A027998 -id:A027998 - cp's OEIS frontend

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

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

A281790 Expansion of Product_{k>=1} (1+x^(k^2))^k.

Original entry on oeis.org

1, 1, 0, 0, 2, 2, 0, 0, 1, 4, 3, 0, 0, 6, 6, 0, 4, 7, 6, 3, 8, 8, 6, 6, 4, 21, 20, 4, 1, 34, 34, 2, 8, 23, 44, 28, 19, 18, 54, 54, 18, 56, 65, 46, 25, 100, 94, 38, 42, 85, 169, 107, 56, 69, 226, 194, 62, 111, 194, 241, 125, 215, 246, 258, 207, 283, 437, 292

Offset: 0

Vaclav Kotesovec, Apr 14 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

Cf. A000219, A026007, A027998, A033461, A255528, A266891, A284896, A285047.

nmax = 100; CoefficientList[Series[Product[(1+x^(k^2))^k, {k,1,nmax}], {x,0,nmax}], x]
nmax = 100; s = 1 + x; Do[s*=Sum[Binomial[k, j]*x^(j*k^2), {j, 0, Floor[nmax/k^2] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; CoefficientList[s, x]

a(n) ~ exp(sqrt(n/6)*Pi) / (2^(11/6) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 15 2017

A291649 Expansion of Product_{k>=1} (1 + x^(k^2))^(k^2).

Original entry on oeis.org

1, 1, 0, 0, 4, 4, 0, 0, 6, 15, 9, 0, 4, 40, 36, 0, 17, 71, 90, 36, 64, 100, 180, 144, 96, 274, 394, 300, 148, 740, 820, 480, 472, 1150, 1851, 1341, 1146, 1318, 3880, 3540, 1704, 3017, 6455, 7134, 3780, 7822, 9574, 12180, 10304, 12057, 19750, 22485, 20558, 15910, 43076, 43236, 31104, 33742, 66895

Offset: 0

Ilya Gutkovskiy, Aug 28 2017

nonn

Number of partitions of n into distinct squares, where k^2 different parts of size k^2 are available (1a, 4a, 4b, 4c, 4d, ...).

			a(8) = 6 because we have [4a, 4b], [4a, 4c], [4a, 4d], [4b, 4c], [4b, 4d] and [4c, 4d].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A000290, A026007, A027998, A033461, A262736, A281790, A284896, A291655, A291692.

nmax = 100; CoefficientList[Series[Product[(1 + x^k^2)^k^2, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; s = 1 + x; Do[s *= Sum[Binomial[k^2, j]*x^(j*k^2), {j, 0, Floor[nmax/k^2] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Aug 28 2017 *)

G.f.: Product_{k>=1} (1 + x^A000290(k))^A000290(k).

a(n) ~ exp(5 * 2^(-9/5) * 3^(-3/5) * (9-4*sqrt(2))^(1/5) * Pi^(1/5) * Zeta(5/2)^(2/5) * n^(3/5)) * 3^(1/5) * (2*sqrt(2)-1)^(1/5) * Zeta(5/2)^(1/5) / (2^(9/10) * sqrt(5) * Pi^(2/5) * n^(7/10)). - Vaclav Kotesovec, Aug 29 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

A261052 Expansion of Product_{k>=1} (1+x^k)^(k!).

Original entry on oeis.org

1, 1, 2, 8, 31, 157, 915, 6213, 48240, 423398, 4147775, 44882107, 531564195, 6837784087, 94909482330, 1413561537884, 22482554909451, 380269771734265, 6815003300096013, 128992737080703803, 2571218642722865352, 53835084737513866662, 1181222084520177393143

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Weigh transform of the factorial numbers. - Alois P. Heinz, Jun 11 2018

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000142, A107895, A168246, A261053, A026007, A027998, A248882, A102866, A256142.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(i!, j)*b(n-i*j,i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 08 2015

nmax=25; CoefficientList[Series[Product[(1+x^k)^(k!),{k,1,nmax}],{x,0,nmax}],x]

seq(n)={Vec(exp(x*Ser(dirmul(vector(n, n, n!), -vector(n, n, (-1)^n/n)))))} \\ Andrew Howroyd, Jun 22 2018

a(n) ~ n! * (1 + 1/n + 2/n^2 + 10/n^3 + 57/n^4 + 401/n^5 + 3382/n^6 + 33183/n^7 + 371600/n^8 + 4685547/n^9 + 65792453/n^10).

