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A194621
Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n in which any two parts differ by at most k.
17
1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 5, 2, 5, 6, 7, 7, 7, 4, 6, 9, 10, 11, 11, 11, 2, 7, 10, 13, 14, 15, 15, 15, 4, 8, 14, 17, 20, 21, 22, 22, 22, 3, 9, 15, 22, 25, 28, 29, 30, 30, 30, 4, 10, 20, 27, 34, 37, 40, 41, 42, 42, 42, 2, 11, 21, 33, 41, 48, 51, 54, 55, 56, 56, 56
OFFSET
0,4
COMMENTS
T(n,k) = A000041(n) for n >= 0 and k >= n.
LINKS
FORMULA
G.f. of column k: 1 + Sum_{j>0} x^j / Product_{i=0..k} (1-x^(i+j)).
EXAMPLE
T(6,0) = 4: [6], [3,3], [2,2,2], [1,1,1,1,1,1].
T(6,1) = 6: [6], [3,3], [2,1,1,1,1], [2,2,1,1], [2,2,2], [1,1,1,1,1,1].
T(6,2) = 9: [6], [4,2], [3,1,1,1], [3,2,1], [3,3], [2,1,1,1,1], [2,2,1,1], [2,2,2], [1,1,1,1,1,1].
Triangle begins:
1;
1, 1;
2, 2, 2;
2, 3, 3, 3;
3, 4, 5, 5, 5;
2, 5, 6, 7, 7, 7;
4, 6, 9, 10, 11, 11, 11;
2, 7, 10, 13, 14, 15, 15, 15;
MAPLE
b:= proc(n, i, k) option remember;
if n<0 or k<0 then 0
elif n=0 then 1
elif i<1 then 0
else b(n, i-1, k-1) +b(n-i, i, k)
fi
end:
T:= (n, k)-> `if`(n=0, 1, 0) +add(b(n-i, i, k), i=1..n):
seq(seq(T(n, k), k=0..n), n=0..20);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n < 0 || k < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, k-1] + b[n-i, i, k]]]]; t[n_, k_] := If[n == 0, 1, 0] + Sum[b[n-i, i, k], {i, 1, n}]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)
CROSSREFS
Columns k=0-10 give (for n>0): A000005, A000027, A117142, A117143, A218506, A218507, A218508, A218509, A218510, A218511, A218512.
Main diagonal gives: A000041.
Sequence in context: A029269 A352166 A272187 * A088004 A070548 A209628
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 30 2011
STATUS
approved