A264394 Triangle read by rows: T(n,k) is the number of partitions of n having k Mersenne number parts (0<=k<=n).

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 2, 0, 1, 1, 3, 0, 2, 0, 1, 3, 1, 4, 0, 2, 0, 1, 1, 6, 1, 4, 0, 2, 0, 1, 5, 2, 7, 1, 4, 0, 2, 0, 1, 3, 9, 2, 8, 1, 4, 0, 2, 0, 1, 8, 4, 12, 2, 8, 1, 4, 0, 2, 0, 1, 5, 15, 5, 13, 2, 8, 1, 4, 0, 2, 0, 1, 12, 9, 19, 5, 14, 2, 8, 1, 4, 0, 2, 0, 1

Offset

0,8

Comments

The Mersenne numbers are of the form 2^n - 1 (n >= 0), i.e., 0, 1, 3, 7, 15, 31, ....; A000225.

Sum of entries in row n = A000041(n) = number of partitions of n.

T(n,0) = A078657(n).

Sum_{k=0..n} k*T(n,k) = A264395(n) = total number of Mersenne number parts in all partitions of n.

Links

Table of n, a(n) for n=0..90.

Formula

G.f.: G(t,x) = Product_{i>0} (1-x^(h(i)))/((1-x^i)*(1-t*x^(h(i)))), where h(i) = 2^i - 1.

Example

T(7,3) = 4 because we have [2,2,1,1,1], [3,2,1,1], [3,3,1], and [4,1,1,1] (the partitions of 7 that have 3 Mersenne number parts).
Triangle starts:
1;
0,1;
1,0,1;
0,2,0,1;
2,0,2,0,1;
1,3,0,2,0,1;

Maple

h := proc (i) options operator, arrow: 2^i-1 end proc: g := product((1-x^h(i))/((1-x^i)*(1-t*x^h(i))), i = 1 .. 30): gser := simplify(series(g, x = 0, 30)): for n from 0 to 18 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 18 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form

Crossrefs

Cf. A000041, A000225, A078657, A264395.

Sequence in context: A321434 A103919 A263234 * A283310 A331534 A035445

Adjacent sequences: A264391 A264392 A264393 * A264395 A264396 A264397

Keyword

nonn,tabl

Author

Emeric Deutsch, Nov 13 2015

Status

approved