A080277 -id:A080277 - cp's OEIS frontend

A360638 Number of sets of nonempty words over binary alphabet where each letter occurs n times.

1, 3, 16, 100, 593, 3497, 20316, 116378, 658214, 3679450, 20350028, 111459648, 605060633, 3257784589, 17408647968, 92378535290, 487031130699, 2552197485757, 13298890952222, 68930923717598, 355507581655752, 1824924721216084, 9326440815314046, 47464093855706540

Offset: 0

Alois P. Heinz, Feb 14 2023

nonn

			a(0) = 1: {}.
a(1) = 3: {ab}, {ba}, {a,b}.
a(2) = 16: {aabb}, {abab}, {abba}, {baab}, {baba}, {bbaa}, {a,abb}, {a,bab}, {a,bba}, {aa,bb}, {aab,b}, {ab,ba}, {aba,b}, {b,baa}, {a,ab,b}, {a,b,ba}.

Alois P. Heinz, Table of n, a(n) for n = 0..380

Cf. A080277, A360626 (the same for multisets), A360634.

g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i), k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
a:= n-> coeff(b(2*n$2), x, n):
seq(a(n), n=0..31);

g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[
    g[n, i - 1, j - k]*x^(i*k)*Binomial[Binomial[n, i], k], {k, 0, j}]]]];
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1,
    If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]];
a[n_] := Coefficient[b[2n, 2n], x, n];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Dec 09 2023, after Alois P. Heinz *)

a(n) = A360634(2n,n).

a(n) mod 2 = 1 <=> n in { A080277 } U {0}.

A327327 Partial sums of the sum of nonpowers of 2 dividing n.

Original entry on oeis.org

0, 0, 3, 3, 8, 17, 24, 24, 36, 51, 62, 83, 96, 117, 140, 140, 157, 193, 212, 247, 278, 311, 334, 379, 409, 448, 487, 536, 565, 634, 665, 665, 712, 763, 810, 894, 931, 988, 1043, 1118, 1159, 1252, 1295, 1372, 1449, 1518, 1565, 1658, 1714, 1804, 1875, 1966, 2019, 2136, 2207, 2312, 2391, 2478, 2537, 2698

Offset: 1

Omar E. Pol, Sep 14 2019

nonn

a(n) can be represented with a diagram since the symmetric diagram of A024916(n) is greater than or equal to the diagram of A080277(n). The difference between both diagrams is a representation of a(n). For more information about the symmetric diagram of A024916 see A236104 and A237593.

			The divisors of 6 are 1, 2, 3, 6. But 1 and 2 are powers of 2, so we only add up 3, 6 to get 9, and add that to the running total of 8 to get a(6) = 17.

Partial sums of A326988.

Cf. A000203, A024916, A038712, A080277, A236104, A237593, A327326.

Accumulate[Table[DivisorSigma[1, n] - Denominator[DivisorSigma[1, 2n]/DivisorSigma[1, n]], {n, 100}]] (* Alonso del Arte, Nov 18 2019, based on Wesley Ivan Hurt's program for A326988 *)

a(n) = A024916(n) - A080277(n).

a(n) = a(n-1) when n is a power of 2.

A122701 a(0)=0, a(n) = 2*a(floor(n/2)) + n - 1 for n > 0.

Original entry on oeis.org

0, 0, 1, 2, 5, 6, 9, 10, 17, 18, 21, 22, 29, 30, 33, 34, 49, 50, 53, 54, 61, 62, 65, 66, 81, 82, 85, 86, 93, 94, 97, 98, 129, 130, 133, 134, 141, 142, 145, 146, 161, 162, 165, 166, 173, 174, 177, 178, 209, 210, 213, 214, 221, 222, 225, 226, 241, 242, 245, 246, 253, 254

Offset: 0

Peter C. Heinig (algorithms(AT)gmx.de), Oct 21 2006

nonn

The recurrence defining this sequence arises in the study of the Merge Sort algorithm. By the master theorem, a(n) is in BigTheta(n*log_2(n)).

Cf. A080277.

a[0]:=0: for n from 1 to 101 do a[n]:=2*a[floor(n/2)] + n-1; od: seq( a[n], n=0..101);

a[0]=0;a[n_]:=2*a[Floor[n/2]]+n-1;Table[a[n],{n,0,61}] (* James C. McMahon, Nov 09 2024 *)

Missing a(0) inserted by James C. McMahon, Nov 09 2024

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A360638 Number of sets of nonempty words over binary alphabet where each letter occurs n times.

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A327327 Partial sums of the sum of nonpowers of 2 dividing n.

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A122701 a(0)=0, a(n) = 2*a(floor(n/2)) + n - 1 for n > 0.

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