)
%o flatten([[A144643(n,k) for k in range(4*n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 11 2023
%Y Row sums give A144508.
%Y See A144644 and A144645 for other versions.
%Y Cf. A144299, A144385.
%K nonn,tabf
%O 0,10
%A _David Applegate_ and _N. J. A. Sloane_, Jan 25 2009
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