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A264033
Triangle read by rows: T(n,k) (n>=0, 0<=k<=A130519(n+1)) is the number of integer partitions of n having k pairs of different size.
2
1, 1, 2, 2, 1, 3, 1, 1, 2, 2, 2, 1, 4, 2, 1, 2, 2, 2, 3, 3, 3, 1, 2, 1, 4, 3, 3, 3, 2, 3, 2, 1, 1, 3, 4, 3, 5, 2, 5, 2, 2, 1, 2, 1, 4, 4, 4, 7, 3, 4, 2, 4, 5, 1, 0, 2, 2, 2, 5, 5, 8, 2, 9, 4, 4, 3, 4, 1, 4, 1, 1, 2, 1, 6, 5, 4, 9, 4, 9, 4, 6, 5, 7, 2, 4, 3, 1, 2, 2, 2, 1, 1
OFFSET
0,3
COMMENTS
Row sums give A000041.
T(n,0) gives A000005(n) for n>0. - Alois P. Heinz, Nov 01 2015
REFERENCES
Richard Stanley, Enumerative combinatorics. Vol. 2 MathSciNet:1676282, page 375.
FORMULA
Sum_{k>0} k * T(n,k) = A271370(n). - Alois P. Heinz, Apr 05 2016
EXAMPLE
Triangle begins:
1;
1;
2;
2,1;
3,1,1;
2,2,2,1;
4,2,1,2,2;
2,3,3,3,1,2,1;
4,3,3,3,2,3,2,1,1;
3,4,3,5,2,5,2,2,1,2,1;
4,4,4,7,3,4,2,4,5,1,0,2,2;
2,5,5,8,2,9,4,4,3,4,1,4,1,1,2,1;
6,5,4,9,4,9,4,6,5,7,2,4,3,1,2,2,2,1,1;
...
MAPLE
b:= proc(n, i, p, t) option remember; expand(
`if`(n=0, x^t, `if`(i<1, 0, add(
b(n-i*j, i-1, p+j, t+j*p), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0$2)):
seq(T(n), n=0..15); # Alois P. Heinz, Nov 01 2015
MATHEMATICA
b[n_, i_, p_, t_] := b[n, i, p, t] = Expand[If[n==0, x^t, If[i<1, 0, Sum[b[n-i*j, i-1, p+j, t+j*p], {j, 0, n/i}]]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Christian Stump, Nov 01 2015
STATUS
approved