A294838 -id:A294838 - cp's OEIS frontend

A294102 Expansion of Product_{k>=1} (1 + x^k)^(k(3k-1)/2).

1, 1, 5, 17, 44, 127, 332, 866, 2182, 5412, 13119, 31292, 73516, 170136, 388829, 877653, 1959111, 4327221, 9464856, 20511598, 44067446, 93901142, 198539477, 416696608, 868448305, 1797890682, 3698350956, 7561361750, 15369154555, 31064311255, 62449795986, 124895635385, 248538538858, 492207649241

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the pentagonal numbers (A000326).

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, Nov 14 2017

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

Cf. A000326, A027998, A028377, A294836, A294837, A294838.

nmax = 33; CoefficientList[Series[Product[(1 + x^k)^(k (3 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 (3 d - 1)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

G.f.: Product_{k>=1} (1 + x^k)^A000326(k).
a(n) ~ exp(-225*Zeta(3)^3 / (98*Pi^8) - 9 * 5^(5/4) * Zeta(3)^2 / (4 * 7^(5/4) * Pi^5) * n^(1/4) - (3*sqrt(5/7) * Zeta(3) / (2*Pi^2)) * sqrt(n) + (2 * (7/5)^(1/4) * Pi / 3) * n^(3/4)) * 7^(1/8) / (2^(47/24) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017
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)^(1+n/d). - Seiichi Manyama, Nov 14 2017

A294837 Expansion of Product_{k>=1} (1 + x^k)^(k(5k-3)/2).

Original entry on oeis.org

1, 1, 7, 25, 73, 236, 688, 1994, 5573, 15272, 40896, 107526, 277999, 707209, 1774067, 4390665, 10734216, 25941541, 62022609, 146793160, 344129900, 799517074, 1841734224, 4208327222, 9542121050, 21477834062, 48005313446, 106579556936, 235107392079, 515441826521, 1123360284127, 2434346065621

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the heptagonal numbers (A000566).

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, Nov 14 2017

Seiichi Manyama, Table of n, a(n) for n = 0..8757
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 Number

Cf. A000566, A027998, A028377, A294102, A294836, A294838.

nmax = 31; CoefficientList[Series[Product[(1 + x^k)^(k (5 k - 3)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (5 d - 3)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 31}]

G.f.: Product_{k>=1} (1 + x^k)^A000566(k).
a(n) ~ 7^(1/8) * exp(2*Pi*7^(1/4) * n^(3/4) / 3^(5/4) - 9*Zeta(3) * sqrt(3*n/7) /(2*Pi^2) - 243*Zeta(3)^2 * (3*n/7)^(1/4) / (28*Pi^5) - 2187*Zeta(3)^3 / (98*Pi^8)) / (2^(15/8) * 3^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017
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)^(1+n/d). - Seiichi Manyama, Nov 14 2017

A294836 Expansion of Product_{k>=1} (1 + x^k)^(k(2k-1)).

Original entry on oeis.org

1, 1, 6, 21, 58, 178, 494, 1365, 3640, 9533, 24401, 61384, 151958, 370335, 890565, 2113913, 4959199, 11505799, 26420628, 60082005, 135386341, 302448477, 670148898, 1473387787, 3215519032, 6968266907, 14999453058, 32079714584, 68187859040, 144083404856, 302727633735, 632579826174

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the hexagonal numbers (A000384).

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

Seiichi Manyama, Table of n, a(n) for n = 0..9392
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 Number

Cf. A000384, A027998, A028377, A294102, A294837, A294838.

nmax = 31; CoefficientList[Series[Product[(1 + x^k)^(k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 31}]

G.f.: Product_{k>=1} (1 + x^k)^A000384(k).
a(n) ~ 7^(1/8) * exp(Pi*2^(3/2) * (7/15)^(1/4) * n^(3/4)/3 - 3*Zeta(3)*sqrt(15*n/7) / (2*Pi^2) - 135*Zeta(3)^2 * (15*n/7)^(1/4) / (28*sqrt(2)*Pi^5) - 2025*Zeta(3)^3 / (196*Pi^8)) / (2^(5/3) * 15^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017
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)^(1+n/d). - Seiichi Manyama, Nov 14 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

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).

A294841 Expansion of Product_{k>=1} (1 + x^(2k-1))^(k(3k-2))(1 + x^(2k))^(k(3*k+2)).

Original entry on oeis.org

1, 1, 5, 13, 34, 87, 212, 504, 1167, 2665, 5933, 13042, 28191, 60148, 126688, 263821, 543414, 1108272, 2239182, 4484482, 8907530, 17555485, 34345465, 66724969, 128772908, 246951514, 470738283, 892159198, 1681544803, 3152656375, 5880839454, 10916463171, 20169007200, 37095527149

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the generalized octagonal numbers (A001082).

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, Octagonal Number

Cf. A001082, A028377, A294838, A294839, A294840.

nmax = 33; CoefficientList[Series[Product[(1 + x^(2 k - 1))^(k (3 k - 2)) (1 + x^(2 k))^(k (3 k + 2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (d^2 + d - Ceiling[d/2]^2), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

G.f.: Product_{k>=1} (1 + x^k)^A001082(k+1).

a(n) ~ exp(Pi/3 * (7/5)^(1/4) * 2^(3/4) * n^(3/4) + 9*Zeta(3) / (2*Pi^2) * sqrt(5*n/14) - (405*Zeta(3)^2 / (56*Pi^5) + Pi/48) * (10*n/7)^(1/4) + (6075*Zeta(3)^2 / (196*Pi^8) + 15/(224*Pi^2)) * Zeta(3)) * 7^(1/8) / (2^(9/4) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017

A318124 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)x^k(1 + (n - 3)x^k)/(k(1 - x^k)^3)).

Original entry on oeis.org

1, 1, 2, 9, 31, 127, 494, 1994, 8040, 32741, 133855, 549775, 2266756, 9372300, 38862245, 161500403, 672538548, 2805669061, 11723319333, 49055511943, 205534846202, 862167483656, 3620429584614, 15217780335870, 64022149180478, 269566679312520, 1135878674712355

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 numbers.

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

Cf. A027998, A028377, A294102, A294836, A294837, A294838, A318118, A318125.

Table[SeriesCoefficient[Exp[Sum[(-1)^(k + 1) x^k (1 + (n - 3) x^k)/(k (1 - x^k)^3), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

a(n) ~ c * d^n / sqrt(n), where d = 4.2950655312028649462400... and c = 0.204576644650802181512... - Vaclav Kotesovec, Aug 19 2018

cp's OEIS Frontend

A294102 Expansion of Product_{k>=1} (1 + x^k)^(k*(3*k-1)/2).

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A294837 Expansion of Product_{k>=1} (1 + x^k)^(k*(5*k-3)/2).

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A294836 Expansion of Product_{k>=1} (1 + x^k)^(k*(2*k-1)).

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

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

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A294841 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(k*(3*k-2))*(1 + x^(2*k))^(k*(3*k+2)).

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A318124 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + (n - 3)*x^k)/(k*(1 - x^k)^3)).

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A294102 Expansion of Product_{k>=1} (1 + x^k)^(k(3k-1)/2).

A294837 Expansion of Product_{k>=1} (1 + x^k)^(k(5k-3)/2).

A294836 Expansion of Product_{k>=1} (1 + x^k)^(k(2k-1)).

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

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

A294841 Expansion of Product_{k>=1} (1 + x^(2k-1))^(k(3k-2))(1 + x^(2k))^(k(3*k+2)).

A318124 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)x^k(1 + (n - 3)x^k)/(k(1 - x^k)^3)).