A295739 -id:A295739 - cp's OEIS frontend

A294363 E.g.f.: exp(Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

1, 1, 5, 25, 193, 1481, 16021, 167665, 2220065, 30004273, 468585541, 7560838121, 138355144225, 2589359765305, 53501800316693, 1146089983207681, 26457132132638401, 632544682981967585, 16171678558995779845, 426926324177655018553, 11938570457328874969601

Offset: 0

Seiichi Manyama, Oct 29 2017

From Peter Bala, Nov 13 2017: (Start)
The terms of the sequence appear to be of the form 4*m + 1.
It appears that the sequence taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the sequence a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k would be periodic with the exact period dividing k. (End)
From Peter Bala, Mar 28 2022: (Start)
The above conjectures are true. See the Bala link.
a(5*n+2) == 0 (mod 5); a(5*n+3) == 0 (mod 5); a(13*n+9) == 0 (mod 13). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..438
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): this sequence (k=0), A294361 (k=1), A294362 (k=2).

Cf. A000005, A028342, A295739, A006171, A107742.

nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 05 2018 *)
a[n_] := a[n] = If[n == 0, 1, Sum[k*DivisorSigma[0, k]*a[n-k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 06 2018 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, numdiv(k)*x^k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000005(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(x^k/(1 - x^k)). - Ilya Gutkovskiy, Nov 27 2017
Conjecture: log(a(n)/n!) ~ sqrt(2*n*log(n)). - Vaclav Kotesovec, Sep 07 2018

A340903 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * sigma_0(k) * a(n-k).

Original entry on oeis.org

1, 1, 4, 20, 139, 1192, 12318, 148318, 2041754, 31616757, 544005172, 10296204096, 212589150300, 4755177958104, 114545293676588, 2956316416222300, 81386676426000157, 2380590235918735576, 73729207700492304684, 2410324868012471929670, 82944575892433740648996

Offset: 0

Ilya Gutkovskiy, Jan 26 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..413

Cf. A000005, A000670, A129921, A295739, A340904.

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

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, numdiv(k)*x^k/k!)))) \\ Seiichi Manyama, Mar 29 2022

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

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

E.g.f.: 1 / (1 - Sum_{k>=1} sigma_0(k) * x^k/k!). - Seiichi Manyama, Mar 29 2022

A300011 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k/k!), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, 1, 2, 6, 20, 80, 362, 1820, 10084, 60522, 391864, 2714514, 20001700, 156107224, 1284705246, 11112088358, 100698613720, 953478331288, 9410963022318, 96614921664444, 1029705968813656, 11373102766644372, 129972789566984682, 1534638410054873892, 18696544357738885720

Offset: 0

Ilya Gutkovskiy, Mar 09 2018

nonn

Exponential transform of A000010.

			E.g.f.: A(x) = 1 + x/1! + 2*x^2/2! + 6*x^3/3! + 20*x^4/4! + 80*x^5/5! + 362*x^6/6! + 1820*x^7/7! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..552
N. J. A. Sloane, Transforms

Cf. A000010, A050392, A159929, A274804, A295739.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      binomial(n-1, j-1)*numtheory[phi](j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 09 2018

nmax = 24; CoefficientList[Series[Exp[Sum[EulerPhi[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[EulerPhi[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}]

a(n) = if(n==0, 1, sum(k=1, n, eulerphi(k)*binomial(n-1, k-1)*a(n-k))); \\ Seiichi Manyama, Feb 27 2022

E.g.f.: exp(Sum_{k>=1} A000010(k)*x^k/k!).

a(0) = 1; a(n) = Sum_{k=1..n} phi(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Feb 27 2022

A295794 Expansion of e.g.f. Product_{k>=1} exp(x^k/(1 + x^k)).

Original entry on oeis.org

1, 1, 1, 13, 25, 241, 2761, 14701, 153553, 1903105, 27877681, 263555821, 4788201001, 65083782193, 1040877257785, 24098794612621, 373918687272481, 7393663746307201, 164894196647876833, 3504497611085823565, 81863829346282866361, 2257321249626793901041, 49755091945025205954601

Offset: 0

Ilya Gutkovskiy, Nov 27 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A028342, A048272, A168243, A206303, A294363, A294392, A295739.

a:=series(mul(exp(x^k/(1+x^k)),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019

nmax = 22; CoefficientList[Series[Product[Exp[x^k/(1 + x^k)], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[x D[Log[Product[(1 + x^k)^(1/k), {k, 1, nmax}]], x]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[-k Sum[(-1)^d, {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 22}]

E.g.f.: exp(Sum_{k>=1} A048272(k)*x^k).
E.g.f.: exp(x*f'(x)), where f(x) = log(Product_{k>=1} (1 + x^k)^(1/k)).
a(n) ~ exp(2*sqrt(n*log(2)) - 1/4 - n) * n^(n - 1/4) * log(2)^(1/4) / sqrt(2). - Vaclav Kotesovec, Sep 07 2018

A338870 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = exp(Sum_{n>0} ud(n)x^n/n!), where d(n) is the number of divisors of n.

