A266964 -id:A266964 - cp's OEIS frontend

A307462 Expansion of Product_{k>=1} (1+x^k)^((-1)^k*k^2).

1, -1, 5, -14, 36, -97, 246, -593, 1423, -3351, 7699, -17432, 38901, -85545, 185862, -399220, 848080, -1783682, 3716584, -7675916, 15722127, -31951330, 64452707, -129102947, 256876062, -507854808, 997954125, -1949631802, 3787674152, -7319306458, 14071371173

Offset: 0

Seiichi Manyama, Apr 09 2019

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = (-1)^(n+1) * n^2, g(n) = -1.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k>=1} (1+x^k)^((-1)^k*k^b): A083365 (b=0), A284474 (b=1), this sequence (b=2).

Cf. A027998, A177155, A266964, A294749, A294750, A307460.

nmax = 40; CoefficientList[Series[Product[(1 + x^k)^((-1)^k*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 09 2019 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k))^(4*k^2) / (1 + x^(2*k - 1))^((2*k - 1)^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 09 2019 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+x^k)^((-1)^k*k^2)))

a(n) ~ (-1)^n * exp(2*Pi*n^(3/4)/3 + 3*Zeta(3)/(4*Pi^2)) / (4*n^(5/8)). - Vaclav Kotesovec, Apr 09 2019

A322199 Expansion of Product_{k>=1} 1/(1 - (2^k + 1) * x^k).

Original entry on oeis.org

1, 3, 14, 51, 195, 663, 2345, 7707, 25744, 82980, 267812, 846150, 2676163, 8337189, 25947281, 80053128, 246468551, 754366239, 2305139065, 7014997404, 21317567297, 64606020012, 195557995054, 590855420007, 1783577678925, 5377112705874, 16199746640340, 48763788775530, 146712079122114, 441146762285301, 1326002750336702, 3984148679940612, 11967872331787643

Offset: 0

Paul D. Hanna, Dec 01 2018

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = 2^n + 1. - Seiichi Manyama, Apr 11 2025

			G.f.: A(x) = 1 + 3*x + 14*x^2 + 51*x^3 + 195*x^4 + 663*x^5 + 2345*x^6 + 7707*x^7 + 25744*x^8 + 82980*x^9 + 267812*x^10 + 846150*x^11 + 2676163*x^12 + ...
such that
A(x) = 1/( (1 - 3*x) * (1 - 5*x^2) * (1 - 9*x^3) * (1 - 17*x^4) * (1 - 33*x^5) * (1 - 65*x^6) * (1 - 129*x^7) * ... * (1 - (2^n+1)*x^n) * ... ).
RELATED SERIES.
log( A(x) ) = 3*x + 19*x^2/2 + 54*x^3/3 + 199*x^4/4 + 408*x^5/5 + 1612*x^6/6 + 3090*x^7/7 + 11023*x^8/8 + 26487*x^9/9 + 80994*x^10/10 + 199686*x^11/11 + 676540*x^12/12 + ... + A322209(n)*x^n/n + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Seiichi Manyama, Generalized Euler transform.

Cf. A322209, A322200, A322210.

Cf. A000051, A266964.

{a(n) = polcoeff( 1/prod(m=1,n, 1 - (2^m+1)*x^m +x*O(x^n)),n)}
for(n=0,30, print1(a(n),", "))

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} A322200(n-k,k) * 2^k ).
a(n) ~ c * 3^n, where c = Product_{k>=2} 1/(1 - (2^k + 1)/3^k) = 6.49344992975096517443610066284481821741772051973643441550853873760083... - Vaclav Kotesovec, Oct 04 2020
a(n) = Sum_{k=0..n} 2^k * A322210(k,n-k). - Seiichi Manyama, Apr 11 2025

A294757 Expansion of Product_{k>=1} 1/(1 - k^k*x^k)^(k^k).

Original entry on oeis.org

1, 1, 17, 746, 66442, 9843731, 2187951485, 680615166718, 282199710311343, 150389915850565698, 100155578811552469018, 81505577529171038120173, 79580089696277797740768316, 91814299717377746850767747558

Offset: 0

Seiichi Manyama, Nov 08 2017

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = g(n) = n^n.

