A258343 -id:A258343 - cp's OEIS frontend

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

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

A343200 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,3).

Original entry on oeis.org

1, 4, 16, 64, 221, 736, 2338, 7132, 21093, 60652, 170172, 467140, 1257571, 3325824, 8654576, 22189340, 56116043, 140122760, 345769094, 843827436, 2038017983, 4874329024, 11550814704, 27134195608, 63215468883, 146120097736, 335227455982, 763592477104, 1727482413548

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..7650

Cf. A000292, A033488, A219555, A255050, A258343, A338645, A344097, A344098.

nmax = 28; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 3, 3], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 3, 3], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 28}]

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * A033488(d) ) * a(n-k).

a(n) ~ (3*zeta(5))^(1/10) / (2^(7/10) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-469*log(2)/720 - 2401*Pi^16 / (656100000000*zeta(5)^3) + 539*Pi^8*zeta(3) / (8100000*zeta(5)^2) - 7*Pi^6 / (27000*zeta(5)) - 121*zeta(3)^2 / (600*zeta(5)) + (343*Pi^12 / (303750000 * 2^(3/5) * 15^(1/5) * zeta(5)^(11/5)) - 77*Pi^4*zeta(3) / (4500 * 2^(3/5) * 15^(1/5) * zeta(5)^(6/5)) + Pi^2 / (6*2^(3/5) * (15*zeta(5))^(1/5))) * n^(1/5) + (-49*Pi^8 / (270000 * 2^(1/5) * 15^(2/5) * zeta(5)^(7/5)) + 11*zeta(3) / (4*2^(1/5) * (15*zeta(5))^(2/5))) * n^(2/5) + (7*Pi^4 / (90*2^(4/5) * (15*zeta(5))^(3/5))) * n^(3/5) + (5*(15*zeta(5))^(1/5) / (4*2^(2/5))) * n^(4/5)). - Vaclav Kotesovec, May 12 2021

A292387 Expansion of Product_{k>=1} (1 - x^k)^(k(k+1)(k+2)/6).

Original entry on oeis.org

1, -1, -4, -6, -4, 19, 60, 131, 149, -4, -572, -1764, -3485, -4716, -2658, 7606, 32944, 77152, 132586, 161275, 75150, -281687, -1111029, -2560293, -4470415, -5922117, -4603551, 3799070, 25573251, 67259095, 130430051, 201158707, 232853019, 124749892, -295134275, -1260897993, -2995361708, -5515840117

Offset: 0

Ilya Gutkovskiy, Sep 15 2017

sign

Convolution inverse of A000335 (Euler transform of the tetrahedral numbers).

Cf. A000335, A258343, A292386.

nmax = 37; CoefficientList[Series[Product[(1 - x^k)^(k (k + 1) (k + 2)/6), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^k)^(k*(k+1)*(k+2)/6).

A295180 Expansion of Product_{k>=1} (1 + x^k)^(3k(k-1)/2+1).

Original entry on oeis.org

1, 1, 4, 14, 35, 96, 242, 609, 1483, 3565, 8376, 19389, 44254, 99584, 221470, 486810, 1058914, 2280519, 4866492, 10294313, 21598679, 44966391, 92930485, 190721585, 388828094, 787710401, 1586166758, 3175548134, 6322372729, 12520759979, 24669499432, 48367447687, 94381633962, 183331308393

Offset: 0

Ilya Gutkovskiy, Nov 16 2017

nonn

Weigh transform of the centered triangular numbers (A005448).

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -(3*n*(n-1)/2+1), g(n) = -1. - Seiichi Manyama, Nov 16 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, Centered Triangular Number
Index entries for sequences related to centered polygonal numbers

Cf. A005448, A028377, A258343, A295179.

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

G.f.: Product_{k>=1} (1 + x^k)^A005448(k).
a(n) ~ exp(15*Zeta(3) / (28*Pi^2) - 6075*Zeta(3)^3 / (98*Pi^8) + (Pi/6 - 405*Zeta(3)^2 / (28*Pi^5)) * (5*n/7)^(1/4) - (9*sqrt(5/7) * Zeta(3) / (2*Pi^2)) * sqrt(n) + (2*Pi * (7/5)^(1/4)/3) * n^(3/4)) * 7^(1/8) / (2^(19/8) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 16 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*(3*d*(d-1)/2+1)*(-1)^(1+n/d). - Seiichi Manyama, Nov 16 2017

A344099 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,4).

