A188277 T(n,k) is the number of nondecreasing strings of numbers x(i=1..n) in -k..k with sum x(i)^3 equal to 0, read by antidiagonals.

1, 1, 2, 1, 3, 2, 1, 4, 3, 3, 1, 5, 4, 6, 3, 1, 6, 5, 10, 6, 4, 1, 7, 6, 15, 10, 10, 4, 1, 8, 7, 21, 15, 20, 10, 5, 1, 9, 8, 30, 21, 37, 22, 15, 5, 1, 10, 9, 38, 30, 62, 41, 37, 17, 6, 1, 11, 10, 47, 42, 106, 68, 84, 45, 23, 6, 1, 12, 11, 59, 51, 148, 114, 170, 106, 68, 27, 7, 1, 13, 12, 70, 65

Offset

1,3

Comments

Table starts:

.1..1..1...1...1....1....1....1....1.....1.....1.....1.....1.....1.....1......1

.2..3..4...5...6....7....8....9...10....11....12....13....14....15....16.....17

.2..3..4...5...6....7....8....9...10....11....12....13....14....15....16.....17

.3..6.10..15..21...30...38...47...59....70....82....99...113...128...144....163

.3..6.10..15..21...30...42...51...65....78....92...111...129...152...172....193

.4.10.20..37..62..106..148..197..280...366...470...637...778...922..1098...1327

.4.10.22..41..68..114..202..273..402...548...720...979..1248..1660..2072...2525

.5.15.37..84.170..346..552..817.1319..1951..2817..4262..5776..7388..9688..12753

.5.17.45.106.216..422..890.1415.2401..3809..5725..8810.12622.18662.26200..35595

.6.23.68.186.450.1070.2020.3505.6456.10987.18010.30214.46352.66586.98330.142605

Links

R. H. Hardin, Table of n, a(n) for n = 1..1196

Formula

T(n,k) = [y^0] [x^n] Product_{j=-k..k} 1/(1-x*y^(j^3)). - Robert Israel, Feb 06 2019

Example

Some solutions for n=9, k=8:
  -8  -6  -8  -6  -8  -6  -8  -7  -8  -7  -8  -5  -5  -7  -7  -7
  -4  -6  -7  -5  -6  -6  -6  -5  -7  -7  -1  -2  -1  -2  -7  -4
  -1  -4  -2  -3  -6  -6  -3  -3  -6  -4  -1  -2  -1  -1  -4  -3
   0  -3  -1  -1  -4  -3   0  -2  -5  -4   2   0  -1   0   1   1
   0  -3   2  -1  -2   0   0  -2  -1   0   2   0  -1   0   1   1
   0  -2   4  -1  -2   3   0  -1  -1   4   3   0   0   2   4   3
   1  -1   4   1   0   6   3  -1   7   4   4   2   1   4   5   4
   4   6   6   3   8   6   6   1   7   7   4   2   4   4   6   5
   8   7   8   7   8   6   8   8   8   7   7   5   4   6   7   6

Maple

T:= proc(n, k) local P, S, j;
  P:= 1/mul(1-x*y^(j^3), j=-k..k);
  S:= series(P, x, n+1);
  coeff(coeff(S, x, n), y, 0)
end proc:
seq(seq(T(i, m-i), i=1..m-1), m=2..16); # Robert Israel, Feb 06 2019

Crossrefs

Sequence in context: A278118 A255238 A212536 * A135227 A104325 A204925

Adjacent sequences: A188274 A188275 A188276 * A188278 A188279 A188280

Keyword

nonn,tabl

Author

R. H. Hardin, Mar 26 2011

Status

approved