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
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! +...
-
{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),", "))
A329258
Expansion of e.g.f. -log(1 - Sum_{k>=1} x^(k*(k + 1)/2) / (k*(k + 1)/2)!).
Original entry on oeis.org
0, 1, 1, 3, 10, 44, 251, 1707, 13496, 122108, 1243201, 14060771, 174932274, 2374268974, 34910039164, 552782630401, 9378254813944, 169714311278784, 3263200704705648, 66434349885323328, 1427653109477475098, 32294539445483981821, 767051977023372086530
Offset: 0
-
nmax = 22; CoefficientList[Series[-Log[1 - Sum[x^(k (k + 1)/2)/(k (k + 1)/2)!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Boole[IntegerQ[(8 n + 1)^(1/2)]] + Sum[Binomial[n, k] Boole[IntegerQ[(8 (n - k) + 1)^(1/2)]] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 22}]
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
-
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 1-3 of 3 results.