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Search: id:A134431
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| A134431 |
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Triangle read by rows: T(n,k) is the number of arrangements of the set {1,2,...,n} in which the sum of the entries is equal to k (n>=0, k>=0; to n=0 there corresponds the empty set). |
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2
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| 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 6, 1, 1, 1, 3, 3, 4, 8, 8, 6, 6, 24, 1, 1, 1, 3, 3, 5, 10, 10, 14, 14, 36, 30, 30, 24, 24, 120, 1, 1, 1, 3, 3, 5, 11, 12, 16, 22, 44, 44, 66, 60, 78, 174, 168, 144, 144, 120, 120, 720, 1, 1, 1, 3, 3, 5, 11, 13, 18, 24, 52, 52, 80, 98, 120, 234
(list; graph; listen)
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OFFSET
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0,7
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COMMENT
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Row n has 1 + n(n+1)/2 terms (n>=0). Row sums yield the arrangement numbers (A000522). T(n,n(n+1)/2)=n!. Sum(k*T(n,k),k=0..n(n+1)/2)=A134432(n)
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FORMULA
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The row generating polynomials P[n](t) are equal to Q[n](t,1), where the polynomials Q[n](t,x) are defined by Q[0]=1 and Q[n]=Q[n-1] + xt^n diff(xQ[n-1], x). [Q[n](t,x) is the bivariate generating polynomial of the arrangements of {1,2,...,n}, where t (x) marks the sum (number) of the entries; for example, Q[2](t,x)=1+tx + t^2*x + 2t^3*x^2, corresponding to: empty, 1, 2, 12, and 21, respectively.]
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EXAMPLE
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T(4,7)=8 because we have 34,43, and the six permutations of {1,2,4}.
Triangle starts:
1;
1,1;
1,1,1,2;
1,1,1,3,2,2,6;
1,1,1,3,3,4,8,8,6,6,24;
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MAPLE
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Q[0]:=1: for n to 7 do Q[n]:=sort(simplify(Q[n-1]+t^n*x*(diff(x*Q[n-1], x))), t) end do: for n from 0 to 7 do P[n]:=sort(subs(x=1, Q[n])) end do: for n from 0 to 7 do seq(coeff(P[n], t, j), j=0..(1/2)*n*(n+1)) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A000522, A134432.
Sequence in context: A110963 A106348 A029332 this_sequence A070879 A125644 A048821
Adjacent sequences: A134428 A134429 A134430 this_sequence A134432 A134433 A134434
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 16 2007
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