Seiichi Manyama, Table of n, a(n) for n = 0..214

Column k=1 of A294756.

Cf. A294704, A294773.

N=20; x='x+O('x^N); Vec(1/prod(k=1, N, (1-k^k*x^k)^k^k))

sd(n) = sumdiv(n, d, d^(d+n+1));
a(n) = if (n==0, 1, sum(k=1, n, sd(k)*a(n-k))/n); \\ Michel Marcus, Nov 10 2017

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} A294773(k)*a(n-k) for n > 0.

a(n) ~ n^(2*n). - Vaclav Kotesovec, Nov 08 2017

A294813 Expansion of Product_{k>=1} 1/(1 - k^k*x^k)^k.

Original entry on oeis.org

1, 1, 9, 90, 1162, 17435, 310193, 6286826, 144750451, 3717959194, 105725550762, 3293914191401, 111659484775650, 4089936343858976, 160992739588472076, 6776415674628574634, 303714862444753023205, 14439925495117621425535

Offset: 0

Seiichi Manyama, Nov 09 2017

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n, g(n) = n^n.

Seiichi Manyama, Table of n, a(n) for n = 0..385

Cf. A294809, A294810.

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-k^k*x^k)^k))

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} A294810(k)*a(n-k) for n > 0.

a(n) ~ n^(n+1). - Vaclav Kotesovec, Nov 10 2017

A294953 Expansion of Product_{k>=1} (1 - k^(2k)x^k)^k.

Original entry on oeis.org

1, -1, -32, -2155, -259701, -48496253, -13001952944, -4732375549802, -2246504006429898, -1348407213767476321, -998562531571744073815, -894380298523142455736017, -953030939828900988652689704, -1191547999931410291515116161158

Offset: 0

Seiichi Manyama, Nov 12 2017

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n, g(n) = n^(2*n).

Seiichi Manyama, Table of n, a(n) for n = 0..213

Column k=2 of A294808.

Cf. A294954, A294955.

N=20; x='x+O('x^N); Vec(prod(k=1, N, (1-k^(2*k)*x^k)^k))

a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} A294955(k)*a(n-k) for n > 0.

A294954 Expansion of Product_{k>=1} 1/(1 - k^(2k)x^k)^k.

Original entry on oeis.org

1, 1, 33, 2220, 265132, 49163241, 13121450895, 4762820449382, 2257130616816421, 1353302193751862072, 1001440612663683369940, 896481723303781965832069, 954894526385647926192875010, 1193519555165192704579377833814

Offset: 0

Seiichi Manyama, Nov 12 2017

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n, g(n) = n^(2*n).

Seiichi Manyama, Table of n, a(n) for n = 0..214

Column k=2 of A294950.

Cf. A294953, A294955.

nmax = 20; CoefficientList[Series[Product[1/(1 - k^(2*k)*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 15 2017 *)

N=20; x='x+O('x^N); Vec(1/prod(k=1, N, (1-k^(2*k)*x^k)^k))

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} A294955(k)*a(n-k) for n > 0.

a(n) ~ n^(2*n+1). - Vaclav Kotesovec, Nov 15 2017

A295086 Expansion of Product_{k>=1} 1/(1 + x^k)^(k(3k-1)/2).

Original entry on oeis.org

1, -1, -4, -8, 1, 24, 78, 111, 75, -249, -876, -1847, -2251, -871, 5170, 17052, 34742, 47176, 34576, -44016, -224561, -530104, -875149, -1030871, -475480, 1488315, 5658668, 12109163, 19411024, 22693048, 12926630, -24000623, -102605376, -230257606, -386964449

Offset: 0

Seiichi Manyama, Nov 15 2017

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n*(3*n-1)/2, g(n) = -1.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A294846 (b=3), A284896 (b=4), this sequence (b=5), A295121 (b=6), A295122 (b=7), A295123 (b=8).

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1+x^k)^(k*(3*k-1)/2)))

Convolution inverse of A294102.
G.f.: Product_{k>=1} 1/(1 + x^k)^A000326(k).
a(0) = 1 and a(n) = (1/(2*n)) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(3*d-1)*(-1)^(n/d).

A295121 Expansion of Product_{k>=1} 1/(1 + x^k)^(k(2k-1)).

