A180305 -id:A180305 - cp's OEIS frontend

A319083 Coefficients of polynomials related to the D'Arcais polynomials and Dedekind's eta(q) function, triangle read by rows, T(n,k) for 0 <= k <= n.

1, 0, 1, 0, 3, 1, 0, 4, 6, 1, 0, 7, 17, 9, 1, 0, 6, 38, 39, 12, 1, 0, 12, 70, 120, 70, 15, 1, 0, 8, 116, 300, 280, 110, 18, 1, 0, 15, 185, 645, 885, 545, 159, 21, 1, 0, 13, 258, 1261, 2364, 2095, 942, 217, 24, 1, 0, 18, 384, 2262, 5586, 6713, 4281, 1498, 284, 27, 1

Offset: 0

Peter Luschny, Oct 03 2018

Column k is the k-fold self-convolution of sigma (A000203). - Alois P. Heinz, Feb 01 2021

For fixed k, Sum_{j=1..n} T(j,k) ~ Pi^(2*k) * n^(2*k) / (6^k * (2*k)!). - Vaclav Kotesovec, Sep 20 2024

			Triangle starts:
[0] 1;
[1] 0,  1;
[2] 0,  3,   1;
[3] 0,  4,   6,    1;
[4] 0,  7,  17,    9,    1;
[5] 0,  6,  38,   39,   12,    1;
[6] 0, 12,  70,  120,   70,   15,   1;
[7] 0,  8, 116,  300,  280,  110,  18,   1;
[8] 0, 15, 185,  645,  885,  545, 159,  21,  1;
[9] 0, 13, 258, 1261, 2364, 2095, 942, 217, 24, 1;

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0..6 give: A000007, A000203, A000385, A374951, A374977, A374978, A374979.
Row sums are A180305.
T(2n,n) gives A340993.
Cf. A008298, A078521, A283334, A319933.

P := proc(n, x) option remember; if n = 0 then 1 else
x*add(numtheory:-sigma(n-k)*P(k,x), k=0..n-1) fi end:
Trow := n -> seq(coeff(P(n, x), x, k), k=0..n):
seq(Trow(n), n=0..9);
# second Maple program:
T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
       add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Feb 01 2021
# Uses function PMatrix from A357368.
PMatrix(10, NumberTheory:-sigma); # Peter Luschny, Oct 19 2022

T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
     If[k == 1, If[n == 0, 0, DivisorSigma[1, n]],
     With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n-j, k-q], {j, 0, n}]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 11 2021, after Alois P. Heinz *)

The polynomials are defined by recurrence: p(0,x) = 1 and for n > 0 by
p(n, x) = x*Sum_{k=0..n-1} sigma(n-k)*p(k, x).
Sum_{k=0..n} (-1)^k * T(n,k) = A283334(n). - Alois P. Heinz, Feb 07 2025

A320649 Expansion of 1/(1 - Sum_{k>=1} k^2*x^k/(1 - x^k)).

Original entry on oeis.org

1, 1, 6, 21, 82, 294, 1116, 4103, 15326, 56833, 211454, 785441, 2920058, 10851016, 40331874, 149892024, 557098510, 2070493098, 7695228038, 28600012305, 106294901116, 395055313662, 1468262641770, 5456942875386, 20281270503914, 75377349437075, 280147395367820

Offset: 0

Ilya Gutkovskiy, Oct 18 2018

nonn

Invert transform of A001157.

Seiichi Manyama, Table of n, a(n) for n = 0..1754
N. J. A. Sloane, Transforms

Cf. A000219, A001157, A180305.

a:=series(1/(1-add(k^2*x^k/(1-x^k),k=1..100)),x=0,27): seq(coeff(a,x,n),n=0..26); # Paolo P. Lava, Apr 02 2019

nmax = 26; CoefficientList[Series[1/(1 - Sum[k^2 x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 26; CoefficientList[Series[1/(1 + x D[Log[Product[(1 - x^k)^k, {k, 1, nmax}]], x]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[2, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 26}]

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

a(0) = 1; a(n) = Sum_{k=1..n} sigma_2(k)*a(n-k).

