A305104 -id:A305104 - cp's OEIS frontend

A305122 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).

0, 0, 1, 2, 4, 8, 16, 28, 47, 78, 126, 198, 306, 464, 694, 1024, 1490, 2146, 3061, 4322, 6052, 8408, 11592, 15872, 21592, 29192, 39242, 52468, 69788, 92376, 121716, 159664, 208569, 271372, 351732, 454228, 584546, 749720, 958472, 1221560, 1552210, 1966698

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Convolution of A305121 and A000009.

The g.f. Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) * Product_{k >= 1} (1 + x ^k)/(1 - x^k) = Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) * Product_{k >= 1} (1 + x ^k)/(1 + x^k - 2*x^k) is congruent mod 2 to Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) = -G(-x^2), where G(x) is the g.f. of A112329. It follows that a(n) is odd iff n = 2*k^2 for some positive integer k. - Peter Bala, Jan 07 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A305102, A305101, A305104, A305105, A305124.

Cf. A116680, A112329, A305121.

nmax = 50; CoefficientList[Series[Sum[x^(2*k)/(1+x^(2*k)), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) = A305101(n) - A305124(n).

a(n) ~ exp(sqrt(n)*Pi) * log(2) / (8*Pi*sqrt(n)).

A305124 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

Original entry on oeis.org

0, 1, 1, 4, 7, 14, 24, 42, 69, 113, 178, 276, 420, 630, 930, 1360, 1963, 2804, 3969, 5568, 7746, 10700, 14672, 19986, 27060, 36423, 48754, 64928, 86038, 113478, 149012, 194842, 253737, 329172, 425452, 547952, 703343, 899858, 1147680, 1459364, 1850310, 2339432

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Convolution of A305123 and A000009.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A305102, A305101, A305104, A305122, A305105.

Cf. A116676, A305123.

nmax = 50; CoefficientList[Series[Sum[x^(2*k-1)/(1+x^(2*k-1)), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) = A305101(n) - A305122(n).

a(n) ~ exp(sqrt(n)*Pi) * log(2) / (8*Pi*sqrt(n)).

A305105 G.f.: Sum_{k>=1} x^(2k-1)/(1-x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

Original entry on oeis.org

0, 1, 3, 8, 17, 34, 64, 114, 195, 325, 526, 832, 1292, 1970, 2958, 4384, 6413, 9276, 13283, 18836, 26478, 36924, 51096, 70210, 95844, 130019, 175350, 235192, 313802, 416618, 550540, 724250, 948719, 1237732, 1608508, 2082600, 2686857, 3454590, 4427144, 5655652

Offset: 0

Vaclav Kotesovec, May 25 2018

Convolution of A066897 and A000009.
Convolution of A067588 and A000041.
Let A(x) = Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) * Product_{k >= 1} (1 + x^k)/(1 - x^k). Then A(x) = Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) * Product_{k >= 1} (1 + x^k)/(1 + x^k - 2*x^k) == Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) (mod 2). It follows from the comment in A001227 by Juri-Stepan Gerasimov, dated Jul 17 2016, that a(n) is odd iff n is a square or twice a square. - Peter Bala, Jan 10 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001227, A015128, A066897, A067588, A305102, A305104.

nmax = 50; CoefficientList[Series[Sum[x^(2*k-1)/(1-x^(2*k-1)), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (2*gamma + log(16*n/Pi^2)) * exp(Pi*sqrt(n)) / (16*Pi*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620.

cp's OEIS Frontend

A305122 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).

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A305124 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

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A305105 G.f.: Sum_{k>=1} x^(2k-1)/(1-x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

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cp's OEIS Frontend

A305122 G.f.: Sum_{k>=1} x^(2*k)/(1+x^(2*k)) * Product_{k>=1} (1+x^k)/(1-x^k).

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A305124 G.f.: Sum_{k>=1} x^(2*k-1)/(1+x^(2*k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

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A305105 G.f.: Sum_{k>=1} x^(2*k-1)/(1-x^(2*k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

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Formula

A305122 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).

A305124 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

A305105 G.f.: Sum_{k>=1} x^(2k-1)/(1-x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).