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),", "))
A249592
E.g.f.: exp(1)*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 - x^k/k).
Original entry on oeis.org
1, 1, 6, 64, 1192, 32360, 1257880, 64644520, 4315649600, 360332919360, 36979925855040, 4564758983929920, 668857835862650880, 114624254940995404800, 22742780483191398589440, 5169745984444274224143360, 1335478685859609449305006080, 388956774210908224056394014720
Offset: 0
E.g.f.: 1 + x + 6*x^2/2!^2 + 64*x^3/3!^2 + 1192*x^4/4!^2 + 32360*x^5/5!^2 +...
such that A(x) = exp(1)*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 - x^k/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)*(2-x^2/2)) + 1/((1-x)*(2-x^2/2)*(3-x^3/3)) + 1/((1-x)*(2-x^2/2)*(3-x^3/3)*(4-x^4/4)) + 1/((1-x)*(2-x^2/2)*(3-x^3/3)*(4-x^4/4)*(5-x^5/5)) +...
We can illustrate the initial terms a(n) in the following manner.
The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
q(0) = 1.718281828459045235360287471352662...
q(1) = 1.718281828459045235360287471352662...
q(2) = 7.591409142295226176801437356763312...
q(3) = 69.19580959449321653265408609628046...
q(4) = 1134.849245160942721468406075477879...
q(5) = 28464.27419359959618642179245898717...
q(6) = 1032370.298622570136419515164963586...
q(7) = 50636398.83839730972810740431058131...
q(8) = 3247132530.854165002836403983556004...
q(9) = 263126229989.7260044371780752021631...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n^2) begin:
A007841 = [1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, ...];
from which we can generate this sequence like so:
a(0) = exp(1)*1 - q(0) = 1;
a(1) = exp(1)*1 - q(1) = 1;
a(2) = exp(1)*5 - q(2) = 6;
a(3) = exp(1)*49 - q(3) = 64;
a(4) = exp(1)*856 - q(4) = 1192;
a(5) = exp(1)*22376 - q(5) = 32360;
a(6) = exp(1)*842536 - q(6) = 1257880;
a(7) = exp(1)*42409480 - q(7) = 64644520;
a(8) = exp(1)*2782192064 - q(8) = 4315649600; ...
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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-x^k/k +O(x^31))))))); }
for(n=0, 30, print1(round(exp(1)*P[n+1]-Q[n+1]), ", "))
A249607
E.g.f.: BesselJ(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)^n/Product_{k=1..n} (k^2 - x^k).
Original entry on oeis.org
1, 1, 4, 37, 600, 15229, 554868, 27444786, 1770376080, 144306428161, 14507072762052, 1762845211827574, 254794661274061848, 43191427238728121445, 8488249087135630544628, 1914196040519793284483542, 491024013925643339847990144, 142153433027873627036756565313
Offset: 0
E.g.f.: 1 + x + 4*x^2/2!^2 + 37*x^3/3!^2 + 600*x^4/4!^2 + 15229*x^5/5!^2 +...
such that A(x) = BesselJ(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)^n/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) = 0.776109220858764331948172545350051...
q(1) = 0.776109220858764331948172545350051...
q(2) = 2.880546104293821659740862726750256...
q(3) = 26.02935182207945226546045472215251...
q(4) = 408.3494930551022681476356988196439...
q(5) = 10219.21992593571069167230887475274...
q(6) = 366231.9585054598651822855036690508...
q(7) = 17949694.47982534876046938459857209...
q(8) = 1147468931.070477389192467314975593...
q(9) = 92955330843.11376518199210023477232...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n^2) begin:
A249588 = [1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, ...];
from which we can generate this sequence like so:
a(0) = BesselJ(0,2)*1 + q(0) = 1;
a(1) = BesselJ(0,2)*1 + q(1) = 1;
a(2) = BesselJ(0,2)*5 + q(2) = 4;
a(3) = BesselJ(0,2)*49 + q(3) = 37;
a(4) = BesselJ(0,2)*856 + q(4) = 600;
a(5) = BesselJ(0,2)*22376 + q(5) = 15229;
a(6) = BesselJ(0,2)*842536 + q(6) = 554868;
a(7) = BesselJ(0,2)*42409480 + q(7) = 27444786;
a(8) = BesselJ(0,2)*2782192064 + q(8) = 1770376080; ...
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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, -(-1)^n*prod(k=1, n, 1./(k^2-x^k +O(x^31))))))); }
for(n=0, 30, print1(round(besselj(0,2)*P[n+1]+Q[n+1]), ", "))
Showing 1-3 of 3 results.
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