A006906 -id:A006906 - cp's OEIS frontend

A022740 Expansion of Product (1-m*q^m)^-16; m=1..inf.

1, 16, 168, 1376, 9604, 59488, 335904, 1758816, 8646986, 40281296, 179065184, 763837600, 3140732344, 12494160288, 48236274976, 181203877248, 663837626163, 2376282980272, 8325497904672

Offset: 0

N. J. A. Sloane

nonn

Robert Israel, Table of n, a(n) for n = 0..3000

Cf. A006906, A022611, A022644, A023014.

Column k=16 of A297328.

N:= 30: # for a(0)..a(N)
P:= mul(1-m*q^m,m=1..N):
S:= series(P,q,N+1):
S16:= series(S^(-16),q,N+1):
seq(coeff(S16,q,i),i=0..N); # Robert Israel, Dec 22 2019

nmax = 20; CoefficientList[Series[Product[1/(1 - k*x^k)^16, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 23 2019 *)

A163318 Expansion of g.f.: Product_{k>=1} 1+k*x^k/(1-x^k)^2.

Original entry on oeis.org

1, 1, 4, 8, 19, 36, 76, 142, 272, 496, 900, 1592, 2784, 4792, 8138, 13688, 22703, 37380, 60838, 98310, 157298, 250162, 394332, 618032, 961512, 1487563, 2286610, 3496776, 5316666, 8044598, 12110538, 18147166, 27068692, 40203306, 59459998, 87587428, 128522850

Offset: 0

Vladeta Jovovic, Jul 24 2009

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A006906, A077285, A162506.

b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-1)*(j*i), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 25 2013

terms = 40;
CoefficientList[Product[1 + k x^k/(1 - x^k)^2, {k, 1, terms}] + O[x]^terms, x] (* Jean-François Alcover, Nov 12 2020 *)

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.

A294589 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^(j^k).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 9, 14, 14, 1, 1, 17, 36, 42, 25, 1, 1, 33, 98, 140, 103, 56, 1, 1, 65, 276, 498, 481, 289, 97, 1, 1, 129, 794, 1844, 2419, 1774, 690, 198, 1, 1, 257, 2316, 7002, 12745, 12173, 5925, 1771, 354, 1, 1, 513, 6818, 27020, 69283, 89706, 56974, 20076, 4206, 672, 1

Offset: 0

Seiichi Manyama, Nov 03 2017

			Square array begins:
    1,  1,   1,   1,    1, ...
    1,  1,   1,   1,    1, ...
    3,  5,   9,  17,   33, ...
    6, 14,  36,  98,  276, ...
   14, 42, 140, 498, 1844, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A006906, A266941, A285241, A294590.
Rows n=0-1 give A000012.
Cf. A144048, A294587.

A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(k+1+j/d)) * A(n-j,k) for n > 0.

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)).

A317912 Expansion of Product_{k>=2} 1/(1 - k*x^k).

Original entry on oeis.org

1, 0, 2, 3, 8, 11, 31, 41, 101, 156, 318, 498, 1037, 1555, 3024, 4889, 8849, 14112, 25622, 40322, 71314, 113926, 194677, 310819, 530030, 835484, 1400523, 2226307, 3668998, 5797558, 9521310, 14942262, 24298136, 38187102, 61384028, 96161997, 154078991, 239891926, 381723396

Offset: 0

Ilya Gutkovskiy, Aug 10 2018

nonn

First differences of A006906.

Sum of products of terms in all partitions of n into parts >= 2.

			a(6) = 31 because we have [6], [4, 2], [3, 3], [2, 2, 2] and 6 + 4*2 + 3*3 + 2*2*2 = 31.

Seiichi Manyama, Table of n, a(n) for n = 0..5000
Index entries for sequences related to partitions

Cf. A002865, A006906, A191659, A302830.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1)+ i*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2) -`if`(n=0, 0, b(n-1$2)):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 10 2018

nmax = 38; CoefficientList[Series[Product[1/(1 - k x^k), {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 38; CoefficientList[Series[Exp[Sum[Sum[k^j x^(j k)/j, {k, 2, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]
b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d^(k/d + 1), {d, Divisors[k]}] b[n - k], {k, 1, n}]/n]; Differences[Table[b[n], {n, -1, 38}]]

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

A318481 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - ijk*x^(ijk)).

