A073119 -id:A073119 - cp's OEIS frontend

A276430 Triangle read by rows: T(n,k) is the number of partitions of n having k parts that are powers of 2 with positive exponent (n>=0).

1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 3, 1, 5, 3, 2, 1, 6, 5, 3, 1, 7, 8, 4, 2, 1, 10, 10, 6, 3, 1, 13, 13, 9, 4, 2, 1, 16, 18, 12, 6, 3, 1, 22, 22, 16, 10, 4, 2, 1, 27, 29, 22, 13, 6, 3, 1, 33, 40, 28, 17, 10, 4, 2, 1, 43, 49, 37, 24, 13, 6, 3, 1, 52, 63, 50, 31, 18, 10, 4, 2, 1

Offset: 0

Emeric Deutsch, Sep 19 2016

Sum of entries in row n is A000041(n) (the partition numbers).
T(n,0) = A276431(n).
Sum(k*T(n,k), k>=0) = A073119(n).

			T(6,1) = 3, counting [1,1,4], [1,2,3], [1,1,1,1,2];
T(6,2) = 2, counting [2,4], [1,1,2,2];
T(6,3) = 1, counting [2,2,2];
Triangle starts:
1;
1;
1,1;
2,1;
2,2,1;
3,3,1;
...

Alois P. Heinz, Rows n = 0..300, flattened

Cf. A000041, A073119, A276431.

h := proc (i) options operator, arrow: 2^i 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, 25)): for n from 0 to 20 do P[n] := sort(coeff(Gser, x, n)) end do: for n from 0 to 20 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
p2:= proc(n) p2(n):= is(n=2^ilog2(n)) end:
b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, 1,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i)*`if`(p2(i), x, 1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Sep 20 2016

p2[n_] := n == 2^Floor[Log[2, n]]; b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, 1, b[n, i-1] + If[i>n, 0, b[n-i, i]*If[p2[i], x, 1]]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)

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

A342230 Total number of parts which are powers of 2 in all partitions of n.

Original entry on oeis.org

0, 1, 3, 5, 11, 17, 29, 44, 71, 102, 153, 216, 311, 429, 599, 810, 1108, 1475, 1974, 2595, 3421, 4441, 5776, 7422, 9542, 12147, 15459, 19513, 24617, 30838, 38590, 48012, 59662, 73754, 91056, 111916, 137357, 167922, 204982, 249349, 302873, 366732, 443390, 534573

Offset: 0

Ilya Gutkovskiy, Mar 06 2021

nonn

			For n = 4 we have:
------------------------------------
Partitions        Number of parts
.              which are powers of 2
------------------------------------
4 ..................... 1
3 + 1 ................. 1
2 + 2 ................. 2
2 + 1 + 1 ............. 3
1 + 1 + 1 + 1 ......... 4
------------------------------------
Total ................ 11
So a(4) = 11.

Cf. A000041, A000070, A000079, A001511, A006128, A073119, A342231.

nmax = 43; CoefficientList[Series[Sum[x^(2^k)/(1 - x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}]/Product[(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[IntegerExponent[2 k, 2] PartitionsP[n - k], {k, 1, n}], {n, 0, 43}]

G.f.: Sum_{k>=0} x^(2^k)/(1 - x^(2^k)) / Product_{j>=1} (1 - x^j).
a(n) = Sum_{k=1..n} A001511(k) * A000041(n-k).
a(n) = A000070(n-1) + A073119(n).

cp's OEIS Frontend

A276430 Triangle read by rows: T(n,k) is the number of partitions of n having k parts that are powers of 2 with positive exponent (n>=0).

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