A340904 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * sigma_1(k) * a(n-k).

Original entry on oeis.org

1, 1, 5, 28, 225, 2206, 26174, 361278, 5704401, 101297701, 1998893240, 43386854622, 1027353587730, 26353742447280, 728030940612638, 21548668265211778, 680330296613877761, 22821706122361385354, 810587673640374442445, 30390159250481750866640

Offset: 0

Ilya Gutkovskiy, Jan 26 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..404

Cf. A000203, A006153, A180305, A274804, A340903.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] DivisorSigma[1, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[1/(1 - Sum[Sum[i x^(i j)/(i j)!, {j, 1, nmax}], {i, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, sigma(k)*x^k/k!)))) \\ Seiichi Manyama, Mar 29 2022

a(n) = if(n==0, 1, sum(k=1, n, sigma(k)*binomial(n, k)*a(n-k))); \\ Seiichi Manyama, Mar 29 2022

E.g.f.: 1 / (1 - Sum_{i>=1} Sum_{j>=1} i * x^(i*j) / (i*j)!).

E.g.f.: 1 / (1 - Sum_{k>=1} sigma_1(k) * x^k/k!). - Seiichi Manyama, Mar 29 2022

A283334 G.f.: 1/(1 - x*d/dx log(eta(x))), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.

Original entry on oeis.org

1, -1, -2, 1, 2, 4, -6, -5, 4, 1, 18, -13, -26, 4, 22, 66, -76, -78, 66, 37, 122, -136, -144, 10, 54, 599, -368, -746, 196, 568, 744, -938, -156, -312, -1428, 2720, 3340, -2324, -8588, 1520, 8814, 4846, 1380, -16565, -16966, -6324, 79170, 47250, -160346

Offset: 0

Seiichi Manyama, Mar 05 2017

sign

			G.f.: A(x) = 1 - x - 2*x^2 + x^3 + 2*x^4 + 4*x^5 - 6*x^6 - 5*x^7 + ...
1/A(x) = 1 - x*d/dx log(eta(x)) = 1 + x + 3*x^2 + 4*x^3 + 7*x^4 + ... + sigma(n)*x^n + ...
eta(x)^3/A(x) = 1 - 2*x + 7*x^6 - 21*x^10 + 44*x^15 - 78*x^21 + ... + A184366(n)*x^n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Bernhard Heim, Markus Neuhauser, and Robert Troeger, Zeros Transfer For Recursively defined Polynomials, arXiv:2304.02694 [math.NT], 2023. See Table 5 p. 14.

Cf. A180305 (1/(1 + x*d/dx log(eta(x)))), A184366, A319083.

lista(nn) = {va = vector(nn); va[1] = 1; for (n=1, nn-1, va[n+1] = -sum(k=0, n-1, sigma(n-k)*va[k+1]);); va;} \\ Michel Marcus, Mar 05 2017

a(n) = -Sum_{k=0..n-1} sigma(n-k)*a(k) for n>0 with a(0) = 1.
G.f.: 1/(1 + Sum_{k>=1} k*x^k/(1 - x^k)). - Ilya Gutkovskiy, Oct 18 2018
a(n) = Sum_{k=0..n} (-1)^k * A319083(n,k). - Alois P. Heinz, Feb 07 2025

A300671 Expansion of 1/(1 - Sum_{k>=1} x^prime(k)/(1 - x^prime(k))).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 8, 15, 23, 40, 63, 108, 172, 290, 471, 782, 1280, 2119, 3474, 5741, 9432, 15557, 25590, 42180, 69413, 114371, 188276, 310136, 510637, 841045, 1384883, 2280831, 3755862, 6185457, 10185941, 16774695, 27624215, 45492412, 74916559, 123374127, 203172520, 334587577

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

nonn

Invert transform of A001221.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Cf. A001221, A013939, A112965, A129921, A180305, A293548, A300672.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*nops(ifactors(i)[2]), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 11 2021

nmax = 42; CoefficientList[Series[1/(1 - Sum[x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 42; CoefficientList[Series[1/(1 - Sum[PrimeNu[k] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PrimeNu[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}]

G.f.: 1/(1 - Sum_{k>=2} A001221(k)*x^k).

