A258343 -id:A258343 - cp's OEIS frontend

A028377 Expansion of Product_{m>0} (1+q^m)^(m(m+1)/2).

1, 1, 3, 9, 19, 46, 100, 218, 460, 965, 1975, 3993, 7975, 15712, 30650, 59150, 113093, 214300, 402812, 751165, 1390714, 2557004, 4670770, 8479232, 15302657, 27462424, 49021252, 87057783, 153850769, 270614429, 473850031, 826125184, 1434286323, 2480145226

Offset: 0

N. J. A. Sloane

nonn

Convolved with aerated A000294: [1, 0, 2, 0, 4, 0, 10, 0, 26, ...] = A000294. - Gary W. Adamson, Jun 13 2009

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 (terms 0..1000 from Alois P. Heinz)

Cf. A000294. - Gary W. Adamson, Jun 13 2009

Cf. A027999, A258341, A258342, A258343, A258344, A258345, A258346.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(i*(i+1)/2, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 03 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[i*(i+1)/2, j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)

a(n) ~ 7^(1/8) * exp(2 * 7^(1/4) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) + 3^(3/2) * 5^(1/2) * Zeta(3) * n^(1/2) / (2 * 7^(1/2) * Pi^2) - 3^(13/4) * 5^(5/4) * Zeta(3)^2 * n^(1/4) / (4 * 7^(5/4) * Pi^5) + 2025 * Zeta(3)^3 / (98*Pi^8)) / (2^(49/24) * 15^(1/8) * n^(5/8)), where Zeta(3) = A002117. - Vaclav Kotesovec, Mar 11 2015
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)^(1+n/d). - Seiichi Manyama, Nov 14 2017
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^3)). - Ilya Gutkovskiy, May 28 2018

A258341 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)).

Original entry on oeis.org

1, 2, 7, 24, 65, 184, 487, 1254, 3145, 7706, 18480, 43490, 100692, 229472, 515802, 1144416, 2508948, 5439642, 11671859, 24801738, 52221911, 109013538, 225718717, 463769652, 945915199, 1915895576, 3854803572, 7706786958, 15314564282, 30255672820, 59440488874

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A027998, A028377, A027999, A258342, A258343, A258344, A258345, A258346, A258347.

nmax=30; CoefficientList[Series[Product[(1+x^k)^(k*(k+1)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 7^(1/8) / (2^(47/24) * 15^(1/8) * n^(5/8)) * exp(2025*Zeta(3)^3 / (49*Pi^8) - 135*(15/14)^(1/4) * Zeta(3)^2 / (14*Pi^5) * n^(1/4) + 3*sqrt(15/14) * Zeta(3) / Pi^2 * sqrt(n) + 2*(14/15)^(1/4)*Pi/3 * n^(3/4)), where Zeta(3) = A002117.

A258344 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)).

Original entry on oeis.org

1, 0, 2, 6, 13, 32, 69, 160, 344, 760, 1601, 3384, 7022, 14434, 29361, 59140, 118089, 233754, 459293, 895382, 1733904, 3334914, 6374654, 12111632, 22881777, 42993244, 80362496, 149464404, 276657082, 509740278, 935046158, 1707916988, 3106810873, 5629121054

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A027998, A028377, A027999, A258341, A258342, A258343, A258345, A258346, A258348.

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

a(n) ~ 7^(1/8) / (2^(43/24) * 15^(1/8) * n^(5/8)) * exp(-2025*Zeta(3)^3 / (49*Pi^8) - 135*(15/14)^(1/4) * Zeta(3)^2 / (14*Pi^5) * n^(1/4) - 3*sqrt(15/14) * Zeta(3) / Pi^2 * sqrt(n) + 2*(14/15)^(1/4)*Pi/3 * n^(3/4)), where Zeta(3) = A002117.

A258342 Expansion of Product_{k>=1} (1+x^k)^(k(k+1)(k+2)).

Original entry on oeis.org

1, 6, 39, 224, 1131, 5412, 24411, 105078, 435048, 1740312, 6755877, 25533330, 94205738, 340064322, 1203313782, 4180514846, 14279610417, 48013553310, 159086287869, 519912616614, 1677331973910, 5345927500226, 16843574682291, 52494817082952, 161923200857711

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A248882, A028377, A258341, A258343, A258344, A258345, A258346, A258350.

nmax=30; CoefficientList[Series[Product[(1+x^k)^(k*(k+1)*(k+2)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 3^(1/5) * Zeta(5)^(1/10) / (2^(91/120) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401*Pi^16 / (1749600000000 * Zeta(5)^3) + 49*Pi^8 * Zeta(3) / (2700000 * Zeta(5)^2) - Zeta(3)^2 / (25*Zeta(5)) + (343*Pi^12/(405000000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) - 7*Pi^4 * Zeta(3) / (750 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (180000 * 2^(3/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + 3^(1/5) * Zeta(3) / (2^(3/5) * (5*Zeta(5))^(2/5))) * n^(2/5) + 7*Pi^4 / (180 * 2^(2/5) * 3^(1/5) * (5*Zeta(5))^(3/5)) * n^(3/5) + 5*3^(2/5) * (5*Zeta(5)/2)^(1/5)/4 * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.

A258345 Expansion of Product_{k>=1} (1+x^k)^(k(k-1)(k-2)).

