A284896 -id:A284896 - cp's OEIS frontend

A281590 Indices k such that A284896(k-1) and A284896(k) have a different sign.

1, 4, 9, 15, 20, 27, 33, 41, 48, 56, 64, 72, 80, 89, 98, 107, 116, 126, 136, 146, 156, 166, 176, 187, 198, 208, 219, 231, 242, 253, 265, 276, 288, 300, 312, 324, 337, 349, 362, 374, 387, 400, 413, 426, 439, 452, 465, 479, 492, 506, 519, 533, 547, 561, 575

Offset: 1

Vaclav Kotesovec, Apr 14 2017

nonn

			A284896(8) = 73, A284896(9) = -45, sign is changed, so 9 is in the sequence.
A284896(14) = -66, A284896(15) = 2794, sign is changed, so 15 is in the sequence.

Vaclav Kotesovec, Table of n, a(n) for n = 1..2612

Cf. A284896, A281591.

nmax = 1000; A284896 = Rest[CoefficientList[Series[Product[1/(1 + x^k)^(k^2), {k, 1, nmax}], {x, 0, nmax}], x]]; csign = {1}; Do[If[(A284896[[n]] < 0 && A284896[[n+1]] >= 0) || (A284896[[n]] >= 0 && A284896[[n+1]] < 0), csign = Flatten[{csign, n + 1}]], {n, 1, Length[A284896] - 1}]; csign

A281790 Expansion of Product_{k>=1} (1+x^(k^2))^k.

Original entry on oeis.org

1, 1, 0, 0, 2, 2, 0, 0, 1, 4, 3, 0, 0, 6, 6, 0, 4, 7, 6, 3, 8, 8, 6, 6, 4, 21, 20, 4, 1, 34, 34, 2, 8, 23, 44, 28, 19, 18, 54, 54, 18, 56, 65, 46, 25, 100, 94, 38, 42, 85, 169, 107, 56, 69, 226, 194, 62, 111, 194, 241, 125, 215, 246, 258, 207, 283, 437, 292

Offset: 0

Vaclav Kotesovec, Apr 14 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

Cf. A000219, A026007, A027998, A033461, A255528, A266891, A284896, A285047.

nmax = 100; CoefficientList[Series[Product[(1+x^(k^2))^k, {k,1,nmax}], {x,0,nmax}], x]
nmax = 100; s = 1 + x; Do[s*=Sum[Binomial[k, j]*x^(j*k^2), {j, 0, Floor[nmax/k^2] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; CoefficientList[s, x]

a(n) ~ exp(sqrt(n/6)*Pi) / (2^(11/6) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 15 2017

A291649 Expansion of Product_{k>=1} (1 + x^(k^2))^(k^2).

Original entry on oeis.org

1, 1, 0, 0, 4, 4, 0, 0, 6, 15, 9, 0, 4, 40, 36, 0, 17, 71, 90, 36, 64, 100, 180, 144, 96, 274, 394, 300, 148, 740, 820, 480, 472, 1150, 1851, 1341, 1146, 1318, 3880, 3540, 1704, 3017, 6455, 7134, 3780, 7822, 9574, 12180, 10304, 12057, 19750, 22485, 20558, 15910, 43076, 43236, 31104, 33742, 66895

Offset: 0

Ilya Gutkovskiy, Aug 28 2017

nonn

Number of partitions of n into distinct squares, where k^2 different parts of size k^2 are available (1a, 4a, 4b, 4c, 4d, ...).

			a(8) = 6 because we have [4a, 4b], [4a, 4c], [4a, 4d], [4b, 4c], [4b, 4d] and [4c, 4d].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A000290, A026007, A027998, A033461, A262736, A281790, A284896, A291655, A291692.

nmax = 100; CoefficientList[Series[Product[(1 + x^k^2)^k^2, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; s = 1 + x; Do[s *= Sum[Binomial[k^2, j]*x^(j*k^2), {j, 0, Floor[nmax/k^2] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Aug 28 2017 *)

G.f.: Product_{k>=1} (1 + x^A000290(k))^A000290(k).

a(n) ~ exp(5 * 2^(-9/5) * 3^(-3/5) * (9-4*sqrt(2))^(1/5) * Pi^(1/5) * Zeta(5/2)^(2/5) * n^(3/5)) * 3^(1/5) * (2*sqrt(2)-1)^(1/5) * Zeta(5/2)^(1/5) / (2^(9/10) * sqrt(5) * Pi^(2/5) * n^(7/10)). - Vaclav Kotesovec, Aug 29 2017

A294846 Expansion of Product_{k>=1} 1/(1 + x^k)^(k*(k+1)/2).

