A249588 -id:A249588 - cp's OEIS frontend

A007841 Number of factorizations of permutations of n letters into cycles in nondecreasing length order.

1, 1, 3, 11, 56, 324, 2324, 18332, 167544, 1674264, 18615432, 223686792, 2937715296, 41233157952, 623159583552, 10008728738304, 171213653641344, 3092653420877952, 59086024678203264, 1185657912197967744, 25015435198774723584, 552130504313534175744

Offset: 0

Arnold Knopfmacher

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
Vaclav Kotesovec, Graph - The asymptotic ratio
A. Knopfmacher, J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.

Cf. A007837, A007838, A249078, A249480, A249588, A249593, A269791, A269793, A269794.

p := product(1/(1-x^m/m), m=1..100):
s := series(p,x,100):
for i from 0 to 100 do printf(`%.0f,`,i!*coeff(s,x,i)) od:
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (i-1)!^j*b(n-i*j, i-1)*multinomial(n, n-i*j, i$j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 21 2014

nmax = 25; CoefficientList[Series[1/Product[(1 - x^k/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 24 2019 *)
nmax = 25; CoefficientList[Series[Exp[Sum[PolyLog[j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 24 2019 *)

R(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 09 2014 */

N=66; q='q+O('q^N);
f=1/prod(n=1,N, 1-1/n*q^n );
egf=serlaplace(f);
Vec(egf)
/* Joerg Arndt, Oct 06 2012 */

E.g.f.: prod{m >= 1} 1/(1-x^m/m).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} d^(1-k/d) and a(0) = 1. - Vladeta Jovovic, Oct 14 2002
a(n) = R(n,1), R(n,m) = R(n,m+1)+binomial(n,m)*(m-1)!*R(n-m,m), R(n,n)=(n-1)!, R(n,m)=0 for nVladimir Kruchinin, Sep 09 2014
a(n) ~ c * n! * n, where c = exp(-gamma) = 0.56145948..., where gamma is the Euler-Mascheroni constant A001620 [Lehmer, 1972]. - Vaclav Kotesovec, Mar 05 2016
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*j^k)). - Ilya Gutkovskiy, May 27 2018

More terms from James Sellers, Jan 09 2001

Prepended a(0) = 1, Joerg Arndt, Oct 06 2012

A249593 G.f.: Product_{n>=1} 1/(1 - x^n/n^3) = Sum_{n>=0} a(n)*x^n/n!^3.

Original entry on oeis.org

1, 1, 9, 251, 16496, 2083824, 453803984, 156304214576, 80272385155584, 58631012094472704, 58713787327403063808, 78225670182020153384448, 135277046518915274471718912, 297374407080303931562525442048, 816367902369725640298981464096768

Offset: 0

Paul D. Hanna, Nov 02 2014

nonn

			G.f.: A(x) = 1 + x + 9*x^2/2!^3 + 251*x^3/3!^3 + 16496*x^4/4!^3 +...
where
A(x) = 1/((1-x)*(1-x^2/2^3)*(1-x^3/3^3)*(1-x^4/4^3)*(1-x^5/5^3)*...).

Vaclav Kotesovec, Table of n, a(n) for n = 0..180

Cf. A007841, A249588, A269791, A269793, A269794.

Table[n!^3 * SeriesCoefficient[Product[1/(1 - x^m/m^3), {m, 1, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 05 2016 *)

{a(n)=n!^3*polcoeff(prod(k=1, n, 1/(1-x^k/k^3 +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))

/* Using logarithmic derivative: */
{b(k) = sumdiv(k, d, d^(1-3*k/d))}
{a(n) = if(n==0, 1, sum(k=1, n, n!^2*(n-1)!/(n-k)!^3 * b(k) * a(n-k)))}
for(n=0, 20, print1(a(n), ", "))

a(n) = Sum_{k=1..n} n!^2*(n-1)!/(n-k)!^3 * b(k) * a(n-k), where b(k) = Sum_{d|k} d^(1-3*k/d) and a(0) = 1 (after Vladeta Jovovic in A007841).

a(n) ~ c * n!^3, where c = Product_{k>=2} 1/(1-1/k^3) = 3*Pi/cosh(sqrt(3)*Pi/2) = 1.235488267746513477155075624616837... . - Vaclav Kotesovec, Mar 05 2016

Name clarified by Vaclav Kotesovec, Mar 05 2016

A269791 G.f.: Product_{n>=1} 1/(1 - x^n/n^4) = Sum_{n>=0} a(n)*x^n/n!^4.

Original entry on oeis.org

1, 1, 17, 1393, 359200, 224991776, 291968881696, 701412781560352, 2873957814268080128, 18859650596161401139200, 188619789441121624152354816, 2761804817165898231731040301056, 57271995555712767650976765232545792, 1635810412682066454426684822491878391808

Offset: 0

Vaclav Kotesovec, Mar 05 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..140

Cf. A007841, A249588, A249593, A269793, A269794.

Table[n!^4 * SeriesCoefficient[Product[1/(1 - x^k/k^4), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

{a(n)=n!^4*polcoeff(prod(k=1, n, 1/(1-x^k/k^4 +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * n!^4, where c = Product_{k>=2} 1/(1-1/k^4) = 4*Pi/sinh(Pi) = 4*A090986 = 1.08811621992853265180094633468815...

