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

Author

N. J. A. Sloane, May 21 2014

Keywords

• nonn
• tabl

Examples

			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, ...
...
		

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.

Crossrefs

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

Programs

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 *)