Original entry on oeis.org

1, -1, -2, -4, 0, 3, 17, 24, 40, 9, -24, -149, -250, -435, -395, -281, 514, 1528, 3542, 5127, 6920, 5416, 1368, -11136, -28533, -57051, -82846, -107315, -95655, -43646, 107826, 345877, 727771, 1150968, 1601729, 1766547, 1495154, 183944, -2339567, -6770991, -12701854

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

sign

Convolution inverse of A028377.

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n*(n+1)/2, g(n) = -1. - Seiichi Manyama, Nov 14 2017

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

Cf. A000217, A000294, A028377, A255528, A284896, A292386.

nmax = 40; CoefficientList[Series[Product[1/(1 + x^k)^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[(-1)^(k/d + 1) d^2 (d + 1)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]

G.f.: Product_{k>=1} 1/(1 + x^k)^A000217(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*(d+1)*(-1)^(n/d). - Seiichi Manyama, Nov 14 2017

A284897 Expansion of Product_{k>=1} 1/(1+x^k)^(k^3) in powers of x.

Original entry on oeis.org

1, -1, -7, -20, -8, 99, 455, 958, 715, -3606, -17450, -44157, -61852, 19546, 419786, 1442212, 3084950, 3756436, -2155907, -27112107, -88277693, -187777531, -251308697, -5153980, 1182558343, 4299818445, 9988792754, 16075200671, 12020651310, -29802956283

Offset: 0

Seiichi Manyama, Apr 05 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..7180

Cf. A248882.

Product_{k>=1} 1/(1+x^k)^(k^m): A081362 (m=0), A255528 (m=1), A284896 (m=2), this sequence (m=3), A284898 (m=4), A284899 (m=5).

CoefficientList[Series[Product[1/(1 + x^k)^(k^3) , {k, 40}], {x, 0, 40}], x] (* Indranil Ghosh, Apr 05 2017 *)

x= 'x + O('x^40); Vec(prod(k=1, 40, 1/(1 + x^k)^(k^3))) \\ Indranil Ghosh, Apr 05 2017

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A284900(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A284898 Expansion of Product_{k>=1} 1/(1+x^k)^(k^4) in powers of x.

Original entry on oeis.org

1, -1, -15, -66, -54, 725, 4580, 12739, 3346, -149076, -791226, -2182124, -1656973, 16553206, 100646954, 318795473, 506196578, -818806580, -9148048880, -36415709566, -87180585636, -70923559814, 484810027389, 2992082912770, 9866919438716, 19936695359140

Offset: 0

Seiichi Manyama, Apr 05 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..3995

Cf. A248883.

Product_{k>=1} 1/(1+x^k)^(k^m): A081362 (m=0), A255528 (m=1), A284896 (m=2), A284897 (m=3), this sequence (m=4), A284899 (m=5).

CoefficientList[Series[Product[1/(1 + x^k)^(k^4) , {k, 40}], {x, 0, 40}], x] (* Indranil Ghosh, Apr 05 2017 *)

x= 'x + O('x^40); Vec(prod(k=1, 40, 1/(1 + x^k)^(k^4))) \\ Indranil Ghosh, Apr 05 2017

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A284926(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A284899 Expansion of Product_{k>=1} 1/(1+x^k)^(k^5) in powers of x.

Original entry on oeis.org

1, -1, -31, -212, -284, 4935, 43719, 160002, -96747, -4914512, -31358932, -94515285, 97642670, 2823746182, 16834776254, 51617810512, -11233909783, -1137004349695, -7267899354808, -25263858110877, -24537905293857, 319397811973578, 2523465326904492

Offset: 0

Seiichi Manyama, Apr 05 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..2647

Cf. A248884.

Product_{k>=1} 1/(1+x^k)^(k^m): A081362 (m=0), A255528 (m=1), A284896 (m=2), A284897 (m=3), A284898 (m=4), this sequence (m=5).

CoefficientList[Series[Product[1/(1 + x^k)^(k^5) , {k, 40}], {x, 0, 40}], x] (* Indranil Ghosh, Apr 05 2017 *)

x= 'x + O('x^40); Vec(prod(k=1, 40, 1/(1 + x^k)^(k^5))) \\ Indranil Ghosh, Apr 05 2017

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A284927(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 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).

cp's OEIS Frontend

A281590 Indices k such that A284896(k-1) and A284896(k) have a different sign.

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A281790 Expansion of Product_{k>=1} (1+x^(k^2))^k.

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A291649 Expansion of Product_{k>=1} (1 + x^(k^2))^(k^2).

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A294846 Expansion of Product_{k>=1} 1/(1 + x^k)^(k*(k+1)/2).

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A284897 Expansion of Product_{k>=1} 1/(1+x^k)^(k^3) in powers of x.

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A284898 Expansion of Product_{k>=1} 1/(1+x^k)^(k^4) in powers of x.

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A284899 Expansion of Product_{k>=1} 1/(1+x^k)^(k^5) in powers of x.

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

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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).