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

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, 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, 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

Offset: 1

Author

N. J. A. Sloane, May 21 2014

Keywords

• nonn
• tabl

Comments

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

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

Examples

			Array begins:
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, ...
1, 1, 3, 5, 10, 16, 29, 45, 75, 115, 181, 271, 413, ...
1, 1, 3, 6, 12, 21, 40, 67, 117, 193, 319, 510, 818, ...
1, 1, 3, 6, 13, 23, 45, 78, 141, 239, 409, 674, 1116, ...
1, 1, 3, 6, 13, 24, 47, 83, 152, 263, 457, 768, 1292, ...
1, 1, 3, 6, 13, 24, 48, 85, 157, 274, 481, 816, 1388, ...
1, 1, 3, 6, 13, 24, 48, 86, 159, 279, 492, 840, 1436, ...
1, 1, 3, 6, 13, 24, 48, 86, 160, 281, 497, 851, 1460, ...
1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 499, 856, 1471, ...
1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 858, 1476, ...
1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1478, ...
1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1479, ...
...
		

Links

• Alois P. Heinz, Antidiagonals 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

Rows give A000041, A000990, A000991, A002799, A001452, A225196, A225197, A225198, A225199.

Main diagonal = A000219.

See A242642 for the upper triangle of the array.

Programs

Maple

# Maple code for the square array:
M:=100:
F:=s->mul((1-q^i)^(-i),i=1..s)*mul((1-q^j)^(-s),j=s+1..M);
A:=(s,n)->coeff(series(F(s),q,M),q,n);
for s from 1 to 12 do lprint( [seq(A(s,j),j=0..12)]); od:
# second Maple program:
B:= proc(s, n) option remember; `if`(n=0, 1, add(add(min(d, s)
      *d, d=numtheory[divisors](j))*B(s, n-j), j=1..n)/n)
    end:
seq(seq(B(d-n, n), n=0..d-1), d=1..14);  # Alois P. Heinz, Oct 02 2018

Mathematica

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 *)

Formula

G.f. for row s: Product_{i=1..s} (1-q^i)^(-i) * Product_{j >= s+1} (1-q^j)^(-s). [MacMahon]