A129921 -id:A129921 - cp's OEIS frontend

A304964 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1, l>=1} x^(ijk*l)).

1, 1, 5, 13, 47, 133, 443, 1333, 4263, 13143, 41419, 128791, 403815, 1259639, 3941579, 12310299, 38492034, 120271953, 375964616, 1174935195, 3672413322, 11477465221, 35872928244, 112117013835, 350417746650, 1095202995267, 3422999582632, 10698350241417, 33437065631262, 104505382585023

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Invert transform of A007426.

N. J. A. Sloane, Transforms
Index entries for sequences related to compositions

Cf. A000005, A007426, A011782, A129921, A280486, A280487, A304963.

A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(A(j, 4)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, May 22 2018

nmax = 29; CoefficientList[Series[1/(1 - Sum[x^(i j k l), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}, {l, 1, nmax/i/j/k}]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[Sum[DivisorSigma[0, d] DivisorSigma[0, k/d], {d, Divisors[k]}] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[DivisorSigma[0, d] DivisorSigma[0, k/d], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]

G.f.: 1/(1 - Sum_{k>=1} A007426(k)*x^k).

A327738 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*j^2)).

Original entry on oeis.org

1, 1, 2, 4, 9, 18, 37, 76, 158, 326, 672, 1386, 2862, 5906, 12187, 25148, 51900, 107103, 221023, 456110, 941256, 1942423, 4008481, 8272094, 17070712, 35227975, 72698206, 150023632, 309596255, 638898274, 1318462339, 2720844607, 5614870612, 11587126980

Offset: 0

Ilya Gutkovskiy, Sep 23 2019

nonn

Invert transform of A046951.

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

Cf. A004101, A046951, A129921, A280451.

a:= proc(n) option remember; `if`(n<1, 1, add(a(n-i)*
      nops(select(issqr, numtheory[divisors](i))), i=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 23 2019

nmax = 33; CoefficientList[Series[1/(1 - Sum[x^(k^2)/(1 - x^(k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Length[Select[Divisors[k], IntegerQ[Sqrt[#]] &]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

G.f.: 1 / (1 - Sum_{k>=1} x^(k^2) / (1 - x^(k^2))).
G.f.: 1 / (1 - Sum_{k>=1} (theta_3(x^k) - 1) / 2), where theta_() is the Jacobi theta function.
a(0) = 1; a(n) = Sum_{k=1..n} A046951(k) * a(n-k).

A307241 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1)d(k+1)a(n-k), where d() is the number of divisors (A000005).

Original entry on oeis.org

1, 2, 2, 3, 6, 12, 23, 42, 75, 135, 248, 460, 849, 1554, 2837, 5192, 9527, 17490, 32083, 58809, 107781, 197578, 362280, 664320, 1218069, 2233202, 4094289, 7506602, 13763219, 25234674, 46266927, 84828138, 155528132, 285154061, 522819002, 958568628, 1757496665, 3222295912

Offset: 0

Ilya Gutkovskiy, Mar 30 2019

nonn

Cf. A000005, A002039, A129921, A307242.

a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) DivisorSigma[0, k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 37}]
nmax = 37; CoefficientList[Series[-x/Sum[(-x)^k/(1 - (-x)^k), {k, 1, nmax + 1}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[1/D[Log[Product[(1 - (-x)^k)^(1/k), {k, 1, nmax + 1}]], x], {x, 0, nmax}], x]

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

G.f.: 1 / (d/dx) log(Product_{k>=1} (1 - (-x)^k)^(1/k)).

A321190 a(n) = [x^n] 1/(1 - Sum_{k>=1} k^n*x^k/(1 - x^k)).

Original entry on oeis.org

1, 1, 6, 47, 778, 25476, 1752936, 242632397, 70015221566, 41446777283255, 49999934258165654, 125272856707074638221, 641938223803783115191706, 6731818441446626626586172740, 146378489075644780343627471981694, 6505906463580477520696075719916583118

Offset: 0

Ilya Gutkovskiy, Oct 29 2018

nonn

Cf. A023887, A129921, A180305, A319647, A320649, A321042.

seq(coeff(series((1-add(k^n*x^k/(1-x^k),k=1..n))^(-1),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 29 2018