A269793 G.f.: Product_{n>=1} 1/(1 - x^n/n^5) = Sum_{n>=0} a(n)*x^n/n!^5.

Original entry on oeis.org

1, 1, 33, 8051, 8259776, 25822962624, 200839327164224, 3375758721819353792, 110621043661751405543424, 6532189550762931700406452224, 653226327065916563182761815212032, 105203470361723800472334968046839365632, 26178104032796403698593899646317901702496256

Offset: 0

Vaclav Kotesovec, Mar 05 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..120

Cf. A007841, A249588, A249593, A269791, A269794.

Table[n!^5 * SeriesCoefficient[Product[1/(1-x^k/k^5), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

{a(n)=n!^5*polcoeff(prod(k=1, n, 1/(1-x^k/k^5 +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * n!^5, where c = Product_{k>=2} 1/(1-1/k^5) = abs(Gamma((9+sqrt(5) + i*sqrt(10-2*sqrt(5)))/4) * Gamma((9-sqrt(5) + i*sqrt(10+2*sqrt(5)))/4))^2 = 1.03814501733099931382497266723652151296563..., where Gamma is the Gamma function and i is the imaginary unit. - Vaclav Kotesovec, Mar 05 2016

A269794 G.f.: Product_{n>=1} 1/(1 - x^n/n^6) = Sum_{n>=0} a(n)*x^n/n!^6.

Original entry on oeis.org

1, 1, 65, 47449, 194444416, 3038449102976, 141766192358448256, 16678817447073033946240, 4372271021740050216976646144, 2323608852183697867526563204694016, 2323611343146528421975097303187359268864, 4116421685969107286571222251382158945547976704

Offset: 0

Vaclav Kotesovec, Mar 05 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A007841, A249588, A249593, A269791, A269793.

Table[n!^6 * SeriesCoefficient[Product[1/(1-x^k/k^6), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

{a(n)=n!^6*polcoeff(prod(k=1, n, 1/(1-x^k/k^6 +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * n!^6, where c = Product_{k>=2} 1/(1-1/k^6) = 6*Pi^2 / cosh(sqrt(3)*Pi/2)^2 = 1.0176208398261870492814795459985... . - Vaclav Kotesovec, Mar 05 2016

A326864 G.f.: Product_{k>=1} (1 + x^k/k^2) = Sum_{n>=0} a(n)*x^n/n!^2.

Original entry on oeis.org

1, 1, 1, 13, 100, 1876, 57636, 2051316, 104640768, 6819033600, 576652089600, 57187381536000, 7057192160793600, 1014733052692300800, 172646881540527744000, 33848454886497227289600, 7637231669166956976537600, 1948418678155880277481881600

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

			a(n) ~ c * (n-1)!^2, where c = A156648 = Product_{k>=1} (1 + 1/k^2) = sinh(Pi)/Pi = 3.67607791037497772...

Alois P. Heinz, Table of n, a(n) for n = 0..254

Cf. A007838, A249588, A326865.

Cf. A087132.

b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 27 2023

nmax = 20; CoefficientList[Series[Product[(1+x^k/k^2), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!^2

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

Paul D. Hanna, Nov 01 2014

nonn

Here BesselI(0,2) = Sum_{n>=0} 1/n!^2 = 2.2795853023360672... (A070910).

			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; ...

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A249607, A249592, A249588, A249078, A070910.

\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]), ", "))

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

Paul D. Hanna, Nov 01 2014

nonn

			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; ...

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A249590, A249078, A249588.

\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

Paul D. Hanna, Nov 02 2014

nonn

Here BesselJ(0,2) = Sum_{n>=0} (-1)^n/n!^2 = 0.223890779141235668... (A091681).

			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; ...

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A249590, A249474, A249588, A091681.

\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]), ", "))

A336295 a(n) = (n!)^n * [x^n] Product_{k>=1} 1/(1 - x^k/k^n).

Original entry on oeis.org

1, 1, 5, 251, 359200, 25822962624, 141766192358448256, 83301485967496541735457536, 7013555995366382867427754604471779328, 109330254486209621988088555707809713786027354619904, 396335044092985772297627538614627390881554195217999599121962369024

Offset: 0

Ilya Gutkovskiy, Jul 16 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..30

Cf. A007841, A215910, A249588, A249593, A269791, A269793, A269794.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k)+b(n-i, min(n-i, i), k)*((i-1)!*binomial(n, i))^k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 27 2023

Table[(n!)^n SeriesCoefficient[Product[1/(1 - x^k/k^n), {k, 1, n}], {x, 0, n}], {n, 0, 10}]

cp's OEIS Frontend

A007841 Number of factorizations of permutations of n letters into cycles in nondecreasing length order.

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A249593 G.f.: Product_{n>=1} 1/(1 - x^n/n^3) = Sum_{n>=0} a(n)*x^n/n!^3.

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A269791 G.f.: Product_{n>=1} 1/(1 - x^n/n^4) = Sum_{n>=0} a(n)*x^n/n!^4.

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A269793 G.f.: Product_{n>=1} 1/(1 - x^n/n^5) = Sum_{n>=0} a(n)*x^n/n!^5.

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A269794 G.f.: Product_{n>=1} 1/(1 - x^n/n^6) = Sum_{n>=0} a(n)*x^n/n!^6.

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A326864 G.f.: Product_{k>=1} (1 + x^k/k^2) = Sum_{n>=0} a(n)*x^n/n!^2.

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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).

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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).

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