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A242641 Array read by antidiagonals upwards: B(s,n) ( s>=1, n >= 0) = number of s-line partitions of n.

%I A242641 #33 Apr 07 2025 15:39:05
%S A242641 1,1,1,1,1,2,1,1,3,3,1,1,3,5,5,1,1,3,6,10,7,1,1,3,6,12,16,11,1,1,3,6,
%T A242641 13,21,29,15,1,1,3,6,13,23,40,45,22,1,1,3,6,13,24,45,67,75,30,1,1,3,6,
%U A242641 13,24,47,78,117,115,42,1,1,3,6,13,24,48,83,141,193,181,56,1,1,3,6,13,24,48,85,152,239,319,271,77
%N A242641 Array read by antidiagonals upwards: B(s,n) ( s>=1, n >= 0) = number of s-line partitions of n.
%C A242641 An s-line partition is a planar partition into at most s rows. s-line partitions of n are equinumerous with partitions of n with min(k,s) sorts of part k (cf. the g.f.). - _Joerg Arndt_, Feb 18 2015
%C A242641 Row s is asymptotic to (Product_{j=1..s-1} j!) * Pi^(s*(s-1)/2) * s^((s^2 + 1)/4) * exp(Pi*sqrt(2*n*s/3)) / (2^((s*(s+2)+5)/4) * 3^((s^2 + 1)/4) * n^((s^2 + 3)/4)). - _Vaclav Kotesovec_, Oct 28 2015
%H A242641 Alois P. Heinz, <a href="/A242641/b242641.txt">Antidiagonals n = 1..200, flattened</a>
%H A242641 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.
%F A242641 G.f. for row s: Product_{i=1..s} (1-q^i)^(-i) * Product_{j >= s+1} (1-q^j)^(-s). [MacMahon]
%e A242641 Array begins:
%e A242641 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, ...
%e A242641 1, 1, 3, 5, 10, 16, 29, 45, 75, 115, 181, 271, 413, ...
%e A242641 1, 1, 3, 6, 12, 21, 40, 67, 117, 193, 319, 510, 818, ...
%e A242641 1, 1, 3, 6, 13, 23, 45, 78, 141, 239, 409, 674, 1116, ...
%e A242641 1, 1, 3, 6, 13, 24, 47, 83, 152, 263, 457, 768, 1292, ...
%e A242641 1, 1, 3, 6, 13, 24, 48, 85, 157, 274, 481, 816, 1388, ...
%e A242641 1, 1, 3, 6, 13, 24, 48, 86, 159, 279, 492, 840, 1436, ...
%e A242641 1, 1, 3, 6, 13, 24, 48, 86, 160, 281, 497, 851, 1460, ...
%e A242641 1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 499, 856, 1471, ...
%e A242641 1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 858, 1476, ...
%e A242641 1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1478, ...
%e A242641 1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1479, ...
%e A242641 ...
%p A242641 # Maple code for the square array:
%p A242641 M:=100:
%p A242641 F:=s->mul((1-q^i)^(-i),i=1..s)*mul((1-q^j)^(-s),j=s+1..M);
%p A242641 A:=(s,n)->coeff(series(F(s),q,M),q,n);
%p A242641 for s from 1 to 12 do lprint( [seq(A(s,j),j=0..12)]); od:
%p A242641 # second Maple program:
%p A242641 B:= proc(s, n) option remember; `if`(n=0, 1, add(add(min(d, s)
%p A242641       *d, d=numtheory[divisors](j))*B(s, n-j), j=1..n)/n)
%p A242641     end:
%p A242641 seq(seq(B(d-n, n), n=0..d-1), d=1..14);  # _Alois P. Heinz_, Oct 02 2018
%t A242641 M=100; F[s_] := Product[(1-q^i)^-i, {i, 1, s}]*Product[(1-q^j)^-s, {j, s+1, M}]; A[s_, n_] := Coefficient[Series[F[s], {q, 0, M}], q, n]; Table[A[s-j, j], {s, 1, 12}, {j, 0, s-1}] // Flatten (* _Jean-François Alcover_, Feb 18 2015, after Maple code *)
%Y A242641 Rows give A000041, A000990, A000991, A002799, A001452, A225196, A225197, A225198, A225199.
%Y A242641 Main diagonal = A000219.
%Y A242641 See A242642 for the upper triangle of the array.
%K A242641 nonn,tabl
%O A242641 1,6
%A A242641 _N. J. A. Sloane_, May 21 2014