A294842 -id:A294842 - cp's OEIS frontend

A294843 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(4*k-1)/6).

1, 1, 7, 29, 93, 320, 1026, 3256, 9995, 30102, 88722, 257042, 732876, 2058370, 5703858, 15606076, 42203027, 112882223, 298849221, 783574536, 2035876825, 5244191462, 13398463986, 33967008194, 85476285603, 213583335753, 530099612487, 1307195997381, 3203555001240, 7804386224233

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the hexagonal pyramidal numbers (A002412).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number

Cf. A002412, A028377, A258343, A281156, A294836, A294842, A294844, A294845.

nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1)(4 k - 1)/6), {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)(4 d - 1)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 29}]

G.f.: Product_{k>=1} (1 + x^k)^A002412(k).

a(n) ~ exp(-2401 * Pi^16 / (671846400000000 * Zeta(5)^3) - 49*Pi^8 * Zeta(3) / (518400000 * Zeta(5)^2) - Zeta(3)^2 / (2400*Zeta(5)) + (343 * Pi^12 / (77760000000 * 15^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4*Zeta(3) / (72000 * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) - (49*Pi^8 / (8640000 * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (8 * (15*Zeta(5))^(2/5))) * n^(2/5) + (7*Pi^4 / (720 * (15*Zeta(5))^(3/5))) * n^(3/5) + (5*(15*Zeta(5))^(1/5)/4) * n^(4/5)) * (3*Zeta(5))^(1/10) / (2^(173/360) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - Vaclav Kotesovec, Nov 10 2017

A294844 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(5*k-2)/6).

Original entry on oeis.org

1, 1, 8, 34, 114, 411, 1380, 4573, 14650, 45995, 141296, 426364, 1265443, 3698011, 10657134, 30312395, 85183177, 236681860, 650686538, 1771098691, 4775571943, 12762628737, 33821018537, 88909273699, 231945942992, 600700301298, 1544897610261, 3946762859175, 10018454809275, 25274880698255

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the heptagonal pyramidal numbers (A002413).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number

Cf. A002413, A258343, A281156, A294837, A294840, A294842, A294843, A294845.

nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1) (5 k - 2)/6), {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) (5 d - 2)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 29}]

G.f.: Product_{k>=1} (1 + x^k)^A002413(k).

a(n) ~ (3*Zeta(5))^(1/10) / (2^(479/720) * 5^(3/10) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (1312200000000000 * Zeta(5)^3) - 49 * Pi^8 * Zeta(3) / (405000000 * Zeta(5)^2) - Zeta(3)^2 / (750*Zeta(5)) + (343*Pi^12 / (60750000000 * 2^(3/5) * 3^(1/5) * 5^(2/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (22500 * 2^(3/5) * 3^(1/5) * 5^(2/5) * Zeta(5)^(6/5))) * n^(1/5) - (49*Pi^8 / (5400000 * 2^(1/5) * 3^(2/5) * 5^(4/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(6/5) * 5^(4/5) * (3*Zeta(5))^(2/5))) * n^(2/5) + (7*Pi^4 / (900 * 2^(4/5) * 5^(1/5) * (3*Zeta(5))^(3/5))) * n^(3/5) + (5^(7/5) * (3*Zeta(5))^(1/5) / 2^(12/5)) * n^(4/5)). - Vaclav Kotesovec, Nov 10 2017

A294845 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(2*k-1)/2).

Original entry on oeis.org

1, 1, 9, 39, 136, 511, 1785, 6139, 20404, 66406, 211418, 660752, 2030172, 6139231, 18300573, 53823451, 156344596, 448886205, 1274840165, 3583595734, 9976530997, 27520998775, 75262394273, 204130567402, 549318633095, 1467178746342, 3890697051314, 10246833932820, 26809705578787, 69702402930045

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the octagonal pyramidal numbers (A002414).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A002414, A258343, A281156, A294838, A294841, A294842, A294843, A294844.

nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1) (2 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 - 1)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 29}]

G.f.: Product_{k>=1} (1 + x^k)^A002414(k).

