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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

Alois P. Heinz, Rows n = 0..140, flattened

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-Fran├žois Alcover, Dec 27 2013, translated from Maple *)

CROSSREFS

Columns k=0-10 give: A000004, A015723, A320372, A320373, A320374, A320375, A320376, A320377, A320378, A320379, A320380.

Row sums give: A006128.

Main diagonal and first lower diagonal give: A001477, A063524.

T(2n,n) gives A320381.

Cf. A091602, A210485.

Sequence in context: A214000 A161123 A035442 * A265017 A035376 A259708

Adjacent sequences:  A213174 A213175 A213176 * A213178 A213179 A213180

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Feb 27 2013

STATUS

approved

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Last modified August 9 13:57 EDT 2020. Contains 336323 sequences. (Running on oeis4.)