A316961 Expansion of Product_{k>=1} 1/(1 - sigma(k)*x^k), where sigma(k) is the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 1, 4, 8, 24, 42, 118, 208, 524, 961, 2191, 3994, 9020, 16142, 34500, 62814, 130496, 234474, 478334, 855982, 1712012, 3061230, 6003546, 10689178, 20783796, 36789875, 70540531, 124812892, 237022708, 417422168, 786509778, 1381137702, 2583046168, 4526024200, 8402928681

Offset: 0

Ilya Gutkovskiy, Jul 17 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..4150

Cf. A000203, A061256, A180305, A279784, A316962.

with(numtheory): a:=series(mul(1/(1-sigma(k)*x^k),k=1..100),x=0,35): seq(coeff(a,x,n),n=0..34); # Paolo P. Lava, Apr 02 2019

nmax = 34; CoefficientList[Series[Product[1/(1 - DivisorSigma[1, k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 34; CoefficientList[Series[Exp[Sum[Sum[DivisorSigma[1, j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d DivisorSigma[1, d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} sigma(j)^k*x^(j*k)/k).
From Vaclav Kotesovec, Jul 28 2018: (Start)
a(n) ~ c * 3^(n/2), where
c = 133.83151651318934683776776253692818185240361972305... if n is even and
c = 131.63961163168586786976253326691345807212512512772... if n is odd.
In closed form, a(n) ~ ((3 + sqrt(3)) * Product_{k>=3} (1/(1 - sigma(k) / 3^(k/2))) + (-1)^n * (3 - sqrt(3)) * Product_{k>=3} (1/(1 - (-1)^k * sigma(k) / 3^(k/2)))) * 3^(n/2) / 4. (End)

A300672 Expansion of 1/(1 - Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k))).

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 8, 10, 23, 32, 64, 98, 187, 296, 543, 891, 1595, 2660, 4694, 7924, 13854, 23556, 40940, 69939, 121122, 207490, 358517, 615292, 1061635, 1824013, 3144404, 5406257, 9314645, 16021922, 27595176, 47478950, 81757104, 140691461, 242232918, 416890645, 717712748, 1235289624

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

nonn

Invert transform of A001222.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Cf. A001222, A022559, A112967, A129921, A180305, A293549, A300671.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*numtheory[bigomega](i), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 11 2021

nmax = 41; CoefficientList[Series[1/(1 - Sum[Boole[PrimePowerQ[k]] x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 41; CoefficientList[Series[1/(1 - Sum[PrimeOmega[k] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PrimeOmega[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 41}]

G.f.: 1/(1 - Sum_{k>=2} A001222(k)*x^k).

A305305 Expansion of e.g.f. 1/(1 - Sum_{k>=1} x^k/(k*(1 - x^k))).

Original entry on oeis.org

1, 1, 5, 32, 292, 3174, 42758, 659028, 11725656, 233646240, 5183599152, 126353158656, 3362529785712, 96896454983184, 3007687250735568, 100017757744279584, 3547903924884082176, 133715849506895518848, 5336112511923188151168, 224772952826373341478912, 9966476790792153522756864

Offset: 0

Ilya Gutkovskiy, May 29 2018

nonn

a(n)/n! is the invert transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 32*x^3/3! + 292*x^4/4! + 3174*x^5/5! + 42758*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..397
N. J. A. Sloane, Transforms

Cf. A000041, A000203, A017665, A017666, A038048, A180305.

