A249588
G.f.: Product_{n>=1} 1/(1 - x^n/n^2) = Sum_{n>=0} a(n)*x^n/n!^2.
Original entry on oeis.org
1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, 229357803456, 23289083584704, 2851295406197184, 414855423241758720, 70695451937596732416, 13958230719814052097024, 3159974451734082088897536, 813380358295803762813321216, 236172126115504055456155975680
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2/2!^2 + 49*x^3/3!^2 + 856*x^4/4!^2 +...
where
A(x) = 1/((1-x)*(1-x^2/4)*(1-x^3/9)*(1-x^4/16)*(1-x^5/25)*...).
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+b(n-i, min(i, n-i))*((i-1)!*binomial(n, i))^2 ))
end:
a:= n-> b(n$2):
seq(a(n), n=0..17); # Alois P. Heinz, Jul 27 2023
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b[k_] := b[k] = DivisorSum[k, #^(1-2*k/#) &]; a[0] = 1; a[n_] := a[n] = Sum[n!*(n-1)!/(n-k)!^2*b[k]*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 23 2015, adapted from PARI *)
Table[n!^2 * SeriesCoefficient[Product[1/(1 - x^m/m^2), {m, 1, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 05 2016 *)
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{a(n)=n!^2*polcoeff(prod(k=1, n, 1/(1-x^k/k^2 +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))
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/* Using logarithmic derivative: */
{b(k) = sumdiv(k,d, d^(1-2*k/d))}
{a(n) = if(n==0,1,sum(k=1,n, n!*(n-1)!/(n-k)!^2 * b(k) * a(n-k)))}
for(n=0,20,print1(a(n),", "))
A249590
E.g.f.: BesselI(0,2)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n^2) and Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k^2 - x^k).
Original entry on oeis.org
1, 1, 6, 63, 1162, 31263, 1207344, 61719326, 4103067834, 341454828363, 34946904263560, 4304483416099530, 629558493157805370, 107728435291299602135, 21346960361800584031800, 4847223770735591212039818, 1250978551922243595690043914, 364052135715732457875255719691
Offset: 0
E.g.f.: 1 + x + 6*x^2/2!^2 + 63*x^3/3!^2 + 1162*x^4/4!^2 + 31263*x^5/5!^2 +...
such that A(x) = BesselI(0,2)*P(x) - Q(x), where
P(x) = 1/Product_{n>=1} (1 - x^n/n^2) = Sum_{n>=0} A249588(n)*x^n/n!^2, and
Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k^2 - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/4)*(1-x^3/9)*(1-x^4/16)*(1-x^5/25)*...);
Q(x) = 1/(1-x) + 1/((1-x)*(4-x^2)) + 1/((1-x)*(4-x^2)*(9-x^3)) + 1/((1-x)*(4-x^2)*(9-x^3)*(16-x^4)) + 1/((1-x)*(4-x^2)*(9-x^3)*(16-x^4)*(25-x^5)) +...
We can illustrate the initial terms a(n) in the following manner.
The coefficients q(n) in Q(x) = Sum_{n>=0} q(n)*x^n/n!^2 begin:
q(0) = 1.279585302336067267437204440811533...
q(1) = 1.279585302336067267437204440811533...
q(2) = 5.397926511680336337186022204057666...
q(3) = 48.69967981446729610442301759976513...
q(4) = 789.3250187996735809262470013346725...
q(5) = 19745.00072507184117617488656759887...
q(6) = 713288.6822890207712374724807435860...
q(7) = 34956701.28771539805703277298850790...
q(8) = 2239176303.370447012433955813571405...
q(9) = 181385849371.3820539848573249577420...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n^2) begin:
A249078 = [1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, ...];
from which we can generate this sequence like so:
a(0) = BesselI(0,2)*1 - q(0) = 1;
a(1) = BesselI(0,2)*1 - q(1) = 1;
a(2) = BesselI(0,2)*5 - q(2) = 6;
a(3) = BesselI(0,2)*49 - q(3) = 63;
a(4) = BesselI(0,2)*856 - q(4) = 1162;
a(5) = BesselI(0,2)*22376 - q(5) = 31263;
a(6) = BesselI(0,2)*842536 - q(6) = 1207344;
a(7) = BesselI(0,2)*42409480 - q(7) = 61719326;
a(8) = BesselI(0,2)*2782192064 - q(8) = 4103067834; ...
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\p100 \\ set precision
{P=Vec(serlaplace(serlaplace(prod(k=1, 31, 1/(1-x^k/k^2 +O(x^31)))))); } \\ A249588
{Q=Vec(serlaplace(serlaplace(sum(n=1, 201, prod(k=1, n, 1./(k^2-x^k +O(x^31))))))); }
for(n=0, 30, print1(round(besseli(0,2)*P[n+1]-Q[n+1]), ", "))
Showing 1-2 of 2 results.
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