A317855 -id:A317855 - cp's OEIS frontend

A301585 G.f.: Sum_{n>=0} ((1+x)^(3*n) - 1)^n.

1, 3, 39, 910, 29949, 1271751, 66116065, 4066082856, 288701376912, 23240635243591, 2091554595246705, 208085119389952134, 22676957610808295192, 2686515300821612112411, 343760257348413122290260, 47248346582443326267328400, 6942339982115290619799947901, 1085919469129099832397573088863, 180160797497273341662653292624309, 31598815412054398239059538582525618

Offset: 0

Paul D. Hanna, Mar 24 2018

nonn

			G.f.: A(x) = 1 + 3*x + 39*x^2 + 910*x^3 + 29949*x^4 + 1271751*x^5 + 66116065*x^6 + 4066082856*x^7 + 288701376912*x^8 + ...
such that
A(x) = 1 + ((1+x)^3-1) + ((1+x)^6-1)^2 + ((1+x)^9-1)^3 + ((1+x)^12-1)^4 + ((1+x)^15-1)^5 + ((1+x)^18-1)^6 + ((1+x)^21-1)^7 + ...
Also,
A(x) = 1/2 + (1+x)^3/(1 + (1+x)^3)^2 + (1+x)^12/(1 + (1+x)^6)^3 + (1+x)^27/(1 + (1+x)^9)^4 + (1+x)^48/(1 + (1+x)^12)^5 + (1+x)^75/(1 + (1+x)^15)^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..325

Cf. A122400, A301584, A301586.

{a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, ((1+x +o)^(3*m) - 1)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

G.f.: Sum_{n>=0} (1+x)^(3*n^2) /(1 + (1+x)^(3*n))^(n+1).

a(n) ~ c * d^n * n! / sqrt(n), where d = 3*A317855 = 9.4832659615962864414905166077643974751791483225656690248818346226130911776579... and c = 0.3108017465925995208675813879173750641359609... - Vaclav Kotesovec, Aug 09 2018

A301586 G.f.: Sum_{n>=0} ((1+x)^(4*n) - 1)^n.

Original entry on oeis.org

1, 4, 70, 2180, 95729, 5422192, 375951144, 30833206304, 2919367902648, 313380517364324, 37606931999739230, 4988933437333555060, 724960700435104219679, 114519163835687116024256, 19538926882901715534673728, 3580844611314789257667535968, 701546780854024941112271649610, 146318317830136401429653726419700, 32367591848747955557013839920695374, 7569528177000020896435962191564396740

Offset: 0

Paul D. Hanna, Mar 24 2018

nonn

			G.f.: A(x) = 1 + 4*x + 70*x^2 + 2180*x^3 + 95729*x^4 + 5422192*x^5 + 375951144*x^6 + 30833206304*x^7 + ...
such that
A(x) = 1 + ((1+x)^4-1) + ((1+x)^8-1)^2 + ((1+x)^12-1)^3 + ((1+x)^16-1)^4 + ((1+x)^20-1)^5 + ((1+x)^24-1)^6 + ((1+x)^28-1)^7 + ...
Also,
A(x) = 1/2 + (1+x)^4/(1 + (1+x)^4)^2 + (1+x)^16/(1 + (1+x)^8)^3 + (1+x)^36/(1 + (1+x)^12)^4 + (1+x)^64/(1 + (1+x)^16)^5 + (1+x)^100/(1 + (1+x)^20)^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..314

Cf. A122400, A301584, A301585.

{a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, ((1+x +o)^(4*m) - 1)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

G.f.: Sum_{n>=0} (1+x)^(4*n^2) /(1 + (1+x)^(4*n))^(n+1).

a(n) ~ c * d^n * n! / sqrt(n), where d = 4*A317855 = 12.64435461546171525532068881035252996690553109675422536650911283015078823687... and c = 0.31492557816516652573983016205911709623053... - Vaclav Kotesovec, Aug 09 2018

A304642 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^(n+1) - A(x) )^n.

