A366813 -id:A366813 - cp's OEIS frontend

A366659 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(floor(n/k)+3,4).

1, 4, 15, 30, 66, 115, 200, 295, 471, 659, 946, 1259, 1715, 2194, 2920, 3591, 4561, 5585, 6916, 8216, 10082, 11823, 14124, 16389, 19350, 22174, 26004, 29435, 33931, 38445, 43902, 48925, 55767, 61941, 69831, 77275, 86415, 94968, 106094, 115874, 128216, 140214, 154405

Offset: 1

Seiichi Manyama, Oct 24 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Partial sums of A366813.

Cf. A078471, A366395, A366723.

Array[Sum[(-1)^(k - 1)*Binomial[Floor[#/k] + 3, 4], {k, #}] &, 56] (* Michael De Vlieger, Oct 25 2023 *)

a(n) = sum(k=1, n, (-1)^(k-1)*binomial(n\k+3, 4));

a(n) = Sum_{k=1..n} binomial(k+2,3) * (floor(n/k) mod 2).

G.f.: -1/(1-x) * Sum_{k>=1} (-x)^k/(1-x^k)^4 = 1/(1-x) * Sum_{k>=1} binomial(k+2,3) * x^k/(1+x^k).

A365007 a(n) = Sum_{d|n} (-1)^(n/d-1) * binomial(d+1,2).

Original entry on oeis.org

1, 2, 7, 6, 16, 17, 29, 22, 52, 42, 67, 57, 92, 79, 142, 86, 154, 143, 191, 146, 266, 189, 277, 217, 341, 262, 430, 279, 436, 402, 497, 342, 634, 444, 674, 507, 704, 553, 878, 562, 862, 766, 947, 677, 1222, 807, 1129, 857, 1254, 992, 1486, 942, 1432, 1250, 1622, 1079

Offset: 1

Seiichi Manyama, Oct 24 2023

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Partial sums give A366395.
Cf. A000593, A366813, A366814.
Cf. A007437.

Table[DivisorSum[n, (-1)^(n/# - 1)*Binomial[# + 1, 2] &], {n, 56}] (* Michael De Vlieger, Oct 25 2023 *)

a(n) = sumdiv(n, d, (-1)^(n/d-1)*binomial(d+1, 2));

G.f.: -Sum_{k>=1} (-x)^k/(1-x^k)^3 = Sum_{k>=1} binomial(k+1,2) * x^k/(1+x^k).

A366814 a(n) = Sum_{d|n} (-1)^(n/d-1) * binomial(d+3,4).

Original entry on oeis.org

1, 4, 16, 29, 71, 115, 211, 289, 511, 649, 1002, 1253, 1821, 2174, 3146, 3505, 4846, 5605, 7316, 8099, 10852, 11653, 14951, 16333, 20546, 21935, 27916, 28904, 35961, 38620, 46377, 48113, 59922, 61204, 74096, 77024, 91391, 93959, 113766, 114059, 135752, 140654, 163186

Offset: 1

Seiichi Manyama, Oct 24 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Partial sums give A366723.

Cf. A000593, A365007, A366813.

Table[DivisorSum[n, (-1)^(n/# - 1)*Binomial[# + 3, 4] &], {n, 56}] (* Michael De Vlieger, Oct 25 2023 *)

a(n) = sumdiv(n, d, (-1)^(n/d-1)*binomial(d+3, 4));

G.f.: -Sum_{k>=1} (-x)^k/(1-x^k)^5 = Sum_{k>=1} binomial(k+3,4) * x^k/(1+x^k).

cp's OEIS Frontend

A366659 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(floor(n/k)+3,4).

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A365007 a(n) = Sum_{d|n} (-1)^(n/d-1) * binomial(d+1,2).

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A366814 a(n) = Sum_{d|n} (-1)^(n/d-1) * binomial(d+3,4).

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