A294749 Expansion of Product_{k>=1} (1 + x^(2*k - 1))^(k^2).

Original entry on oeis.org

1, 1, 0, 4, 4, 9, 15, 22, 52, 65, 129, 190, 335, 534, 814, 1399, 2074, 3462, 5135, 8303, 12658, 19562, 30182, 45542, 70620, 105034, 161223, 239532, 362929, 539252, 805320, 1197589, 1769483, 2624604, 3847755, 5681787, 8291848, 12165978, 17696362, 25796820

Offset: 0

Vaclav Kotesovec, Nov 08 2017

nonn

In general, if g.f. = Product_{k>=1} (1 + x^(2*k-1))^(c2*k^2 + c1*k + c0) and c2>0, then a(n) ~ exp(Pi*sqrt(2)/3 * (7*c2/15)^(1/4) * n^(3/4) + 3*(c1+c2) * Zeta(3) / (2*Pi^2) * sqrt(15*n/(7*c2)) + (Pi*(4*c0 + 2*c1 + c2) * (15/(7*c2))^(1/4) / (24*sqrt(2)) - 9*(c1+c2)^2 * Zeta(3)^2 * (15/(7*c2))^(5/4) / (2^(3/2) * Pi^5)) * n^(1/4) + 2025*(c1+c2)^3 * Zeta(3)^3 / (49 * c2^2 * Pi^8) - 15*(c1+c2) * (4*c0 + 2*c1 + c2) * Zeta(3) / (112 * c2 * Pi^2)) * (7/15)^(1/8) * 2^((c1+c2)/24 - 9/4) * c2^(1/8) / n^(5/8).

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A027998, A263140, A294750, A294755.

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

a(n) ~ exp(Pi/3 * (7/15)^(1/4) * sqrt(2) * n^(3/4) + 3*Zeta(3) * sqrt(15*n/7) / (2*Pi^2) + (Pi * (15/7)^(1/4) / (24*sqrt(2)) - 9*Zeta(3)^2 * (15/7)^(5/4) / (2^(3/2) * Pi^5)) * n^(1/4) + 2025*Zeta(3)^3 / (49*Pi^8) - 15*Zeta(3) / (112*Pi^2)) * (7/15)^(1/8) / (2^(53/24) * n^(5/8)).

A307462 Expansion of Product_{k>=1} (1+x^k)^((-1)^k*k^2).

Original entry on oeis.org

1, -1, 5, -14, 36, -97, 246, -593, 1423, -3351, 7699, -17432, 38901, -85545, 185862, -399220, 848080, -1783682, 3716584, -7675916, 15722127, -31951330, 64452707, -129102947, 256876062, -507854808, 997954125, -1949631802, 3787674152, -7319306458, 14071371173

Offset: 0

Seiichi Manyama, Apr 09 2019

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = (-1)^(n+1) * n^2, g(n) = -1.

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

Product_{k>=1} (1+x^k)^((-1)^k*k^b): A083365 (b=0), A284474 (b=1), this sequence (b=2).