Original entry on oeis.org

1, 2, 2, 10, 112, 1670, 30682, 663606, 16443254, 458349374, 14184612446, 482476888374, 17892738705864, 718662489646314, 31085968593760190, 1441017859748316954, 71281146361450601326, 3748236082140499881942, 208808936226479892694126, 12286084218797404915838902, 761413942238514103243322732, 49577303456014047226843229946, 3383829651598944830489407813422

Offset: 0

Paul D. Hanna, May 16 2018

nonn

			G.f.: A(x) = 1 + 2*x + 2*x^2 + 10*x^3 + 112*x^4 + 1670*x^5 + 30682*x^6 + 663606*x^7 + 16443254*x^8 + 458349374*x^9 + 14184612446*x^10 + 482476888374*x^11 + ...
such that
1 = 1  +  ((1+x)^2 - A(x))  +  ((1+x)^3 - A(x))^2  +  ((1+x)^4 - A(x))^3  +  ((1+x)^5 - A(x))^4  +  ((1+x)^6 - A(x))^5  +  ((1+x)^7 - A(x))^6  +  ((1+x)^8 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)^2/(1 + (1+x)*A(x))^2  +  (1+x)^6/(1 + (1+x)^2*A(x))^3  +  (1+x)^12/(1 + (1+x)^3*A(x))^4  +  (1+x)^20/(1 + (1+x)^4*A(x))^5  +  (1+x)^30/(1 + (1+x)^5*A(x))^6  +  (1+x)^42/(1 + (1+x)^6*A(x))^7 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A303056, A304639.

{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(m+1) - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^(n+1) - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(n*(n+1)) / (1 + (1+x)^n*A(x))^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.16108865386542881383... and c = 0.154769618099522133628... - Vaclav Kotesovec, Oct 14 2020

A227619 G.f.: A(x) = 1+x + Sum_{n>=2} (A(x)^n - 1)^n.

Original entry on oeis.org

1, 1, 4, 63, 1278, 29764, 758065, 20611793, 590579518, 17707907024, 553879330720, 18066513887790, 615744470668778, 22014659625607877, 830262409494773896, 33243718957578687811, 1422095813097928147636, 65311403344808947050730, 3227884786251446164710376

Offset: 0

Paul D. Hanna, Aug 21 2013

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 63*x^3 + 1278*x^4 + 29764*x^5 +...
where
A(x) = 1+x + (A(x)^2 - 1)^2 + (A(x)^3 - 1)^3 + (A(x)^4 - 1)^4 + (A(x)^5 - 1)^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A122400.

{a(n)=local(A=1+x);for(i=1,n,A=1+x+sum(k=2,n,(A^k-1 +x*O(x^n))^k));polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.9913753087... . - Vaclav Kotesovec, May 07 2014

A303057 G.f. A(x) satisfies: A(x) = Sum_{n>=0} ((1+x)^n - 1)^n / A(x)^n.

Original entry on oeis.org

1, 1, 3, 21, 221, 3117, 54597, 1136127, 27293715, 742143113, 22512196673, 753402861159, 27571631761077, 1095346704175755, 46948527167219957, 2159638211148320085, 106129271000784614099, 5549226963359699829711, 307623817602110038648839, 18022345501064909362595723, 1112657716434830018636702797

Offset: 0

Paul D. Hanna, Apr 20 2018

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 221*x^4 + 3117*x^5 + 54597*x^6 + 1136127*x^7 + 27293715*x^8 + 742143113*x^9 + 22512196673*x^10 + ...
such that
A(x) = 1 + ((1+x)-1)/A(x) + ((1+x)^2-1)^2/A(x)^2 + ((1+x)^3-1)^3/A(x)^3 + ((1+x)^4-1)^4/A(x)^4 + ((1+x)^5-1)^5/A(x)^5 + ((1+x)^6-1)^6/A(x)^6 + ...
also,
1 = 1/(A(x) + 1) + (1+x)/(A(x) + (1+x))^2 + (1+x)^4/(A(x) + (1+x)^2)^3 + (1+x)^9/(A(x) + (1+x)^3)^4 + (1+x)^16/(A(x) + (1+x)^4)^5 + (1+x)^25/(A(x) + (1+x)^5)^6 + (1+x)^36/(A(x) + (1+x)^6)^7 + ... + (1+x)^(n^2) / (A(x) + (1+x)^n)^(n+1) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A303058.

{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); A[#A] = Vec(sum(n=0,#A, ((1+x)^n - 1 +x*O(x^#A))^n / Ser(A)^(n+1) ) )[#A] );A[n+1]}
for(n=0,30, print1(a(n),", "))

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^n - 1)^n / A(x)^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) / (A(x) + (1+x)^n)^(n+1). - Paul D. Hanna, Dec 13 2018
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.16108865386542881383... and c = 0.212154215724410476311... - Vaclav Kotesovec, Oct 06 2020

A121886 a(n) = (1/n!)* Sum_{k=0..n} |Stirling1(n,k)|*A122399(k).