Original entry on oeis.org

1, 1, 5, 20, 60, 190, 561, 1651, 4720, 13300, 36716, 99872, 267836, 708890, 1854255, 4796273, 12279445, 31135188, 78236006, 194921680, 481758832, 1181675902, 2877646681, 6959866116, 16723591530, 39934902812, 94795718409, 223741936855, 525206126933, 1226393510220

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Cf. A000332, A000391, A028377, A258343, A305206, A344100, A344101.

nmax = 29; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 3, 4], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 3, 4], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 29}]

G.f.: exp( Sum_{k>=1} (-1)^(k+1) * x^k / (k*(1 - x^k)^5) ).

A344100 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+4,5).

Original entry on oeis.org

1, 1, 6, 27, 92, 323, 1070, 3527, 11314, 35708, 110478, 336629, 1011097, 2997233, 8778761, 25424358, 72867447, 206804742, 581573340, 1621407554, 4483701126, 12303384015, 33514076529, 90656680725, 243603875523, 650444927010, 1726229294595, 4554686670838, 11950683658941

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Cf. A000389, A000417, A028377, A258343, A305206, A344099, A344101.

nmax = 28; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 4, 5], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 4, 5], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 28}]

G.f.: exp( Sum_{k>=1} (-1)^(k+1) * x^k / (k*(1 - x^k)^6) ).

A344101 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+5,6).

Original entry on oeis.org

1, 1, 7, 35, 133, 511, 1869, 6797, 24095, 83938, 286734, 964348, 3196984, 10460310, 33813984, 108076908, 341821250, 1070484009, 3321584021, 10217036263, 31169524988, 94351439060, 283498600776, 845848778722, 2506779443603, 7381617323598, 21603241378334, 62853440151768

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Cf. A000428, A000579, A028377, A258343, A305206, A344099, A344100.

nmax = 27; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 5, 6], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 5, 6], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 27}]

G.f.: exp( Sum_{k>=1} (-1)^(k+1) * x^k / (k*(1 - x^k)^7) ).

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

A281157 Expansion of Product_{k>=1} (1 + x^k)^(k(2k^2+1)/3).

Original entry on oeis.org

1, 1, 6, 25, 78, 258, 800, 2463, 7344, 21511, 61677, 173980, 483319, 1323470, 3577605, 9553658, 25227727, 65918419, 170552866, 437196640, 1110945961, 2799689792, 7000246591, 17372882671, 42809388080, 104774554942, 254771953179, 615667051237, 1478934870484, 3532347875968

Offset: 0

Ilya Gutkovskiy, Jan 16 2017

nonn

Weigh transform of octahedral numbers (A005900).

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, Octahedral Number

Cf. A005900, A248882, A258343.

nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (2 k^2 + 1)/3), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^k)^(k*(2*k^2+1)/3).

a(n) ~ exp(-Zeta(3)^2 / (600*Zeta(5)) + (Zeta(3) / (4*(15*Zeta(5))^(2/5))) * n^(2/5) + (5*(15*Zeta(5))^(1/5) / 4) * n^(4/5)) * (3*Zeta(5))^(1/10) / (sqrt(Pi) * 2^(47/90) * 5^(2/5) * n^(3/5)). - Vaclav Kotesovec, Nov 09 2017

cp's OEIS Frontend

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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A343200 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,3).

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A292387 Expansion of Product_{k>=1} (1 - x^k)^(k*(k+1)*(k+2)/6).

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A295180 Expansion of Product_{k>=1} (1 + x^k)^(3*k*(k-1)/2+1).

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A344099 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,4).

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A344100 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+4,5).

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A344101 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+5,6).

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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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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).

A292387 Expansion of Product_{k>=1} (1 - x^k)^(k(k+1)(k+2)/6).

A295180 Expansion of Product_{k>=1} (1 + x^k)^(3k(k-1)/2+1).

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

A281157 Expansion of Product_{k>=1} (1 + x^k)^(k(2k^2+1)/3).