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%I #17 Dec 10 2019 12:10:45
%S 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,
%T 83,85,86,22,75,117,141,152,157,159,160,30,115,193,239,263,274,279,
%U 281,282,42,181,319,409,457,481,492,497,499,500,56,271,510,674,768,816,840,851,856,858,859
%N Triangle read by rows: T(s,n) (n>=1, 1 <= s <= n) = number of s-line partitions of n.
%H Alois P. Heinz, <a href="/A242642/b242642.txt">Rows n = 1..200, flattened</a>
%H P. A. MacMahon, <a href="https://archive.org/stream/messengerofmathe52cambuoft#page/112/mode/2up">The connexion between the sum of the squares of the divisors and the number of partitions of a given number</a>, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table II.
%e Triangle begins:
%e [1]
%e [2, 3]
%e [3, 5, 6]
%e [5, 10, 12, 13]
%e [7, 16, 21, 23, 24]
%e [11, 29, 40, 45, 47, 48]
%e [15, 45, 67, 78, 83, 85, 86]
%e [22, 75, 117, 141, 152, 157, 159, 160]
%e ...
%e The square array (A242641 with n=0 column omitted) begins:
%e 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, ...
%e 1, 3, 5, 10, 16, 29, 45, 75, 115, 181, 271, 413, ...
%e 1, 3, 6, 12, 21, 40, 67, 117, 193, 319, 510, 818, ...
%e 1, 3, 6, 13, 23, 45, 78, 141, 239, 409, 674, 1116, ...
%e 1, 3, 6, 13, 24, 47, 83, 152, 263, 457, 768, 1292, ...
%e 1, 3, 6, 13, 24, 48, 85, 157, 274, 481, 816, 1388, ...
%e 1, 3, 6, 13, 24, 48, 86, 159, 279, 492, 840, 1436, ...
%e ...
%p T:= proc(s, n) option remember; `if`(n=0, 1, add(add(min(d, s)
%p *d, d=numtheory[divisors](j))*T(s, n-j), j=1..n)/n)
%p end:
%p seq(seq(T(s, n), s=1..n), n=1..14); # _Alois P. Heinz_, Oct 02 2018
%t 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];
%t Table[Table[T[s, n], {s, 1, n}], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Dec 10 2019, after _Alois P. Heinz_ *)
%Y Upper triangle of array in A242641 (with the n=0 column omitted).
%K nonn,tabl
%O 1,2
%A _N. J. A. Sloane_, May 21 2014
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