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A079308 For a partition P of a positive integer, let f(P) be the product of k+1, over all parts k in P. Let a(n,r) be the sum of f(P) over all partitions P of n with smallest part r. Sequence gives table of a(n,r) for 1 <= r <= n, in the order a(1,1); a(2,1), a(2,2); a(3,1), a(3,2), a(3,3); ... 2
2, 4, 3, 14, 0, 4, 36, 9, 0, 5, 100, 12, 0, 0, 6, 236, 42, 16, 0, 0, 7, 602, 54, 20, 0, 0, 0, 8, 1368, 195, 24, 25, 0, 0, 0, 9, 3242, 246, 92, 30, 0, 0, 0, 0, 10, 7240, 759, 112, 35, 36, 0, 0, 0, 0, 11, 16386, 1134, 232, 40, 42, 0, 0, 0, 0, 0, 12, 35692, 2859, 528, 170, 48, 49, 0, 0, 0, 0, 0, 13 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

EXAMPLE

The partitions with minimal part 3 begin 3, 3+3, 4+3, 5+3, 6+3, 3+3+3, ... which yield the following values of f: 4, 16, 20, 24, 28, 64, ... therefore the 3rd column of our table begins 4,0,0,16,20,24,(28+64)=92,...

Triangle a(n,r) begins:

:    2;

:    4,   3;

:   14,   0,   4;

:   36,   9,   0,  5;

:  100,  12,   0,  0,  6;

:  236,  42,  16,  0,  0, 7;

:  602,  54,  20,  0,  0, 0, 8;

: 1368, 195,  24, 25,  0, 0, 0, 9;

: 3242, 246,  92, 30,  0, 0, 0, 0, 10;

: 7240, 759, 112, 35, 36, 0, 0, 0,  0, 11;

MAPLE

b:= proc(n, k) option remember; `if`(n=0, 1,

      `if`(k>n, 0, b(n, k+1) +(k+1)*b(n-k, k)))

    end:

a:= (n, k)-> b(n, k)-b(n, k+1):

seq(seq(a(n, k), k=1..n), n=1..12);  # Alois P. Heinz, May 22 2015

MATHEMATICA

a[n_, r_] := Which[r>n, 0, r==n, n+1, True, a[n, r]=(r+1)Sum[a[n-r, s], {s, r, n-r}]]; Flatten[Table[a[n, r], {n, 1, 12}, {r, 1, n}]]

CROSSREFS

Cf. A074139, A074141 (row sums).

Sequence in context: A091861 A200715 A297901 * A189825 A271878 A259476

Adjacent sequences:  A079305 A079306 A079307 * A079309 A079310 A079311

KEYWORD

easy,nonn,tabl

AUTHOR

Alford Arnold, Feb 09 2003

EXTENSIONS

Edited by Dean Hickerson, Feb 11 2003

Offset changed to 1 by Alois P. Heinz, May 22 2015

STATUS

approved

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Last modified April 22 14:20 EDT 2021. Contains 343177 sequences. (Running on oeis4.)