OFFSET
1,4
COMMENTS
From Gary W. Adamson, Mar 13 2010: (Start)
The triangle by rows = finite differences starting from the top, of an array in which row 1 = p(x)/p(x^2), row 2 = p(x)/p(x^3), ... row k = p(x)/p(x^k); such that p(x) = polcoeff A000041: (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...)
Note that p(x)/p(x^2) = polcoeff A000009: (1 + x + x^2 + 2x^3 + 2x^4 + ...).
Refer to the example. (End)
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
FORMULA
G.f.: G = G(t,x) = sum(k>=1, t^k*(prod(j>=1, (1-x^((k+1)*j))/(1-x^j) ) -prod(j>=1, (1-x^(k*j))/(1-x^j) ) ) ). - Emeric Deutsch, Mar 30 2006
Sum_{k=1..n} k * T(n,k) = A264397(n). - Alois P. Heinz, Nov 20 2015
EXAMPLE
Triangle starts:
1: 1;
2: 1, 1;
3: 2, 0, 1;
4: 2, 2, 0, 1;
5: 3, 2, 1, 0, 1;
6: 4, 3, 2, 1, 0, 1;
7: 5, 4, 3, 1, 1, 0, 1;
8: 6, 7, 3, 3, 1, 1, 0, 1;
9: 8, 8, 6, 3, 2, 1, 1, 0, 1;
10: 10, 12, 7, 5, 3, 2, 1, 1, 0, 1;
11: 12, 15, 11, 6, 5, 2, 2, 1, 1, 0, 1;
12: 15, 21, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1;
13: 18, 26, 20, 12, 9, 5, 4, 2, 2, 1, 1, 0, 1;
14: 22, 35, 25, 18, 11, 8, 5, 4, 2, 2, 1, 1, 0, 1;
...
In the partition 5+2+2+2+1+1, 2 is repeated 3 times, no part is repeated more than 3 times.
From Gary W. Adamson, Mar 13 2010: (Start)
First few rows of the array =
...
1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ... = p(x)/p(x^2) = A000009
1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, ... = p(x)/p(x^3)
1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, ... = p(x)/p(x^4)
1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, ... = p(x)/p(x^5)
1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, ... = p(x)/p(x^6)
...
Finally, taking finite differences from the top and deleting the first "1", we obtain triangle A091602 with row sums = A000041 starting with offset 1:
1;
1, 1;
2, 0, 1;
2, 2, 0, 1;
3, 2, 1, 0, 1;
4, 3, 2, 1, 0, 1;
...
(End)
MAPLE
g:=sum(t^k*(product((1-x^((k+1)*j))/(1-x^j), j=1..50)-product((1-x^(k*j))/(1-x^j), j=1..50)), k=1..50): gser:=simplify(series(g, x=0, 20)): for n from 1 to 13 do P[n]:=coeff(gser, x^n) od: for n from 1 to 13 do seq(coeff(P[n], t^j), j=1..n) od;
# yields sequence in triangular form - Emeric Deutsch, Mar 30 2006
b:= proc(n, i, k) option remember; `if`(n=0, 1,
`if`(i>n, 0, add(b(n-i*j, i+1, min(k,
iquo(n-i*j, i+1))), j=0..min(n/i, k))))
end:
T:= (n, k)-> b(n, 1, k) -`if`(k=0, 0, b(n, 1, k-1)):
seq(seq(T(n, k), k=1..n), n=1..20);
# Alois P. Heinz, Nov 27 2013
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i>n, 0, Sum[b[n-i*j, i+1, Min[k, Quotient[n-i*j, i+1]]], {j, 0, Min[n/i, k]}]]]; t[n_, k_] := b[n, 1, k] - If[k == 0, 0, b[n, 1, k-1]]; Table[t[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 17 2014, after Alois P. Heinz's second Maple program *)
CROSSREFS
Cf. A000009. - Gary W. Adamson, Mar 13 2010
T(2n,n) gives: A232697.
KEYWORD
nonn,tabl
AUTHOR
Christian G. Bower, Jan 23 2004
STATUS
approved