Original entry on oeis.org

1, 0, 0, 6, 24, 60, 135, 354, 972, 2684, 6990, 17802, 44627, 111582, 277329, 684164, 1671984, 4050096, 9735209, 23238480, 55120950, 129940442, 304502583, 709464798, 1643920584, 3789158988, 8690016942, 19833550266, 45056952957, 101900481462, 229462378987

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A248882, A028377, A258341, A258342, A258343, A258344, A258346, A258351.

nmax=40; CoefficientList[Series[Product[(1+x^k)^(k*(k-1)*(k-2)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 3^(1/5) * Zeta(5)^(1/10) / (2^(91/120) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (1749600000000*Zeta(5)^3) + 49 * Pi^8 * Zeta(3) / (2700000 * Zeta(5)^2) - Zeta(3)^2 / (25*Zeta(5)) + (-343 * Pi^12 / (405000000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (750 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (180000 * 2^(3/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + 3^(1/5) * Zeta(3) / (2^(3/5) * (5*Zeta(5))^(2/5))) * n^(2/5) - 7*Pi^4 / (180 * 2^(2/5) * 3^(1/5) * (5*Zeta(5))^(3/5)) * n^(3/5) + 5*3^(2/5) * ((5*Zeta(5))/2)^(1/5)/4 * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.

A258346 Expansion of Product_{k>=1} (1+x^k)^(k(k-1)(k-2)/6).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 20, 39, 72, 144, 280, 567, 1112, 2187, 4204, 8073, 15309, 28986, 54548, 102286, 190881, 354717, 656194, 1208712, 2217624, 4052633, 7379630, 13390098, 24215587, 43649482, 78435884, 140513905, 250988186, 447037367, 794031641, 1406585604

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A248882, A028377, A258341, A258342, A258343, A258344, A258345, A258352.

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k*(k-1)*(k-2)/6),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ (3*Zeta(5))^(1/10) / (2^(523/720) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (10497600000000 * Zeta(5)^3) + 49*Pi^8 * Zeta(3) / (16200000 * Zeta(5)^2) - Zeta(3)^2 / (150*Zeta(5)) + (-343*Pi^12 / (2430000000 * 2^(3/5) * 15^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (4500 * 2^(3/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (1080000 * 2^(1/5) * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(6/5) * (15*Zeta(5))^(2/5))) * n^(2/5) - 7*Pi^4 / (180 * 2^(4/5) * (15*Zeta(5))^(3/5)) * n^(3/5) + 5*(15*Zeta(5))^(1/5) / 2^(12/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.

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

A305206 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)x^k/(k(1 - x^k)^n)).

Original entry on oeis.org

1, 1, 2, 9, 36, 190, 1070, 6797, 46942, 350901, 2806187, 23894662, 215598410, 2053090936, 20557071012, 215697357449, 2364810631734, 27023086395647, 321160376470277, 3962047673946906, 50648323260067319, 669819485900273336, 9150740338219903590, 128965789655207156299

Offset: 0

Ilya Gutkovskiy, May 27 2018

nonn

N. J. A. Sloane, Transforms

Cf. A026007, A028377, A258343, A293554, A305205.

Table[SeriesCoefficient[Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[(1 + x^k)^Binomial[n + k - 2, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 23}]

a(n) = [x^n] Product_{k>=1} (1 + x^k)^binomial(n+k-2,n-1).

A281156 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(2*k+1)/6).

Original entry on oeis.org

1, 1, 5, 19, 54, 165, 467, 1317, 3599, 9687, 25519, 66203, 169254, 426750, 1062950, 2616818, 6373911, 15369774, 36716706, 86939235, 204152395, 475631501, 1099874363, 2525418842, 5759549109, 13050991205, 29391523405, 65801951770, 146486952644, 324340095729, 714389015139

Offset: 0

Ilya Gutkovskiy, Jan 16 2017

nonn

Weigh transform of square pyramidal numbers (A000330).

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, Square Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000330, A027998, A258343.

nmax = 30; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1) (2 k + 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]

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

a(n) ~ exp(5*(15*Zeta(5))^(1/5) * n^(4/5) / 2^(11/5) + 7*Pi^4 * n^(3/5) / (360*2^(2/5) * (15*Zeta(5))^(3/5)) + (Zeta(3) / (2^(13/5) * (15*Zeta(5))^(2/5)) - 49*Pi^8 / (2160000 * 2^(3/5) * 15^(2/5) * Zeta(5)^(7/5)))*n^(2/5) + (343*Pi^12 / (9720000000 * 2^(4/5) * 15^(1/5) * Zeta(5)^(11/5)) - 7*Pi^4 * Zeta(3) / (18000 * 2^(4/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + 49*Pi^8 * Zeta(3) / (129600000 * Zeta(5)^2) - 2401 * Pi^16 / (83980800000000 * Zeta(5)^3) - Zeta(3)^2 / (1200*Zeta(5))) * (3*Zeta(5))^(1/10) / (2^(11/18) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - Vaclav Kotesovec, Nov 09 2017

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

Original entry on oeis.org

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

cp's OEIS Frontend

A028377 Expansion of Product_{m>0} (1+q^m)^(m(m+1)/2).

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A258341 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)).

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A258344 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)).

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A258342 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)*(k+2)).

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A258345 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)*(k-2)).

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A258346 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)*(k-2)/6).

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

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A305206 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^n)).

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A281156 Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(2*k+1)/6).

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A258342 Expansion of Product_{k>=1} (1+x^k)^(k(k+1)(k+2)).

A258345 Expansion of Product_{k>=1} (1+x^k)^(k(k-1)(k-2)).

A258346 Expansion of Product_{k>=1} (1+x^k)^(k(k-1)(k-2)/6).

A305206 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)x^k/(k(1 - x^k)^n)).

A281156 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(2*k+1)/6).

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