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

1,6

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

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.

Formula

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

Example

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, ...
...

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 *)

Crossrefs

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

Main diagonal = A000219.

See A242642 for the upper triangle of the array.

Sequence in context: A157458 A174447 A174374 * A347187 A027948 A360335

Adjacent sequences: A242638 A242639 A242640 * A242642 A242643 A242644

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 21 2014

Status

approved