A329256 -id:A329256 - cp's OEIS frontend

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

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

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).

A329259 Expansion of e.g.f. -log(1 - Sum_{k>=1} x^(k^2) / (k^2)!).

Original entry on oeis.org

0, 1, 1, 2, 7, 29, 150, 930, 6755, 56071, 523540, 5430710, 61967070, 771361525, 10402051660, 151065164250, 2350567168951, 39013029955917, 687979755287416, 12845920452293594, 253183788618567525, 5252704310496986070, 114424576082127987830, 2611313756103949479660

Offset: 0

Ilya Gutkovskiy, Nov 09 2019

nonn

Cf. A000629, A010052, A205803, A205804, A329256, A329258.

nmax = 23; CoefficientList[Series[-Log[1 - Sum[x^(k^2)/(k^2)!, {k, 1, Floor[nmax^(1/2)] + 1}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Boole[IntegerQ[n^(1/2)]] + Sum[Binomial[n, k] Boole[IntegerQ[(n - k)^(1/2)]] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 23}]

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

cp's OEIS Frontend

A205800 Expansion of e.g.f. exp( Sum_{n>=1} x^(n^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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A329259 Expansion of e.g.f. -log(1 - Sum_{k>=1} x^(k^2) / (k^2)!).

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