A205802 -id:A205802 - cp's OEIS frontend

A205801 Expansion of e.g.f. exp( Sum_{n>=1} x^(n^2) / (n^2) ).

1, 1, 1, 1, 7, 31, 91, 211, 1681, 52417, 461161, 2427481, 10744471, 219643711, 2619643027, 18939628891, 1410692293921, 23943786881281, 263853697605841, 2237281161036337, 53316533506210471, 900164075618402911, 11265158441537890891, 112769404714319769571

Offset: 0

Paul D. Hanna, Jan 31 2012

nonn

Number of permutations of [n] whose cycle lengths are squares. - Alois P. Heinz, May 12 2016

			E.g.f.: A(x) = 1 + x + x^2/2! + x^3/3! + 7*x^4/4! + 31*x^5/5! + 91*x^6/6! +...
where
log(A(x)) = x + x^4/4 + x^9/9 + x^16/16 + x^25/25 + x^36/36 +...

Alois P. Heinz, Table of n, a(n) for n = 0..451
David Harry Richman and Andrew O'Desky, Derangements and the p-adic incomplete gamma function, arXiv:2012.04615 [math.NT], 2020.
Wikipedia, Liouville Function.

Cf. A000290, A193374, A205800, A205802, A273001, A273997, A308397, A317129, A329945, A353184.

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(issqr(j),
       a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 12 2016

a[n_] := a[n] = If[n==0, 1, Sum[If[IntegerQ @ Sqrt[j], a[n-j]*(j-1)! * Binomial[n-1, j-1], 0], {j, 1, n}]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 19 2017, after Alois P. Heinz *)
nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 17 2019 *)

{a(n)=n!*polcoeff(exp(sum(m=1, sqrtint(n+1), x^(m^2)/(m^2)+x*O(x^n))), n)}

a(n) = if(n==0, 1, (n-1)!*sum(k=1, sqrtint(n), a(n-k^2)/(n-k^2)!)); \\ Seiichi Manyama, Apr 29 2022

The e.g.f. A(x)=1+a(1)x+a(2)x^2/2!+... is equal to the power series expansion of the product of (1-x^n)^{-lambda(n)/n} (n=1,2,...) where lambda(n) is the Liouville function A008836 (follows easily from the Lambert series of lambda(n) - see e. g., the Wikipedia link). - Mamuka Jibladze, Jan 12 2014

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..floor(sqrt(n))} a(n-k^2)/(n-k^2)!. - Seiichi Manyama, Apr 29 2022

A205800 Expansion of e.g.f. exp( Sum_{n>=1} x^(n^2) ).

Original entry on oeis.org

1, 1, 1, 1, 25, 121, 361, 841, 21841, 547345, 4541041, 23292721, 169658281, 7550279881, 95230199065, 692107448761, 25431412450081, 563675083228321, 9791797014753121, 112525775579561185, 3370231071632996281, 65798618669268652441, 1345746844683430533961

Offset: 0

Paul D. Hanna, Jan 31 2012

nonn

			E.g.f.: A(x) = 1 + x + x^2/2! + x^3/3! + 25*x^4/4! + 121*x^5/5! +...
where
log(A(x)) = x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + x^64 +...

Seiichi Manyama, Table of n, a(n) for n = 0..448

Cf. A193375, A205801, A205802, A329256.

seq(coeff(series(factorial(n)*(exp(add(x^(k^2),k=1..n))),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 23 2018

With[{nn=30},CoefficientList[Series[Exp[Sum[x^n^2,{n,nn}]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 01 2020 *)

{a(n)=n!*polcoeff(exp(sum(m=1, sqrtint(n+1), x^(m^2)+x*O(x^n))), n)}

a(n) = if(n==0, 1, (n-1)!*sum(k=1, sqrtint(n), k^2*a(n-k^2)/(n-k^2)!)); \\ Seiichi Manyama, Apr 29 2022

E.g.f.: exp((theta_3(x) - 1)/2), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Oct 23 2018

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..floor(sqrt(n))} k^2 * a(n-k^2)/(n-k^2)!. - Seiichi Manyama, Apr 29 2022

A205799 E.g.f.: exp( Sum_{n>=1} x^(n(n+1)/2) / (n(n+1)/2)! ).

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 32, 113, 365, 1373, 6072, 25279, 115633, 606321, 3051413, 16344785, 98402881, 576283953, 3523586227, 23840955908, 158428389359, 1085566420290, 8128568533790, 60203101002122, 455911264482697, 3734114950288571, 30413492882578846

Offset: 0

Paul D. Hanna, Jan 31 2012

nonn

Number of set partitions of [n] whose block lengths are triangular numbers. - Alois P. Heinz, Jun 10 2018

			E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 11*x^5/5! + 32*x^6/6! +...
where
log(A(x)) = x + x^3/3! + x^6/6! + x^10/10! + x^15/15! + x^21/21! +...