Original entry on oeis.org

1, 1, 5, 40, 444, 6324, 110023, 2261576, 53632424, 1441341350, 43290170494, 1437020742408, 52243864528990, 2064488610832106, 88106523694973953, 4038627301344466648, 197888243609535940091, 10321811633042512528240

Offset: 0

Vladeta Jovovic, Aug 31 2006

Number of square matrices with nonnegative integer entries and without zero rows such that sum of all entries is equal to n. - Vladeta Jovovic, Mar 04 2008

			G.f.: A(x) = 1 + x + 5*x^2 + 40*x^3 + 444*x^4 + 6324*x^5 +...
where
A(x) = 1 + (1/(1-x) - 1) + (1/(1-x)^2 - 1)^2 + (1/(1-x)^3 - 1)^3 + ...
Also,
A(x) = 1/2 + (1-x)/(1 + (1-x))^2 + (1-x)^2/(1 + (1-x)^2)^3 +  + (1-x)^3/(1 + (1-x)^3)^4 + (1-x)^4/(1 + (1-x)^4)^5 + ...

Cf. A104209, A220352.

Flatten[{1,Table[1/n!* Sum[Abs[StirlingS1[n,k]]*Sum[m^k*m!*StirlingS2[k, m], {m, 1, k}],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 07 2014 *)

{a(n)=polcoeff(sum(m=0,n,(1/(1-x+x*O(x^n))^m-1)^m),n)}

G.f.: Sum_{n>=0} ( 1/(1-x)^n - 1 )^n.
G.f.: Sum_{n>=0} (1-x)^n / (1 + (1-x)^n)^(n+1). - Paul D. Hanna, Sep 07 2018
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.38377369607518184186200387319561108... . - Vaclav Kotesovec, May 07 2014

More terms from Max Alekseyev, Feb 01 2007

A232192 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^n - 1)^n.

Original entry on oeis.org

1, 1, 1, 5, 44, 519, 7590, 132347, 2689046, 62644234, 1651650774, 48731341965, 1592908456996, 57173688136781, 2235773294509565, 94608603077007214, 4306708055122614542, 209823573154587335730, 10892496561736261641371, 600171728539156939466278

Offset: 0

Paul D. Hanna, Nov 20 2013

nonn

			G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 44*x^4 + 519*x^5 + 7590*x^6 + 132347*x^7 + 2689046*x^8 + 62644234*x^9 + 1651650774*x^10 +...
where
A(x) = 1 + x + x*(A(x)-1) + x*(A(x)^2-1)^2 + x*(A(x)^3-1)^3 + x*(A(x)^4-1)^4 + x*(A(x)^5-1)^5 + x*(A(x)^6-1)^6 + x*(A(x)^7-1)^7 +...
Also,
A(x) = 1 + x/2  +  x*A(x)/(1 + A(x))^2  +  x*A(x)^4/(1 + A(x)^2)^3  +  x*A(x)^9/(1 + A(x)^3)^4  +  x*A(x)^16/(1 + A(x)^4)^5  +  x*A(x)^25/(1 + A(x)^5)^6  + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..175

Cf. A122400, A192945, A192946, A192947, A192948.

{a(n)=local(A=1+x); for(i=1, n, A=1+x*sum(m=0, n, (A^m-1+x*O(x^n))^m)); polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

G.f. satisfies:
(1) A(x) = 1 + x*Sum_{n>=0} A(x)^(n^2) / (1 + A(x)^n)^(n+1). - Paul D. Hanna, Mar 31 2018
(2) A(x) = 1 + Series_Reversion(x/G(x))
(3) A(x) = 1 + x*G(A(x)-1)
where G(x) is the g.f. of A122400, the number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1.
a(n) ~ c * d^n * n! / n^(3/2), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.12140554666... . - Vaclav Kotesovec, May 07 2014

A303654 G.f. A(x) satisfies: x = Sum_{n>=1} ((1+x)^n - 1)^n / A(x)^n.

