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A265020
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Total sum T(n,k) of number of lambda-parking functions of partitions lambda of n into exactly k distinct parts; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.
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3
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1, 0, 1, 0, 2, 0, 3, 3, 0, 4, 5, 0, 5, 15, 0, 6, 21, 16, 0, 7, 42, 25, 0, 8, 54, 68, 0, 9, 90, 142, 0, 10, 110, 248, 125, 0, 11, 165, 409, 189, 0, 12, 195, 710, 496, 0, 13, 273, 1033, 967, 0, 14, 315, 1562, 2096, 0, 15, 420, 2291, 3265, 1296, 0, 16, 476, 3180
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OFFSET
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0,5
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COMMENTS
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Differs from A265208 first at T(5,2). See example.
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LINKS
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FORMULA
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EXAMPLE
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T(5,2) = 15 because there are two partitions of 5 into 2 distinct parts: [2,3] and [1,4]. And [2,3] has 8 lambda-parking functions: [1,1], [1,2], [1,3], [2,1], [2,2], [2,3], [3,1], [3,2] and [1,4] has 7: [1,1], [1,2], [1,3], [1,4], [2,1], [3,1], [4,1]. So [1,1], [1,2], [1,3], [2,1], [3,1] are counted twice.
Triangle T(n,k) begins:
00 : 1;
01 : 0, 1;
02 : 0, 2;
03 : 0, 3, 3;
04 : 0, 4, 5;
05 : 0, 5, 15;
06 : 0, 6, 21, 16;
07 : 0, 7, 42, 25;
08 : 0, 8, 54, 68;
09 : 0, 9, 90, 142;
10 : 0, 10, 110, 248, 125;
11 : 0, 11, 165, 409, 189;
12 : 0, 12, 195, 710, 496;
13 : 0, 13, 273, 1033, 967;
14 : 0, 14, 315, 1562, 2096;
15 : 0, 15, 420, 2291, 3265, 1296;
16 : 0, 16, 476, 3180, 6057, 1921;
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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^nops(l),
g(n, i-1, l)+`if`(i>n, 0, g(n-i, i-1, [i, l[]])))):
T:= n-> (f-> seq(coeff(f, x, i), i=0..degree(f)))(g(n$2, [])):
seq(T(n), n=0..20);
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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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