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A144385
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Triangle read by rows: T(n,k) is the number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2 or 3 (n >= 0, 0 <= k <= 3n).
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13
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1, 0, 1, 1, 1, 0, 0, 1, 3, 7, 10, 10, 0, 0, 0, 1, 6, 25, 75, 175, 280, 280, 0, 0, 0, 0, 1, 10, 65, 315, 1225, 3780, 9100, 15400, 15400, 0, 0, 0, 0, 0, 1, 15, 140, 980, 5565, 26145, 102025, 323400, 800800, 1401400, 1401400, 0, 0, 0, 0, 0, 0, 1, 21, 266, 2520, 19425, 125895, 695695, 3273270, 12962950, 42042000, 106506400, 190590400, 190590400
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OFFSET
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0,9
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COMMENTS
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Row n has 3n+1 entries.
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LINKS
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FORMULA
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T(n, k) = T(n - 1, k - 1) + (k - 1)*T(n - 1, k - 2) + (1/2)*(k - 1)*(k - 2)*T(n - 1, k - 3).
E.g.f.: Sum_{ n >= 0, k >= 0 } T(n, k) y^n x^k / k! = exp( y*(x+x^2/2+x^3/6) ). That is, the coefficient of y^n is the e.g.f. for row n. E.g. the e.g.f. for row 2 is (1/2)*(x+x^2/2+x^3/6)^2 = 1*x^2/2! + 3*x^3/3! + 7*x^4/4! + 10*x^5/5! + 10*x^6/6!.
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EXAMPLE
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Triangle begins:
[1]
[0, 1, 1, 1]
[0, 0, 1, 3, 7, 10, 10]
[0, 0, 0, 1, 6, 25, 75, 175, 280, 280]
[0, 0, 0, 0, 1, 10, 65, 315, 1225, 3780, 9100, 15400, 15400]
[0, 0, 0, 0, 0, 1, 15, 140, 980, 5565, 26145, 102025, 323400, 800800, 1401400, 1401400]
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MAPLE
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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);
end if;
end proc;
for n from 0 to 12 do lprint([seq(T(n, k), k=0..3*n)]); od:
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MATHEMATICA
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t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; 0 <= k <= 3*n := t[n, k] = t[n-1, k-1] + (k-1)*t[n-1, k-2] + (1/2)*(k-1)*(k-2)*t[n-1, k-3]; t[_, _] = 0; Table[t[n, k], {n, 0, 12}, {k, 0, 3*n}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)
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CROSSREFS
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A generalization of the triangle in A144331 (and in several other entries).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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