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A213177 Number T(n,k) of parts in all partitions of n with largest multiplicity k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 14
0, 0, 1, 0, 1, 2, 0, 3, 0, 3, 0, 3, 5, 0, 4, 0, 5, 6, 4, 0, 5, 0, 8, 9, 7, 5, 0, 6, 0, 10, 13, 13, 5, 6, 0, 7, 0, 13, 23, 14, 15, 6, 7, 0, 8, 0, 18, 30, 27, 16, 13, 7, 8, 0, 9, 0, 25, 44, 33, 30, 18, 15, 8, 9, 0, 10, 0, 30, 58, 55, 36, 34, 15, 17, 9, 10, 0, 11 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
FORMULA
T(n,k) = A210485(n,k) - A210485(n,k-1) for k>0, T(n,0) = 0.
EXAMPLE
T(6,1) = 8: partitions of 6 with largest multiplicity 1 are [3,2,1], [4,2], [5,1], [6], with 3+2+2+1 = 8 parts.
T(6,2) = 9: [2,2,1,1], [3,3], [4,1,1].
T(6,3) = 7: [2,2,2], [3,1,1,1].
T(6,4) = 5: [2,1,1,1,1].
T(6,5) = 0.
T(6,6) = 6: [1,1,1,1,1,1].
Triangle begins:
0;
0, 1;
0, 1, 2;
0, 3, 0, 3;
0, 3, 5, 0, 4;
0, 5, 6, 4, 0, 5;
0, 8, 9, 7, 5, 0, 6;
0, 10, 13, 13, 5, 6, 0, 7;
0, 13, 23, 14, 15, 6, 7, 0, 8;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
end:
T:= (n, k)-> b(n, n, k)[2] -b(n, n, k-1)[2]:
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[b[n-i*j, i-1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]]*j}, {j, 0, Min[n/i, k]}]]]; T[_, 0] = 0; T[n_, k_] := b[n, n, k][[2]] - b[n, n, k-1][[2]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)
CROSSREFS
Row sums give: A006128.
Main diagonal and first lower diagonal give: A001477, A063524.
T(2n,n) gives A320381.
Sequence in context: A214000 A161123 A035442 * A265017 A349136 A035376
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Feb 27 2013
STATUS
approved

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Last modified April 21 10:39 EDT 2024. Contains 371852 sequences. (Running on oeis4.)