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A189804
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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).
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1
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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
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OFFSET
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0,8
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COMMENTS
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Row n has 3*n+1 entries.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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]
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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 =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:
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MATHEMATICA
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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
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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