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A265019
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Total sum T(n,k) of number of lambda-parking functions of partitions lambda of n into distinct parts with largest part k; triangle T(n,k), k>=0, k<=n<=k*(k+1)/2, read by columns.
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4
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1, 1, 2, 3, 3, 5, 8, 16, 4, 7, 12, 40, 34, 50, 125, 5, 9, 16, 55, 73, 132, 281, 351, 307, 432, 1296, 6, 11, 20, 70, 96, 212, 469, 642, 1020, 1361, 3294, 3305, 3910, 3506, 4802, 16807, 7, 13, 24, 85, 119, 267, 644, 959, 1567, 2686, 5570, 7109, 11890, 13234
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OFFSET
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0,3
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LINKS
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FORMULA
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EXAMPLE
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Triangle T(n,k) begins:
00 : 1;
01 : 1;
02 : 2;
03 : 3, 3;
04 : 5, 4;
05 : 8, 7, 5;
06 : 16, 12, 9, 6;
07 : 40, 16, 11, 7;
08 : 34, 55, 20, 13, 8;
09 : 50, 73, 70, 24, 15, 9;
10 : 125, 132, 96, 85, 28, 17, 10;
11 : 281, 212, 119, 100, 32, 19, 11;
12 : 351, 469, 267, 142, 115, 36, 21, 12;
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MAPLE
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p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
-> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
g:= (n, i, l)-> `if`(i*(i+1)/2<n, 0, `if`(n=0, p(l)*x^
`if`(l=[], 0, l[-1]), g(n, i-1, l)+
`if`(i>n, 0, g(n-i, i-1, [i, l[]])))):
b:= proc(n) option remember; g(n$2, []) end:
T:= k-> seq(coeff(b(n), x, k), n=k..k*(k+1)/2):
seq(T(k), k=0..8);
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MATHEMATICA
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p[l_] := With[{n = Length[l]}, n!*Det[Table[t = j-i+1; If[t<0, 0, l[[i]]^t/t!], {i, 1, n}, {j, 1, n}]]]; g[n_, i_, l_] := g[n, i, l] = If[i*(i+1)/2<n, 0, If[n==0, p[l]*x^If[l=={}, 0, l[[-1]]], g[n, i-1, l] + If[i>n, 0, g[n-i, i-1, Join[{i}, l]]]]]; b[n_] := b[n] = g[n, n, {}]; T[0] = {1}; T[k_] := Table[Coefficient[b[n], x, k], {n, k, k*(k+1)/2}]; Table[T[k], {k, 0, 8}] // Flatten (* Jean-François Alcover, Feb 11 2017, translated from Maple *)
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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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STATUS
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approved
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