Table[SeriesCoefficient[1/(1 - Sum[k^n x^k/(1 - x^k), {k, 1, n}]), {x, 0, n}], {n, 0, 15}]
Table[SeriesCoefficient[1/(1 - Sum[DivisorSigma[n, k] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 15}]
Table[SeriesCoefficient[1/(1 - Sum[Sum[j^n x^(i j), {j, 1, n}], {i, 1, n}]), {x, 0, n}], {n, 0, 15}]

a(n) = [x^n] 1/(1 - Sum_{k>=1} sigma_n(k)*x^k).
a(n) = [x^n] 1/(1 - Sum_{i>=1, j>=1} j^n*x^(i*j)).
a(n) = [x^n] 1/(1 + x * (d/dx) log(Product_{k>=1} (1 - x^k)^(k^(n-1)))).

A327798 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*(j + 1))).

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 9, 10, 25, 34, 72, 106, 215, 330, 635, 1025, 1899, 3141, 5713, 9602, 17213, 29292, 51982, 89149, 157249, 271027, 476037, 823386, 1442063, 2500015, 4370386, 7588146, 13248591, 23026728, 40169991, 69865026, 121811765, 211954826, 369412910

Offset: 0

Ilya Gutkovskiy, Sep 25 2019

nonn

Invert transform of A032741.

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

Cf. A032741, A129921, A318783, A327739.

N:= 100: # for a(0)..a(N)
G:= 1/(1-add(x^(2*k)/(1-x^k),k=1..(N+1)/2)):
S:= series(G,x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Jan 10 2023

nmax = 38; CoefficientList[Series[1/(1 - Sum[x^(2 k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(DivisorSigma[0, k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 38}]

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

a(0) = 1; a(n) = Sum_{k=1..n} A032741(k) * a(n-k).

A352839 Expansion of g.f. 1/(1 - Sum_{k>=1} sigma_k(k) * x^k).

Original entry on oeis.org

1, 1, 6, 39, 370, 4132, 59288, 990705, 19577018, 439550259, 11142216938, 313147651821, 9680830606850, 325944181383936, 11875777329091878, 465292113335910106, 19507503314546762246, 871248546067010133794, 41295079536653463057146

Offset: 0

Seiichi Manyama, Apr 05 2022

nonn

Cf. A129921, A180305, A320649.

Cf. A023887, A352841, A352842.

my(N=20, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, sigma(k, k)*x^k)))

a(n) = if(n==0, 1, sum(k=1, n, sigma(k, k)*a(n-k)));

a(0) = 1; a(n) = Sum_{k=1..n} sigma_k(k) * a(n-k).

A353005 Decimal expansion of the root of the equation Sum_{k>0} x^k/(1-x^k) = 1.

Original entry on oeis.org

4, 0, 6, 1, 4, 8, 0, 0, 5, 0, 0, 1, 2, 4, 7, 2, 2, 8, 8, 6, 8, 9, 5, 8, 6, 0, 3, 0, 5, 9, 0, 4, 1, 9, 4, 5, 5, 6, 2, 9, 4, 0, 1, 9, 3, 9, 3, 6, 8, 7, 2, 4, 3, 2, 0, 6, 7, 0, 5, 4, 4, 9, 3, 6, 4, 7, 6, 6, 4, 1, 6, 6, 7, 7, 4, 7, 5, 2, 7, 9, 1, 1, 8, 5, 6, 7, 8, 7, 3, 6, 0, 9, 3, 5, 9, 6, 5, 7, 3, 1, 9, 0, 9, 1, 2, 0

Offset: 0

Vaclav Kotesovec, Apr 15 2022

			0.40614800500124722886895860305904194556294019393687243206705449364766416677475...

Sylvie Corteel and Paweł Hitczenko, Generalizations of Carlitz Compositions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.8., p. 7.

Cf. A129921.

RealDigits[x/.FindRoot[QPolyGamma[0, 1, x]==Log[x/(1-x)], {x, 1/2}, WorkingPrecision->110]][[1]]

Root of the equation Sum_{k>0} A000005(k)*x^k = 1.

Equals lim_{n->infinity} 1/A129921(n)^(1/n).

A305049 Expansion of 1/(1 - Sum_{k>=1} tau_k(k)*x^k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

Original entry on oeis.org

1, 1, 3, 8, 27, 67, 216, 569, 1747, 4812, 14041, 39483, 115408, 326385, 941735, 2684170, 7725097, 22063737, 63354066, 181223899, 519883185, 1488316952, 4266788191, 12219763777, 35023995792, 100326757107, 287503501905, 823654031283, 2360146144917, 6761847714698, 19374935267810

Offset: 0

Ilya Gutkovskiy, May 24 2018

nonn

Invert transform of A163767.