b:= proc(n) option remember; `if`(n=0, 1, add(add(
      1/d, d=numtheory[divisors](j))*b(n-j), j=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..20);  # Alois P. Heinz, May 29 2018

nmax = 20; CoefficientList[Series[1/(1 - Sum[x^k/(k (1 - x^k)), {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[1/(1 - Sum[DivisorSigma[-1, k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[-1, k] a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 20}]

E.g.f.: 1/(1 - Sum_{k>=1} (sigma(k)/k)*x^k), where sigma() = A000203.
E.g.f.: 1/(1 - Sum_{k>=1} (A017665(k)/A017666(k))*x^k).
E.g.f.: 1/(1 - log(f(x))), where f(x) = o.g.f. for A000041, Product_{k>=1} 1/(1 - x^k).

A316962 Expansion of Product_{k>=1} (1 + sigma(k)*x^k), where sigma(k) is the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 1, 3, 7, 11, 25, 51, 87, 129, 286, 462, 760, 1312, 2102, 3470, 5988, 8840, 13884, 22577, 33545, 55961, 85341, 126705, 194317, 293621, 435040, 641472, 971503, 1462483, 2108161, 3124489, 4474579, 6545809, 9561923, 13518678, 19809034, 28387625, 40286631, 57039233

Offset: 0

Ilya Gutkovskiy, Jul 17 2018

nonn

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

Cf. A000203, A180305, A192065, A279786, A316961.

with(numtheory): a:=series(mul(1+sigma(k)*x^k,k=1..100),x=0,39): seq(coeff(a,x,n),n=0..38); # Paolo P. Lava, Apr 02 2019

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

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

A320651 Expansion of 1/(1 - Sum_{k>=1} k*x^k/(1 + x^k)).

Original entry on oeis.org

1, 1, 2, 7, 14, 36, 90, 213, 520, 1271, 3082, 7493, 18238, 44324, 107782, 262142, 637368, 1549870, 3768886, 9164499, 22285034, 54190024, 131771616, 320424614, 779166270, 1894671121, 4607207304, 11203190618, 27242414612, 66244451632, 161084380040, 391703392954

Offset: 0

Ilya Gutkovskiy, Oct 18 2018

nonn

Invert transform of A000593.

N. J. A. Sloane, Transforms

Cf. A000009, A000593, A180305, A320650.

a:=series(1/(1-add(k*x^k/(1+x^k),k=1..100)),x=0,32): seq(coeff(a,x,n),n=0..31); # Paolo P. Lava, Apr 02 2019

nmax = 31; CoefficientList[Series[1/(1 - Sum[k x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 31; CoefficientList[Series[24/(25 - EllipticTheta[2, 0, x]^4 - EllipticTheta[3, 0, x]^4), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[Mod[d, 2] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 31}]

G.f.: 1/(1 - x * (d/dx) log(Product_{k>=1} (1 + x^k))).
G.f.: 24/(25 - theta_2(x)^4 - theta_3(x)^4), where theta_() is the Jacobi theta function.
a(0) = 1; a(n) = Sum_{k=1..n} A000593(k)*a(n-k).

cp's OEIS Frontend

A319083 Coefficients of polynomials related to the D'Arcais polynomials and Dedekind's eta(q) function, triangle read by rows, T(n,k) for 0 <= k <= n.

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A320649 Expansion of 1/(1 - Sum_{k>=1} k^2*x^k/(1 - x^k)).

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A340904 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * sigma_1(k) * a(n-k).

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A283334 G.f.: 1/(1 - x*d/dx log(eta(x))), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.

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A300671 Expansion of 1/(1 - Sum_{k>=1} x^prime(k)/(1 - x^prime(k))).

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A316961 Expansion of Product_{k>=1} 1/(1 - sigma(k)*x^k), where sigma(k) is the sum of the divisors of k (A000203).

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A300672 Expansion of 1/(1 - Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k))).

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A305305 Expansion of e.g.f. 1/(1 - Sum_{k>=1} x^k/(k*(1 - x^k))).

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