OFFSET
1,6
COMMENTS
Row sums = A004111.
LINKS
Alois P. Heinz, Rows n = 1..400, flattened
FORMULA
G.f.: x * Product_{n>=1} (1 + y * x^n)^A004111(n).
From Alois P. Heinz, Aug 25 2017: (Start)
T(n,k) = Sum_{h=0..n-k} A291529(n-1,h,k).
Sum_{k>=1} k * T(n,k) = A291532(n-1). (End)
EXAMPLE
Triangular array T(n,k) begins:
n\k: 0 1 2 3 4 ...
---+---------------------------
01 : 1;
02 : . 1;
03 : . 1;
04 : . 1, 1;
05 : . 2, 1;
06 : . 3, 3;
07 : . 6, 5, 1;
08 : . 12, 11, 2;
09 : . 25, 22, 5;
10 : . 52, 49, 12;
11 : . 113, 104, 28, 2;
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(b((i-1)$2), j)*b(n-i*j, i-1), j=0..n/i)))
end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, expand(
add(x^j*binomial(b((i-1)$2), j)*g(n-i*j, i-1), j=0..n/i))))
end:
T:= n-> `if`(n=1, 1,
(p-> seq(coeff(p, x, k), k=1..degree(p)))(g((n-1)$2))):
seq(T(n), n=1..25); # Alois P. Heinz, Jul 30 2013
MATHEMATICA
nn=20; f[x_]:=Sum[a[n]x^n, {n, 0, nn}]; sol=SolveAlways[0==Series[f[x]-x Product[(1+x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x]; A004111=Drop[ Flatten[Table[a[n], {n, 0, nn}]/.sol], 1]; Map[Select[#, #>0&]&, Drop[CoefficientList[Series[x Product[(1 + y x^i)^A004111[[i]], {i, 1, nn}], {x, 0, nn}], {x, y}], 1]]//Grid
PROG
(Python)
from sympy import binomial, Poly, Symbol
from sympy.core.cache import cacheit
x=Symbol('x')
@cacheit
def b(n, i):return 1 if n==0 else 0 if i<1 else sum([binomial(b(i - 1, i - 1), j)*b(n - i*j, i - 1) for j in range(n//i + 1)])
@cacheit
def g(n, i):return 1 if n==0 else 0 if i<1 else sum([x**j*binomial(b(i - 1, i - 1), j)*g(n - i*j, i - 1) for j in range(n//i + 1)])
def T(n): return [1] if n==1 else Poly(g(n - 1, n - 1)).all_coeffs()[::-1][1:]
for n in range(1, 26): print(T(n)) # Indranil Ghosh, Aug 28 2017
CROSSREFS
KEYWORD
AUTHOR
Geoffrey Critzer, Jul 30 2013
STATUS
approved