Original entry on oeis.org

1, 1, 7, 16, 64, 133, 465, 1008, 3023, 6695, 18206, 40175, 103229, 225470, 549873, 1194620, 2801742, 6015042, 13686306, 29063919, 64424496, 135362432, 293512852, 610061141, 1298516539, 2670738781, 5591712472, 11388116508, 23499720744, 47403692965, 96564236754

Offset: 0

Vaclav Kotesovec, Aug 27 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A006906, A318415.

Cf. A000041, A280540, A318413, A318482.

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

A318483 Expansion of Product_{k>=1} 1/(1 - k*x^k)^sigma(k), where sigma = A000203.

Original entry on oeis.org

1, 1, 7, 19, 71, 173, 583, 1443, 4255, 10648, 28929, 71159, 184740, 445626, 1110122, 2638328, 6369490, 14870194, 35031627, 80465028, 185556696, 419916149, 950785580, 2121471778, 4727971847, 10412230698, 22876886529, 49776871862, 107974178843, 232302695301

Offset: 0

Vaclav Kotesovec, Aug 27 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5920

Cf. A000203, A006906, A266941, A318415, A006171, A061256, A318484.

nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^DivisorSigma[1, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[1, k], j]*(-1)^j*k^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]

a(n) ~ c * n^3 * 3^(n/3), where
c = 280631952508395331283883354935233682635.581151020... if mod(n,3)=0
c = 280631952508395331283883354935233682635.059082354... if mod(n,3)=1
c = 280631952508395331283883354935233682635.088610121... if mod(n,3)=2
In closed form, c = (Product_{k>=4}((1 - k/3^(k/3))^(-sigma(k)))/(18*(57 - 90*3^(1/3) + 35*3^(2/3)))) - Product_{k>=4}((1 + ((-1)^(1 + 2*k/3)*k)/3^(k/3))^(-sigma(k)))/ ((-1)^(2*n/3)*(6*(3 + 2*(-3)^(1/3))^3*(-3 + (-3)^(2/3)))) - ((-1)^(1 - (4*n)/3)*Product_{k>=4}((1 + ((-1)^(1 + 4*k/3)*k)/3^(k/3))^(-sigma(k))))/(486*(1 + (-1/3)^(1/3))* (1 - 2*(-1/3)^(2/3))^3)

A319859 Expansion of Product_{k>0} (1 + (2k-1)x^(2k-1))/(1 - 2kx^(2k)).

Original entry on oeis.org

1, 1, 2, 5, 11, 19, 33, 63, 124, 212, 350, 620, 1107, 1819, 2977, 5076, 8549, 13797, 22199, 36304, 59271, 94406, 148948, 238199, 380653, 595930, 928696, 1460474, 2288948, 3541879, 5460144, 8458886, 13084665, 20046161, 30590724, 46871521, 71711287, 108863135, 164802583

Offset: 0

Seiichi Manyama, Sep 29 2018

nonn

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

Cf. A006906, A067553, A282207, A319860.

seq(coeff(series(mul((1+(2*k-1)*x^(2*k-1))/(1-2*k*x^(2*k)),k=1..n),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Sep 29 2018

nmax = 50; CoefficientList[Series[Product[(1 + (2*k-1)*x^(2*k-1))/(1 - 2*k*x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2018 *)

N=99; x='x+O('x^N); Vec(prod(k=1, N, (1+(2*k-1)*x^(2*k-1))/(1-(2*k)*x^(2*k))))

From Vaclav Kotesovec, Oct 06 2018: (Start)
a(n) ~ c * n * 2^(n/2), where
c = 59.39385182785860961527832575945047265281719... if n is even
c = 59.39502666671757816086328506683601946035153... if n is odd
(End)

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A022740 Expansion of Product (1-m*q^m)^-16; m=1..inf.

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A163318 Expansion of g.f.: Product_{k>=1} 1+k*x^k/(1-x^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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A294589 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^(j^k).

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

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A317912 Expansion of Product_{k>=2} 1/(1 - k*x^k).

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A318481 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - i*j*k*x^(i*j*k)).

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A318483 Expansion of Product_{k>=1} 1/(1 - k*x^k)^sigma(k), where sigma = A000203.

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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).

A318481 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - ijk*x^(ijk)).

A319859 Expansion of Product_{k>0} (1 + (2k-1)x^(2k-1))/(1 - 2kx^(2k)).