N. J. A. Sloane, Transforms
Index entries for sequences related to compositions

Cf. A001906, A055887, A088305, A129373, A129374, A129921, A163767, A304963, A304964, A304965.

A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(A(j$2)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, May 24 2018

nmax = 30; CoefficientList[Series[1/(1 - Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]

G.f.: 1/(1 - Sum_{k>=1} A163767(k)*x^k).

A318493 Expansion of 1/(1 - Sum_{i>=1, j>=1} ijx^(i*j)).

Original entry on oeis.org

1, 1, 5, 15, 53, 165, 561, 1807, 5993, 19586, 64491, 211466, 695101, 2281614, 7494995, 24610588, 80829373, 265437828, 871738976, 2862815763, 9401768055, 30875971366, 101399191222, 333001988025, 1093603789613, 3591473940515, 11794667169894, 38734550365835, 127207121681103, 417757532953031

Offset: 0

Ilya Gutkovskiy, Aug 27 2018

nonn

Cf. A000005, A038040, A088305, A129921, A180305, A280540, A280541.

a:=series(1/(1-add(add(i*j*x^(i*j),j=1..100),i=1..100)),x=0,30): seq(coeff(a,x,n),n=0..29); # Paolo P. Lava, Apr 02 2019

nmax = 29; CoefficientList[Series[1/(1 - Sum[Sum[i j x^(i j), {i, 1, nmax}], {j, 1, nmax}]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[k x^k/(1 - x^k)^2, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[k DivisorSigma[0, k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[k DivisorSigma[0, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]

G.f.: 1/(1 - Sum_{k>=1} k*x^k/(1 - x^k)^2).
G.f.: 1/(1 - Sum_{k>=1} k*d(k)*x^k), where d(k) = number of divisors of k (A000005).
a(0) = 1; a(n) = Sum_{k=1..n} A038040(k)*a(n-k).
a(n) ~ c / r^n, where r = 0.304499876501217750838861744045680232405337905509126... is the root of the equation Sum_{k>=1} k*r^k/(1 - r^k)^2 = 1 and c = 0.44152042515136849968144466258954953693306684400261343177792428746297872748... - Vaclav Kotesovec, Aug 28 2018

A327764 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(ij(j + 1)/2)).

Original entry on oeis.org

1, 1, 2, 5, 10, 21, 47, 99, 211, 455, 973, 2081, 4464, 9558, 20466, 43848, 93914, 201140, 430844, 922818, 1976553, 4233613, 9067960, 19422576, 41601229, 89105550, 190854784, 408791400, 875589076, 1875421302, 4016959325, 8603912899, 18428694036, 39472363286

Offset: 0

Ilya Gutkovskiy, Sep 24 2019

nonn

Invert transform of A007862.

Cf. A007862, A129921, A327738, A327744, A327745.

nmax = 33; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1)/2)/(1 - x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Length[Select[Divisors[k], IntegerQ[Sqrt[8 # + 1]] &]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

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

a(0) = 1; a(n) = Sum_{k=1..n} A007862(k) * a(n-k).

cp's OEIS Frontend

A304964 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1, l>=1} x^(i*j*k*l)).

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A327738 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*j^2)).

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A307241 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1)*d(k+1)*a(n-k), where d() is the number of divisors (A000005).

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A321190 a(n) = [x^n] 1/(1 - Sum_{k>=1} k^n*x^k/(1 - x^k)).

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A327798 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*(j + 1))).

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A352839 Expansion of g.f. 1/(1 - Sum_{k>=1} sigma_k(k) * x^k).

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PARI

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A353005 Decimal expansion of the root of the equation Sum_{k>0} x^k/(1-x^k) = 1.

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A305049 Expansion of 1/(1 - Sum_{k>=1} tau_k(k)*x^k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

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A304964 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1, l>=1} x^(ijk*l)).

A307241 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1)d(k+1)a(n-k), where d() is the number of divisors (A000005).

A318493 Expansion of 1/(1 - Sum_{i>=1, j>=1} ijx^(i*j)).

A327764 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(ij(j + 1)/2)).