a(n) ~ exp(-2401 * Pi^16 / (2267481600000000 * Zeta(5)^3) - 49*Pi^8 * Zeta(3) / (388800000 * Zeta(5)^2) - Zeta(3)^2 / (400*Zeta(5)) + (343*Pi^12 / (87480000000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (18000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) - (49*Pi^8 / (6480000 * 2^(3/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + 3^(1/5)*Zeta(3) / (2^(13/5) * (5*Zeta(5))^(2/5))) * n^(2/5) + (7*Pi^4 / (1080 * 2^(2/5) * 3^(1/5) * (5*Zeta(5))^(3/5))) * n^(3/5) + (3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) / 2^(11/5)) * n^(4/5)) * 3^(1/5) * Zeta(5)^(1/10) / (2^(11/20) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - Vaclav Kotesovec, Nov 10 2017

A327063 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^j).

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 8, 11, 15, 24, 34, 43, 63, 87, 115, 159, 217, 279, 380, 505, 657, 868, 1139, 1458, 1913, 2482, 3162, 4069, 5232, 6628, 8469, 10755, 13544, 17127, 21634, 27061, 33988, 42557, 52985, 66069, 82289, 101862, 126281, 156275, 192655, 237530, 292502

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A028377, A192065, A211856, A294842, A280541, A327064, A327065, A327066.

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

A327064 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^k).

Original entry on oeis.org

1, 1, 2, 5, 10, 18, 35, 62, 110, 197, 339, 573, 975, 1621, 2674, 4385, 7108, 11422, 18277, 28976, 45648, 71531, 111372, 172416, 265695, 407210, 621143, 943392, 1426414, 2147672, 3221271, 4812534, 7163440, 10625651, 15706871, 23141148, 33987287, 49762235

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A028377, A192065, A211856, A294842, A280541, A327063, A327065, A327067.

nmax = 40; CoefficientList[Series[Product[Product[(1+x^(k*j))^k, {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x]

A327065 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(kj))^(kj)).

Original entry on oeis.org

1, 1, 2, 5, 12, 20, 42, 75, 141, 259, 466, 799, 1427, 2443, 4169, 7049, 11863, 19605, 32518, 53184, 86579, 140018, 225380, 359739, 572864, 905903, 1426270, 2234952, 3488313, 5416403, 8383226, 12917257, 19831763, 30334937, 46245977, 70242043, 106371686

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A028377, A192065, A211856, A294842, A280541, A327063, A327064, A327068.

nmax = 40; CoefficientList[Series[Product[Product[(1+x^(k*j))^(k*j), {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x]

A318125 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)x^k(1 + (n - 3)x^k)/(k(1 - x^k)^4)).

Original entry on oeis.org

1, 1, 3, 14, 54, 238, 1026, 4573, 20404, 91902, 415953, 1891908, 8638846, 39569655, 181766878, 836950153, 3861927937, 17853107055, 82668539290, 383360628369, 1780126898575, 8275908734103, 38517137597486, 179442212204245, 836741558761935, 3905012142470483

Offset: 0

Ilya Gutkovskiy, Aug 18 2018

nonn

For n > 2, a(n) is the n-th term of the weigh transform of n-gonal pyramidal numbers.

N. J. A. Sloane, Transforms
Index to sequences related to pyramidal numbers

Cf. A258343, A281156, A294842, A294843, A294844, A294845, A318121, A318124.

Table[SeriesCoefficient[Exp[Sum[(-1)^(k + 1) x^k (1 + (n - 3) x^k)/(k (1 - x^k)^4), {k, 1, n}]], {x, 0, n}], {n, 0, 25}]

a(n) ~ c * d^n / sqrt(n), where d = 4.761510955746025663058811... and c = 0.2241869836397882024713... - Vaclav Kotesovec, Aug 19 2018

cp's OEIS Frontend

A294843 Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(4*k-1)/6).

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A294844 Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(5*k-2)/6).

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A294845 Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(2*k-1)/2).

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A327063 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^j).

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A327064 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^k).

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A327065 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^(k*j)).

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A318125 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + (n - 3)*x^k)/(k*(1 - x^k)^4)).

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A294843 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(4*k-1)/6).

A294844 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(5*k-2)/6).

A294845 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(2*k-1)/2).

A327065 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(kj))^(kj)).

A318125 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)x^k(1 + (n - 3)x^k)/(k(1 - x^k)^4)).