Original entry on oeis.org

1, -1, -5, -10, 3, 42, 124, 160, 15, -677, -1941, -3425, -2807, 3488, 21004, 49547, 77879, 63395, -65104, -406091, -988889, -1655508, -1779329, -145347, 5087175, 15405270, 30158849, 42617486, 36116136, -19457047, -161973496, -418712896, -759063566

Offset: 0

Seiichi Manyama, Nov 15 2017

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n*(2*n-1), g(n) = -1.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A294846 (b=3), A284896 (b=4), A295086 (b=5), this sequence (b=6), A295122 (b=7), A295123 (b=8).

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1+x^k)^(k*(2*k-1))))

Convolution inverse of A294836.
G.f.: Product_{k>=1} 1/(1 + x^k)^A000384(k).
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(2*d-1)*(-1)^(n/d).

A295122 Expansion of Product_{k>=1} 1/(1 + x^k)^(k(5k-3)/2).

Original entry on oeis.org

1, -1, -6, -12, 6, 65, 179, 202, -137, -1392, -3492, -5135, -1325, 15437, 52934, 101787, 116827, -16945, -462603, -1350732, -2475989, -2889620, -343236, 8559858, 26972213, 53099230, 72521956, 47535918, -86985043, -409729146, -952305325, -1577038736

Offset: 0

Seiichi Manyama, Nov 15 2017

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n*(5*n-3)/2, g(n) = -1.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A294846 (b=3), A284896 (b=4), A295086 (b=5), A295121 (b=6), this sequence (b=7), A295123 (b=8).

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1+x^k)^(k*(5*k-3)/2)))

Convolution inverse of A294837.
G.f.: Product_{k>=1} 1/(1 + x^k)^A000566(k).
a(0) = 1 and a(n) = (1/(2*n)) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(5*d-3)*(-1)^(n/d).

A295123 Expansion of Product_{k>=1} 1/(1 + x^k)^(k(3k-2)).

Original entry on oeis.org

1, -1, -7, -14, 10, 93, 242, 229, -410, -2446, -5500, -6458, 4062, 38899, 104715, 165843, 103045, -327200, -1393131, -3075317, -4305200, -2069461, 9129361, 35219829, 75832840, 109569915, 74818084, -143480059, -686408279, -1607860793, -2614721006, -2674073316

Offset: 0

Seiichi Manyama, Nov 15 2017

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n*(3*n-2), g(n) = -1.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A294846 (b=3), A284896 (b=4), A295086 (b=5), A295121 (b=6), A295122 (b=7), this sequence (b=8).

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1+x^k)^(k*(3*k-2))))

Convolution inverse of A294838.
G.f.: Product_{k>=1} 1/(1 + x^k)^A000567(k).
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(3*d-2)*(-1)^(n/d).

cp's OEIS Frontend

A307462 Expansion of Product_{k>=1} (1+x^k)^((-1)^k*k^2).

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A322199 Expansion of Product_{k>=1} 1/(1 - (2^k + 1) * x^k).

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A294757 Expansion of Product_{k>=1} 1/(1 - k^k*x^k)^(k^k).

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A294813 Expansion of Product_{k>=1} 1/(1 - k^k*x^k)^k.

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A294953 Expansion of Product_{k>=1} (1 - k^(2*k)*x^k)^k.

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A294954 Expansion of Product_{k>=1} 1/(1 - k^(2*k)*x^k)^k.

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A295086 Expansion of Product_{k>=1} 1/(1 + x^k)^(k*(3*k-1)/2).

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A295121 Expansion of Product_{k>=1} 1/(1 + x^k)^(k*(2*k-1)).

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A294953 Expansion of Product_{k>=1} (1 - k^(2k)x^k)^k.

A294954 Expansion of Product_{k>=1} 1/(1 - k^(2k)x^k)^k.

A295086 Expansion of Product_{k>=1} 1/(1 + x^k)^(k(3k-1)/2).

A295121 Expansion of Product_{k>=1} 1/(1 + x^k)^(k(2k-1)).

A295122 Expansion of Product_{k>=1} 1/(1 + x^k)^(k(5k-3)/2).

A295123 Expansion of Product_{k>=1} 1/(1 + x^k)^(k(3k-2)).