A242642 Triangle read by rows: T(s,n) (n>=1, 1 <= s <= n) = number of s-line partitions of n.

1, 2, 3, 3, 5, 6, 5, 10, 12, 13, 7, 16, 21, 23, 24, 11, 29, 40, 45, 47, 48, 15, 45, 67, 78, 83, 85, 86, 22, 75, 117, 141, 152, 157, 159, 160, 30, 115, 193, 239, 263, 274, 279, 281, 282, 42, 181, 319, 409, 457, 481, 492, 497, 499, 500, 56, 271, 510, 674, 768, 816, 840, 851, 856, 858, 859

Offset

1,2

Links

Alois P. Heinz, Rows n = 1..200, flattened

P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table II.

Example

Triangle begins:
[1]
[2, 3]
[3, 5, 6]
[5, 10, 12, 13]
[7, 16, 21, 23, 24]
[11, 29, 40, 45, 47, 48]
[15, 45, 67, 78, 83, 85, 86]
[22, 75, 117, 141, 152, 157, 159, 160]
...
The square array (A242641 with n=0 column omitted) begins:
1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, ...
1, 3, 5, 10, 16, 29, 45, 75, 115, 181, 271, 413, ...
1, 3, 6, 12, 21, 40, 67, 117, 193, 319, 510, 818, ...
1, 3, 6, 13, 23, 45, 78, 141, 239, 409, 674, 1116, ...
1, 3, 6, 13, 24, 47, 83, 152, 263, 457, 768, 1292, ...
1, 3, 6, 13, 24, 48, 85, 157, 274, 481, 816, 1388, ...
1, 3, 6, 13, 24, 48, 86, 159, 279, 492, 840, 1436, ...
...

Maple

T:= proc(s, n) option remember; `if`(n=0, 1, add(add(min(d, s)
      *d, d=numtheory[divisors](j))*T(s, n-j), j=1..n)/n)
    end:
seq(seq(T(s, n), s=1..n), n=1..14);  # Alois P. Heinz, Oct 02 2018

Mathematica

T[s_, n_] := T[s, n] = If[n==0, 1, Sum[Sum[Min[d, s]*d, {d, Divisors[j]}]* T[s, n - j], {j, 1, n}]/n];
Table[Table[T[s, n], {s, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 10 2019, after Alois P. Heinz *)

Crossrefs

Upper triangle of array in A242641 (with the n=0 column omitted).

Sequence in context: A183322 A295918 A296834 * A178041 A181805 A369450

Adjacent sequences: A242639 A242640 A242641 * A242643 A242644 A242645

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 21 2014

Status

approved