A294102 -id:A294102 - cp's OEIS frontend

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

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

A294838 Expansion of Product_{k>=1} (1 + x^k)^(k(3k-2)).

Original entry on oeis.org

1, 1, 8, 29, 89, 301, 915, 2763, 8040, 22910, 63776, 174174, 467448, 1233836, 3209679, 8234149, 20857621, 52206847, 129227514, 316543962, 767767628, 1844925743, 4394337797, 10379319118, 24320964976, 56557678603, 130571770387, 299357973400, 681777058604, 1542840256421, 3470045577372

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the octagonal numbers (A000567).

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n*(3*n-2), g(n) = -1. - Seiichi Manyama, Nov 14 2017

Seiichi Manyama, Table of n, a(n) for n = 0..8270
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. A000567, A027998, A028377, A294102, A294836, A294837.

nmax = 30; CoefficientList[Series[Product[(1 + x^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^2 (3 d - 2), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 30}]

G.f.: Product_{k>=1} (1 + x^k)^A000567(k).
a(n) ~ exp(-1800*Zeta(3)^3 / (49*Pi^8) - (9 * 2^(3/4) * 5^(5/4) * Zeta(3)^2 / (7^(5/4)*Pi^5)) * n^(1/4) - (3*sqrt(10/7) * Zeta(3) / Pi^2) * sqrt(n) + (2*(14/5)^(1/4) * Pi/3) * n^(3/4)) * 7^(1/8) / (2^(41/24) * 5^(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*(3*d-2)*(-1)^(1+n/d). - Seiichi Manyama, Nov 14 2017

A294842 Expansion of Product_{k>=1} (1 + x^k)^(k^2*(k+1)/2).

Original entry on oeis.org

1, 1, 6, 24, 73, 238, 722, 2175, 6343, 18177, 50982, 140671, 382227, 1023623, 2706184, 7067324, 18250671, 46635309, 117997008, 295794098, 735030985, 1811435607, 4429226677, 10749552338, 25903858181, 62000039513, 147435739522, 348431110651, 818549931526, 1912010876019, 4441687009798

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the pentagonal pyramidal numbers (A002411).

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
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 Pyramidal Number

Cf. A002411, A258343, A281156, A294102, A294839, A294843, A294844, A294845.

nmax = 30; CoefficientList[Series[Product[(1 + x^k)^(k^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^3 (d + 1)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 30}]

G.f.: Product_{k>=1} (1 + x^k)^A002411(k).

a(n) ~ exp(-2401 * Pi^16 / (2^12 * 3^11 * 5^8 * Zeta(5)^3) + (343 * Pi^12 / (2^(38/5) * 3^(37/5) * 5^(36/5) * Zeta(5)^(11/5))) * n^(1/5) - (49*Pi^8 / (2^(31/5) * 3^(24/5) * 5^(22/5) * Zeta(5)^(7/5))) * n^(2/5) + (7*Pi^4 / (2^(14/5) * 3^(16/5) * 5^(8/5) * Zeta(5)^(3/5))) * n^(3/5) + (5 * 3^(2/5) * (5*Zeta(5))^(1/5) / 2^(12/5)) * n^(4/5)) * 3^(1/5) * Zeta(5)^(1/10) / (2^(167/240) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - Vaclav Kotesovec, Nov 10 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

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).

A294839 Expansion of Product_{k>=1} (1 + x^(2k-1))^(k(3k-1)/2)(1 + x^(2k))^(k(3*k+1)/2).

Original entry on oeis.org

1, 1, 2, 7, 13, 30, 61, 125, 250, 494, 960, 1835, 3487, 6520, 12105, 22239, 40515, 73207, 131315, 233831, 413625, 727100, 1270405, 2207243, 3814155, 6557164, 11217391, 19099932, 32375026, 54640509, 91836697, 153739008, 256379360, 425964293, 705197513, 1163452547, 1913096832, 3135609791

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the generalized pentagonal numbers (A001318).

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. A001318, A028377, A294102, A294840, A294841.

nmax = 37; CoefficientList[Series[Product[(1 + x^(2 k - 1))^(k (3 k - 1)/2) (1 + x^(2 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 Ceiling[d/2] Ceiling[(3 d + 1)/2]/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 37}]

G.f.: Product_{k>=1} (1 + x^k)^A001318(k).

a(n) ~ exp(Pi*sqrt(2) * 7^(1/4) * n^(3/4) / (3*5^(1/4)) + 9*Zeta(3) * sqrt(5*n/7) / (4*Pi^2) + (7*Pi^6 - 2430*Zeta(3)^2) * (5/7)^(1/4) * n^(1/4) / (336 * sqrt(2) * Pi^5) + 15*Zeta(3)*(3240*Zeta(3)^2 - 7*Pi^6) / (3136*Pi^8)) * 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

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

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A294838 Expansion of Product_{k>=1} (1 + x^k)^(k*(3*k-2)).

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A294842 Expansion of Product_{k>=1} (1 + x^k)^(k^2*(k+1)/2).

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

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

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A294839 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(k*(3*k-1)/2)*(1 + x^(2*k))^(k*(3*k+1)/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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A294837 Expansion of Product_{k>=1} (1 + x^k)^(k(5k-3)/2).

A294838 Expansion of Product_{k>=1} (1 + x^k)^(k(3k-2)).

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

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

A294839 Expansion of Product_{k>=1} (1 + x^(2k-1))^(k(3k-1)/2)(1 + x^(2k))^(k(3*k+1)/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)).