A077335 -id:A077335 - cp's OEIS frontend

A265836 Expansion of Product_{k>=1} 1/(1 - k(k+1)x^k).

1, 2, 10, 32, 120, 342, 1206, 3320, 10604, 29578, 88342, 239400, 702020, 1863654, 5262650, 13948824, 38427192, 100244162, 272822282, 703972024, 1883948848, 4839944150, 12779850278, 32548367784, 85335644100, 215826029018, 560407835934, 1412632075328

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

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

Cf. A074141, A077335, A092485, A305204.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      2^n, b(n, i-1)+(1+i)*i*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..33);  # Alois P. Heinz, Aug 16 2019

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

a(n) ~ c * 6^(n/2), where
c = 79.0418032646837469192452349...... if n is even,
c = 78.4480460169710091436913691...... if n is odd.

A292165 Expansion of Product_{k>=1} 1/(1 + k^2*x^k).

Original entry on oeis.org

1, -1, -3, -6, 6, 5, 40, 11, 226, -516, -186, -844, 3731, -3734, 814, -33819, 85660, -46022, 210342, -411678, 593996, -2980156, 2076721, -3445584, 40785410, -37503158, 98085, -271846888, 336918770, -295108832, 2178341296, -2404059340, 6127604258

Offset: 0

Seiichi Manyama, Sep 10 2017

sign

Alois P. Heinz, Table of n, a(n) for n = 0..3145

Cf. A077335, A092484.

b:= proc(n, i) option remember; (m->
      `if`(mn, 0, i^2*b(n-i, i-1)))))(i*(i+1)/2)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i$2)*a(i$2), i=0..n-1))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 10 2017

b[n_, i_] := b[n, i] = Function[m,
     If[m < n, 0, If[n == m, i!^2, b[n, i - 1] +
     If[i > n, 0, i^2*b[n - i, i - 1]]]]][i*(i + 1)/2];
a[n_] := a[n] = If[n == 0, 1, -Sum[b[n - i, n - i]*a[i], {i, 0, n - 1}]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 21 2022, after Alois P. Heinz *)

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

Convolution inverse of A092484.
From Vaclav Kotesovec, Sep 10 2017: (Start)
a(n) ~ (-1)^n * c * 3^(2*n/3), where
c = 0.717271758899891528435966115495396784611147877234945... if mod(n,3)=0
c = 0.387695187106751505296020614217498222070185848125472... if mod(n,3)=1
c = 0.241939482775588594057384356004734639024152664456553... if mod(n,3)=2
(End)
G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018

A292417 a(n) = [x^n] Product_{k>=1} 1/(1 - n^2*x^k).

Original entry on oeis.org

1, 1, 20, 819, 70160, 10188775, 2240751636, 692647082799, 286013768613952, 151994274055319070, 101020305070908050100, 82086758986568812837856, 80056656965795630400382608, 92282612223268812357487227077, 124113156850218393012451734737460

Offset: 0

Vaclav Kotesovec, Sep 16 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..214

Cf. A077335, A124577, A292304.

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

{a(n)= polcoef(prod(k=1, n, 1/(1-n^2*x^k +x*O(x^n))), n)};
for(n=0,20, print1(a(n), ", ")) \\ G. C. Greubel, Feb 02 2019

a(n) ~ n^(2*n) * (1 + 1/n^2 + 2/n^4 + 3/n^6 + 5/n^8 + 7/n^10), for coefficients see A000041.

A294586 Expansion of Product_{k>=1} 1/(1 - k^2*x^k)^(k^2).

Original entry on oeis.org

1, 1, 17, 98, 514, 2435, 12752, 58849, 277362, 1243056, 5523734, 23889860, 102176581, 427458488, 1768064752, 7197695011, 28955246228, 114977761216, 451686925462, 1754581791860, 6749143188662, 25707194720502, 97041994691555, 363121143230292

Offset: 0

Seiichi Manyama, Nov 03 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..2889

Column k=2 of A294585.