Original entry on oeis.org

1, 2, 1, 2, 6, 1, 3, 20, 12, 1, 2, 55, 80, 20, 1, 4, 142, 405, 220, 30, 1, 2, 322, 1792, 1785, 490, 42, 1, 4, 779, 7224, 12152, 5810, 952, 56, 1, 3, 1608, 27323, 73920, 56532, 15498, 1680, 72, 1, 4, 3894, 99690, 414815, 482160, 204204, 35910, 2760, 90, 1

Offset: 1

Seiichi Manyama, Nov 13 2020

Also the Bell transform of A000005.

			exp(Sum_{n>0} u*d(n)*x^n/n!) = 1 + u*x + (2*u+u^2)*x^2/2! + (2*u+6*u^2+u^3)*x^3/3! + ... .
Triangle begins:
  1;
  2,   1;
  2,   6,    1;
  3,  20,   12,     1;
  2,  55,   80,    20,    1;
  4, 142,  405,   220,   30,   1;
  2, 322, 1792,  1785,  490,  42,  1;
  4, 779, 7224, 12152, 5810, 952, 56, 1;
  ...

Seiichi Manyama, Rows n = 1..100, flattened
Peter Luschny, The Bell transform.

Column k=1..2 give A000005, A328681(n-1).
Row sums give A295739.
Cf. A338805, A338864, A338871.

T[n_, 0] := Boole[n == 0]; T[n_, k_] := T[n, k] = Sum[Boole[j > 0] * Binomial[n - 1, j - 1] * DivisorSigma[0, j] * T[n - j, k - 1], {j, 0, n - k + 1}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 13 2020 *)

a(n) = if(n<1, 0, numdiv(n));
T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1)))

T(n; u) = Sum_{k=1..n} T(n,k)*u^k is given by T(n; u) = u * Sum_{k=1..n} binomial(n-1,k-1)*d(k)*T(n-k; u), T(0; u) = 1.

T(n,k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} d(i_j)/(i_j)!.

A352694 Expansion of e.g.f. exp(Sum_{k>=1} sigma_2(k) * x^k/k!).

Original entry on oeis.org

1, 1, 6, 26, 167, 1157, 9372, 82742, 806872, 8487255, 96086764, 1159845766, 14866684968, 201266031865, 2867695938970, 42849364911878, 669517721182731, 10910196881874549, 184997231064875867, 3257297876661453487, 59443905364431491367, 1122496527274459462803

Offset: 0

Seiichi Manyama, Mar 29 2022

nonn

Exponential transform of A001157.

Seiichi Manyama, Table of n, a(n) for n = 0..505

Cf. A295739, A274804.

Cf. A001157, A294362, A352693.

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sigma(k, 2)*x^k/k!))))

a(n) = if(n==0, 1, sum(k=1, n, sigma(k, 2)*binomial(n-1, k-1)*a(n-k)));

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

A322513 Expansion of e.g.f. log(1 + Sum_{k>=1} d(k) * x^k / k!), where d(k) = number of divisors of k (A000005).

Original entry on oeis.org

0, 1, 1, -2, 1, 11, -48, -6, 1241, -6431, -15320, 452970, -2317212, -17584137, 372119776, -1552313624, -31732274313, 565880016193, -1217992446564, -90197542736656, 1400682677566587, 1990004001731140, -384348195167184028, 5109122826021406702

Offset: 0

Ilya Gutkovskiy, Oct 03 2019

sign

Logarithmic transform of A000005.

Alois P. Heinz, Table of n, a(n) for n = 0..470

Cf. A000005, A028342, A274805, A294363, A295739.

a:= proc(n) option remember; `if`(n=0, 0, (b-> b(n)-add(a(j)
     *binomial(n, j)*j*b(n-j), j=1..n-1)/n)(numtheory[tau]))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 06 2019

nmax = 23; CoefficientList[Series[Log[1 + Sum[DivisorSigma[0, k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = DivisorSigma[0, n] - Sum[Binomial[n, k] DivisorSigma[0, n - k] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 23}]

A352843 Expansion of e.g.f. exp(Sum_{k>=1} sigma_k(k) * x^k/k!).

Original entry on oeis.org

1, 1, 6, 44, 491, 6597, 110652, 2144606, 47988524, 1206275925, 33777572464, 1040200674416, 34967153135940, 1273241146218823, 49928549099500206, 2097300313258417056, 93953420539864844743, 4470694981375022862697, 225184078001798318202935

Offset: 0

Seiichi Manyama, Apr 05 2022

nonn

Exponential transform of A023887.

Cf. A295739, A274804, A352694.

Cf. A023887, A352841, A352842, A202477.

nmax = 20; CoefficientList[Series[E^(Sum[DivisorSigma[k, k]*x^k/k!, {k, 1, nmax}]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2022 *)

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

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

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

cp's OEIS Frontend

A294363 E.g.f.: exp(Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

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A340903 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * sigma_0(k) * a(n-k).

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A300011 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k/k!), where phi() is the Euler totient function (A000010).

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A295794 Expansion of e.g.f. Product_{k>=1} exp(x^k/(1 + x^k)).

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A338870 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = exp(Sum_{n>0} u*d(n)*x^n/n!), where d(n) is the number of divisors of n.

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A352694 Expansion of e.g.f. exp(Sum_{k>=1} sigma_2(k) * x^k/k!).

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A322513 Expansion of e.g.f. log(1 + Sum_{k>=1} d(k) * x^k / k!), where d(k) = number of divisors of k (A000005).

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A352843 Expansion of e.g.f. exp(Sum_{k>=1} sigma_k(k) * x^k/k!).

A338870 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = exp(Sum_{n>0} ud(n)x^n/n!), where d(n) is the number of divisors of n.