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
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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}]
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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
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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
A205803
E.g.f.: -log( Sum_{n>=0} (-x)^(n*(n+1)/2) / (n*(n+1)/2)! ).
Original entry on oeis.org
1, 1, 3, 10, 44, 249, 1693, 13384, 120932, 1229759, 13894429, 172677450, 2341111850, 34385290940, 543885462121, 9217332132024, 166621796893824, 3200275345024464, 65082833538054240, 1397097303096855578, 31569176387677926733, 749012642530028865010
Offset: 1
E.g.f.: A(x) = x + x^2/2! + 3*x^3/3! + 10*x^4/4! + 44*x^5/5! + 249*x^6/6! +...
where
exp(-A(x)) = 1 - x - x^3/3! + x^6/6! + x^10/10! - x^15/15! - x^21/21! +...
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{a(n) = n!*polcoeff(-log(sum(m=0, sqrtint(2*n+1), (-x)^(m*(m+1)/2)/(m*(m+1)/2)!+x*O(x^n))), n)}
for(n=1,25,print1(a(n),", "))
A305620
Expansion of e.g.f. log(1 + Sum_{k>=1} x^(k^2)/k^2).
Original entry on oeis.org
1, -1, 2, 0, -6, 60, -540, 3780, 12600, -199080, 3074400, -45738000, 511434000, -5621616000, 55394539200, 960323364000, -24001273296000, 498178528848000, -9994137465312000, 156104172544320000, -2076607873660320000, 18061446353670720000, 206725394268993600000
Offset: 1
E.g.f.: A(x) = x - x^2/2! + 2*x^3/3! - 6*x^5/5! + 60*x^6/6! - 540*x^7/7! + ...
exp(A(x)) = 1 + x + x^4/4 + x^9/9 + x^16/16 + ... + x^A000290(k)/A000290(k) + ...
exp(exp(A(x))-1) = 1 + x + x^2/2! + x^3/3! + 7*x^4/4! + 31*x^5/5! + ... + A205801(k)*x^k/k! + ... = Product_{j>=1} 1/(1 - x^j)^(A008836(j)/j).
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N:= 50: # for a(1)..a(N)
g:= log(1 + add(x^(k^2)/k^2,k=1..floor(sqrt(N)))):
S:= series(g,x,N+1):
seq(coeff(S,x,n)*n!,n=1..N); # Robert Israel, Jun 07 2018
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nmax = 23; Rest[CoefficientList[Series[Log[1 + Sum[x^k^2/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 23; Rest[CoefficientList[Series[Log[1 + Log[Product[1/(1 - x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}]]], {x, 0, nmax}], x] Range[0, nmax]!]
a[n_] := a[n] = Boole[IntegerQ[n^(1/2)]] (n - 1)! - Sum[k Binomial[n, k] Boole[IntegerQ[(n - k)^(1/2)]] (n - k - 1)! a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 23}]
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
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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}]
Showing 1-4 of 4 results.