Cf. A077335, A285241, A285674, A294584.

nmax = 30; CoefficientList[Series[Product[1/(1 - k^2*x^k)^(k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 11 2017 *)
nmax = 30; s = 1 - x; Do[s *= Sum[Binomial[k^2, j]*(-1)^j*k^(2*j)*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 12 2017 *)

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

From Vaclav Kotesovec, Nov 14 2017: (Start)
a(n) ~ c * 3^(2*n/3) * n^8, where
if mod(n,3)=0 then c = 350793443467906700358779160929996923840677857044\
13786172.61998576944425459411592809123023259309183199454386580509531344\
26216683391121761062030679551011342614958936988089343473390138...
if mod(n,3)=1 then c = 350793443467906700358779160929996923840677857044\
13786172.61998576943431618172412821798685989333734080090574886961583670\
65437558779530384541992249698997443314123905740649930258416583...
if mod(n,3)=2 then c = 350793443467906700358779160929996923840677857044\
13786172.61998576943586440772541471067224229278174424709431922476448338\
37991534958575385658058309282842532811502400165735702386411333...
In closed form, a(n) ~ ((Product_{k>=4} ((1 - k^2 / 3^(2*k/3))^(-k^2))) / ((1 - 1/3^(2/3)) * (1 - 4/3^(4/3))^4) + (Product_{k>=4} ((1 - (-1)^(2*k/3) * k^2 / 3^(2*k/3))^(-k^2))) / ((-1)^(2*n/3) * (1 + 4/3 * (-1/3)^(1/3))^4 * (1 - (-1/3)^(2/3))) + (Product_{k>=4} ((1 - (-(-1)^(1/3))^k * k^2 / 3^(2*k/3))^(-k^2))) / ((-(-1)^(1/3))^n * (1 + (-1)^(1/3) / 3^(2/3)) * (1 - 4*(-1)^(2/3) / 3^(4/3))^4)) * 3^(2*n/3) * n^8 / 793618560. - Vaclav Kotesovec, Nov 14 2017 (End)

A266137 Expansion of Product_{k>=1} 1/(1 - k(x^(2k-1))).

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 10, 14, 20, 33, 50, 68, 106, 147, 214, 325, 445, 624, 916, 1259, 1780, 2553, 3477, 4821, 6794, 9340, 12777, 17808, 24266, 32998, 45764, 61770, 83593, 114594, 154039, 208617, 283232, 379040, 509270, 687448, 919709, 1228319, 1650595, 2195745

Offset: 0

Vaclav Kotesovec, Dec 21 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..9850

Cf. A000009, A006906, A077335, A265951, A266138.

nmax=60; CoefficientList[Series[Product[1/(1-k*(x^(2*k-1))), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 2^(n/3), where
c = 2684.3207660224428945778151546260301591494083790... if mod(n,3) = 0
c = 2683.9203893332021512699407898064547843826991184... if mod(n,3) = 1
c = 2683.7635451650373491773203224442103370428384569... if mod(n,3) = 2.

A266138 Expansion of Product_{k>=1} 1/(1 - k(x^(2k+1))).

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 1, 3, 2, 5, 7, 7, 11, 13, 24, 26, 35, 44, 69, 78, 112, 150, 188, 245, 318, 429, 537, 729, 924, 1177, 1534, 1965, 2518, 3287, 4108, 5394, 6857, 8604, 11022, 14073, 17899, 22549, 28900, 36182, 45954, 58395, 72912, 92118, 116201, 146279

Offset: 0

Vaclav Kotesovec, Dec 21 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A006906, A077335, A087897, A265951, A266137.

nmax=80; CoefficientList[Series[Product[1/(1-k*(x^(2*k+1))), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^(n/7), where
c = 617630.638335... if mod(n,7) = 0
c = 617630.321433... if mod(n,7) = 1
c = 617630.360795... if mod(n,7) = 2
c = 617630.429073... if mod(n,7) = 3
c = 617630.357078... if mod(n,7) = 4
c = 617630.421636... if mod(n,7) = 5
c = 617630.341606... if mod(n,7) = 6.