Alois P. Heinz, Table of n, a(n) for n = 0..603

Cf. A193374, A205802, A305824.

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
      issqr(8*j+1), a(n-j)*binomial(n-1, j-1), 0), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 10 2018

m = 30;
CoefficientList[Exp[Sum[x^(n(n+1)/2)/(n(n+1)/2)!, {n, 1, m}]] + O[x]^m, x]* Range[0, m-1]! (* Jean-François Alcover, Mar 05 2021 *)

{a(n)=n!*polcoeff(exp(sum(m=1, sqrtint(2*n+1), x^(m*(m+1)/2)/(m*(m+1)/2)!+x*O(x^n))), n)}

A205804 E.g.f.: -log( Sum_{n>=0} (-x)^(n^2) / (n^2)! ).

Original entry on oeis.org

1, 1, 2, 5, 19, 90, 510, 3395, 25831, 221140, 2104310, 22027170, 251540795, 3111928820, 41460769350, 591847005749, 9011786683883, 145794610986004, 2497443795363566, 45157627509568965, 859494143391347310, 17176870199851102510, 359623890969235361700

Offset: 1

Paul D. Hanna, Jan 31 2012

nonn

			E.g.f.: A(x) = x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 19*x^5/5! + 90*x^6/6! +...
where
exp(-A(x)) = 1 - x + x^4/4! - x^9/9! + x^16/16! - x^25/25! + x^36/36! +...

Cf. A205802, A205803.

{a(n) = n!*polcoeff(-log(sum(m=0, sqrtint(n+1), (-x)^(m^2)/(m^2)!+x*O(x^n))), n)}
for(n=1,25,print1(a(n),", "))

A329256 Expansion of e.g.f. exp(Sum_{k>=1} x^(k^2) / (k^2)!).

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 36, 106, 443, 1796, 6161, 23816, 122266, 643644, 2934296, 14002237, 83835433, 532282819, 3005258539, 17039094646, 115611682810, 848428608644, 5682350940168, 37297365940462, 281594230420802, 2323660209441962, 17929392395804072

Offset: 0

Ilya Gutkovskiy, Nov 09 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..611

Cf. A000110, A010052, A205799, A205800, A205801, A205802, A205804, A353180.

nmax = 27; CoefficientList[Series[Exp[Sum[x^(k^2)/(k^2)!, {k, 1, Floor[nmax^(1/2)] + 1}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Boole[IntegerQ[k^(1/2)]] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 27}]

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, sqrtint(N), x^k^2/(k^2)!)))) \\ Seiichi Manyama, Apr 29 2022

a(n) = if(n==0, 1, sum(k=1, sqrtint(n), binomial(n-1, k^2-1)*a(n-k^2))); \\ Seiichi Manyama, Apr 29 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A010052(k) * a(n-k).

A353180 Expansion of e.g.f. 1/(1 - Sum_{k>=1} x^(k^2) / (k^2)!).

Original entry on oeis.org

1, 1, 2, 6, 25, 130, 810, 5880, 48790, 455491, 4725020, 53915730, 671141130, 9050528630, 131437406100, 2045160117000, 33944105995801, 598591246152934, 11176863039391538, 220287874849834596, 4570225746232479690, 99557506547622369750, 2272028399094852806100

Offset: 0

Seiichi Manyama, Apr 29 2022

nonn

Cf. A006456, A205802, A329256, A353184.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k^2] * a[n - k^2], {k, 1, Floor@Sqrt[n]}]; Array[a, 23, 0] (* Amiram Eldar, Apr 30 2022 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, sqrtint(N), x^k^2/(k^2)!))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, sqrtint(i), binomial(i, j^2)*v[i-j^2+1])); v;

a(0) = 1; a(n) = Sum_{k=1..floor(sqrt(n))} binomial(n,k^2) * a(n-k^2).

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A205801 Expansion of e.g.f. exp( Sum_{n>=1} x^(n^2) / (n^2) ).

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A205800 Expansion of e.g.f. exp( Sum_{n>=1} x^(n^2) ).

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A205799 E.g.f.: exp( Sum_{n>=1} x^(n*(n+1)/2) / (n*(n+1)/2)! ).

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A205804 E.g.f.: -log( Sum_{n>=0} (-x)^(n^2) / (n^2)! ).

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A329256 Expansion of e.g.f. exp(Sum_{k>=1} x^(k^2) / (k^2)!).

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A353180 Expansion of e.g.f. 1/(1 - Sum_{k>=1} x^(k^2) / (k^2)!).

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A205799 E.g.f.: exp( Sum_{n>=1} x^(n(n+1)/2) / (n(n+1)/2)! ).