A185588 Triangular array read by rows. The n-th row is the expansion of (1+x)(1+2x+4x^2)...(1+nx+(nx)^2+(nx)^3+...(nx)^n).

1, 1, 1, 1, 3, 6, 4, 1, 6, 24, 76, 147, 198, 108, 1, 10, 64, 332, 1475, 5074, 14260, 32464, 52032, 57600, 27648, 1, 15, 139, 1027, 6610, 38124, 189255, 822489, 3164477, 10692485, 30443198, 72934740, 141861200, 202056000, 197280000, 86400000

Offset

0,5

Comments

T(n,k) is the sum of products of the elements in the size k submultisets of the multiset {1,2,2,3,3,3,...n} which contains i copies of i, 1<=i<=n.

The n-th row has n*(n+1)/2+1 elements: 0 <= k <= A000217(n).

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

O.g.f. for row n: Product_{j=1..n} Sum_{i=0..j} (j*x)^i.

Example

T(3,2) = 24 because the size 2 submultisets of {1,2,2,3,3,3} are: {1,2},{1,3}, {2,2}, {2,3}, {3,3}. And 1*2 + 1*3 + 2*2 + 2*3 + 3*3 = 24.
Triangle T(n,k) begins:
1;
1,  1;
1,  3,  6,   4;
1,  6, 24,  76,  147,  198,   108;
1, 10, 64, 332, 1475, 5074, 14260, 32464, 52032, 57600, 27648;

Maple

T:= (n, k)-> coeff (mul (add ((j*x)^i, i=0..j), j=1..n), x, k):
seq (seq (T(n, k), k=0..n*(n+1)/2), n=0..7);

Mathematica

Table[CoefficientList[Series[Product[Sum[(j x)^i, {i, 0, j}], {j, 1, n}], {x, 0, 20}], x], {n, 0, 5}]//Grid

Crossrefs

Cf. A000217.

Sequence in context: A334279 A334278 A169842 * A199737 A220397 A021737

Adjacent sequences: A185585 A185586 A185587 * A185589 A185590 A185591

Keyword

nonn,tabf

Author

Geoffrey Critzer, Feb 04 2011

Status

approved