A305204 Expansion of Product_{k>=1} 1/(1 - (k(k + 1)/2)x^k).

Original entry on oeis.org

1, 1, 4, 10, 29, 62, 176, 363, 931, 2029, 4751, 10062, 23749, 48959, 109342, 230981, 500344, 1031667, 2223218, 4531585, 9570395, 19523510, 40411313, 81628389, 168484616, 336850254, 685112670, 1369559157, 2757908932, 5464925114, 10958578421, 21574592680

Offset: 0

Ilya Gutkovskiy, May 27 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..3841

Cf. A000217, A000294, A006906, A077335, A265836.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      1, b(n, i-1)+(1+i)*i/2*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..33);  # Alois P. Heinz, Aug 16 2019

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

G.f.: Product_{k>=1} 1/(1 - A000217(k)*x^k).

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (j*(j + 1))^k*x^(j*k)/(k*2^k)).

A356560 Expansion of e.g.f. Product_{k>0} 1/(1 - k^2 * x^k)^(1/k^2).

Original entry on oeis.org

1, 1, 4, 18, 156, 1020, 16560, 143640, 2898000, 43016400, 926856000, 13749674400, 524416939200, 8626888670400, 284030505158400, 7950850859952000, 284397434953632000, 6752059834744224000, 357295791069689472000, 9098085523917918528000

Offset: 0

Seiichi Manyama, Aug 12 2022

nonn

Cf. A294462, A294469, A356530, A356561.

Cf. A077335, A308688.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-k^2*x^k)^(1/k^2))))

a308688(n) = sumdiv(n, d, d^(2*n/d-1));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, a308688(j)*v[i-j+1]/(i-j)!)); v;

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A308688(k) * a(n-k)/(n-k)!.

A369887 Sum of products of squares of parts , counted without multiplicity, in all partitions of n.

Original entry on oeis.org

1, 1, 5, 14, 34, 95, 208, 537, 1090, 2812, 5566, 12480, 26199, 53486, 112866, 229111, 450800, 885030, 1778190, 3319846, 6624376, 12354288, 23674929, 43485580, 81441398, 149864634, 273431081, 503205344, 906757150, 1630802024, 2920280596, 5166820832

Offset: 0

Seiichi Manyama, Feb 04 2024

nonn

			The partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1. So a(4) = 16 + 9 + 4 + 4 + 1 = 34.

Cf. A162506, A369888.

Cf. A077335.

my(N=40, x='x+O('x^N)); Vec(prod(k=1, N, 1+k^2*x^k/(1-x^k)))

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

cp's OEIS Frontend

A265836 Expansion of Product_{k>=1} 1/(1 - k*(k+1)*x^k).

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A292165 Expansion of Product_{k>=1} 1/(1 + k^2*x^k).

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A292417 a(n) = [x^n] Product_{k>=1} 1/(1 - n^2*x^k).

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A294586 Expansion of Product_{k>=1} 1/(1 - k^2*x^k)^(k^2).

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A266137 Expansion of Product_{k>=1} 1/(1 - k*(x^(2*k-1))).

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A266138 Expansion of Product_{k>=1} 1/(1 - k*(x^(2*k+1))).

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A305204 Expansion of Product_{k>=1} 1/(1 - (k*(k + 1)/2)*x^k).

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A356560 Expansion of e.g.f. Product_{k>0} 1/(1 - k^2 * x^k)^(1/k^2).

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A369887 Sum of products of squares of parts , counted without multiplicity, in all partitions of n.

A265836 Expansion of Product_{k>=1} 1/(1 - k(k+1)x^k).

A266137 Expansion of Product_{k>=1} 1/(1 - k(x^(2k-1))).

A266138 Expansion of Product_{k>=1} 1/(1 - k(x^(2k+1))).

A305204 Expansion of Product_{k>=1} 1/(1 - (k(k + 1)/2)x^k).