Cf. A027998, A177155, A266964, A294749, A294750, A307460.

nmax = 40; CoefficientList[Series[Product[(1 + x^k)^((-1)^k*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 09 2019 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k))^(4*k^2) / (1 + x^(2*k - 1))^((2*k - 1)^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 09 2019 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+x^k)^((-1)^k*k^2)))

a(n) ~ (-1)^n * exp(2*Pi*n^(3/4)/3 + 3*Zeta(3)/(4*Pi^2)) / (4*n^(5/8)). - Vaclav Kotesovec, Apr 09 2019

A261051 Expansion of Product_{k>=1} (1+x^k)^(Lucas(k)).

Original entry on oeis.org

1, 1, 3, 7, 14, 33, 69, 148, 307, 642, 1314, 2684, 5432, 10924, 21841, 43431, 85913, 169170, 331675, 647601, 1259737, 2441706, 4716874, 9083215, 17439308, 33387589, 63749174, 121409236, 230658963, 437198116, 826838637, 1560410267, 2938808875, 5524005110

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015

Cf. A000032, A261031, A261050, A026007, A027998, A248882, A102866, A256142.

L:= n-> (<<0|1>, <1|1>>^n. <<2, 1>>)[1, 1]:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(binomial(L(i), 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 08 2015

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

a(n) ~ phi^n / (2*sqrt(Pi)*n^(3/4)) * exp(-1 + 1/(2*sqrt(5)) + 2*sqrt(n) + s), where s = Sum_{k>=2} (-1)^(k+1) * (2 + phi^k)/((phi^(2*k) - phi^k - 1)*k) = -0.590290697526802161885355317939144642488927381134222996704542... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 2*x^k)/(k*(1 - x^k - x^(2*k)))). - Ilya Gutkovskiy, May 30 2018

A304445 Coefficient of x^n in Product_{k>=1} (1+x^k)^(n^2).

Original entry on oeis.org

1, 1, 10, 174, 4132, 126905, 4802046, 216313349, 11313533064, 674172155790, 45102830448510, 3347925678474168, 273085613904116184, 24282144087974698445, 2337736453663291696624, 242272777074973285192935, 26892702305505020726198800, 3183326728470139531614913088

Offset: 0

Vaclav Kotesovec, May 12 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..330

Cf. A027998, A270913, A304446.

nmax = 20; Table[SeriesCoefficient[Product[(1+x^k)^(n^2), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 20; Table[SeriesCoefficient[(QPochhammer[-1, x]/2)^(n^2), {x, 0, n}], {n, 0, nmax}]

a(n) ~ exp(n + 1/2) * n^(n - 1/2) / sqrt(2*Pi).

A285240 Expansion of Product_{k>=1} (1 + k*x^k)^(k^2).

Original entry on oeis.org

1, 1, 8, 35, 115, 429, 1425, 4803, 15398, 48940, 151046, 459000, 1373219, 4037721, 11723911, 33566828, 94993571, 265722551, 735543433, 2015558930, 5471271099, 14719853084, 39266487114, 103908002173, 272855152096, 711272144097, 1841162650896, 4734074846631

Offset: 0

Vaclav Kotesovec, Apr 15 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A022629, A027998, A266891, A285241, A285242.

nmax = 40; CoefficientList[Series[Product[(1 + k*x^k)^(k^2), {k,1,nmax}], {x,0,nmax}], x]
nmax = 40; s = 1 + x; Do[s*=Sum[Binomial[k^2, j]*k^j*x^(j*k), {j, 0, Floor[nmax/k] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; CoefficientList[s, x]

cp's OEIS Frontend

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

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A281790 Expansion of Product_{k>=1} (1+x^(k^2))^k.

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A291649 Expansion of Product_{k>=1} (1 + x^(k^2))^(k^2).

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

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A261052 Expansion of Product_{k>=1} (1+x^k)^(k!).

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A294749 Expansion of Product_{k>=1} (1 + x^(2*k - 1))^(k^2).

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A307462 Expansion of Product_{k>=1} (1+x^k)^((-1)^k*k^2).

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A261051 Expansion of Product_{k>=1} (1+x^k)^(Lucas(k)).

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

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