Original entry on oeis.org

1, 4, 15, 110, 1319, 21626, 440406, 10567338, 289567071, 8881182506, 300677809182, 11123151634732, 446124758009229, 19278179994562787, 892894885467043254, 44130236756271133940, 2318466084626196805383, 129037338117765390406606, 7585054768393048247917776, 469612308887467564648834414

Offset: 0

Paul D. Hanna, May 04 2018

nonn

			G.f.: A(x) = 1 + 4*x + 15*x^2 + 110*x^3 + 1319*x^4 + 21626*x^5 + 440406*x^6 + 10567338*x^7 + 289567071*x^8 + 8881182506*x^9 + ...
such that
x = ((1+x) - 1)/A(x) + ((1+x)^2 - 1)^2/A(x)^2 + ((1+x)^3 - 1)^3/A(x)^3 + ((1+x)^4 - 1)^4/A(x)^4 + ((1+x)^5 - 1)^5/A(x)^5 + ((1+x)^6 - 1)^6/A(x)^6 + ...
Also,
A(x) = (1+x) / ( 1/(1 + A(x)) + (1+x)/((1+x) + A(x))^2 + (1+x)^4/((1+x)^2 + A(x))^3 + (1+x)^9/((1+x)^3 + A(x))^4 + (1+x)^16/((1+x)^4 + A(x))^5 + (1+x)^25/((1+x)^5 + A(x))^6 + (1+x)^36/((1+x)^6 + A(x))^7 + ... ).

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

{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(m=1, #A, ((1+x)^m - 1 +x*O(x^#A))^m / Ser(A)^m ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f.: A(x) = (1+x) / ( Sum_{n>=0} (1+x)^(n^2) / ((1+x)^n + A(x))^(n+1) ).

a(n) ~ c * d^n * n! * sqrt(n), where d = A317855 = 3.161088... and c = 0.292671... - Vaclav Kotesovec, Jun 09 2025

A326010 G.f. A(x) satisfies: 0 = Sum_{n>=1} n * ((1+x)^n - A(x))^n.

Original entry on oeis.org

1, 1, 2, 20, 282, 5134, 112053, 2823119, 80202565, 2529045393, 87523776013, 3295995672161, 134155142687732, 5869278171065418, 274718037952537674, 13701118397652347442, 725505704889894172448, 40658992718689480518864, 2404662897766073643050293, 149692182669205551972626617, 9784886698908632846522031701

Offset: 0

Paul D. Hanna, Jun 04 2019

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 20*x^3 + 282*x^4 + 5134*x^5 + 112053*x^6 + 2823119*x^7 + 80202565*x^8 + 2529045393*x^9 + 87523776013*x^10 + ...
such that
0 = ((1+x) - A(x)) + 2*((1+x)^2 - A(x))^2 + 3*((1+x)^3 - A(x))^3 + 4*((1+x)^4 - A(x))^4 + 5*((1+x)^5 - A(x))^5 + 6*((1+x)^6 - A(x))^6 + ...
The terms a(n) modulo 2 begin:
1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,
0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,1,1,1,1,1,1,0,0,
0,0,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,1,1,
0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,1,1,1,
1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,1,
0,0,0,0,1,1,1,1,0,0,1,1,1,1,1,1,0,0,1,1,1,1,0,0,
1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0,
1,1,0,0,0,0,0,0,0, ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

{a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = polcoeff( sum(m=1,#A, m* ((1+x)^m - Ser(A))^m ), #A-1));A[n+1]}
for(n=0,25,print1(a(n),", "))

G.f. A(x) satisfies:
(1) 0 = Sum_{n>=1} n * ((1+x)^n - A(x))^n.
(2) A(x) = P(x)/Q(x) where
P(x) = Sum_{n>=0} n * (1+x)^(n^2) / (1 + (1+x)^n*A(x))^(n+2),
Q(x) = Sum_{n>=0} (1+x)^(n*(n+1)) / (1 + (1+x)^n*A(x))^(n+2).
(3) A'(x) = P(x)/Q(x) where
P(x) = Sum_{n>=0} (n+1)^3 * ((1+x)^(n+1) - A(x))^n * (1+x)^n,
Q(x) = Sum_{n>=0} (n+1)^2 * ((1+x)^(n+1) - A(x))^n.
a(n) ~ c * d^n * sqrt(n) * n!, where d = A317855 = 3.16108865386... and c = 0.102568345138... - Vaclav Kotesovec, Jun 05 2019

A321089 G.f.: Sum_{n>=0} ((1+x)^(n+1) - 1)^n.

Original entry on oeis.org

1, 2, 10, 82, 928, 13406, 235690, 4883702, 116548222, 3148151702, 94950591878, 3162966582742, 115334767261792, 4569294561813770, 195438629679894238, 8975996556375735458, 440572146080811981406, 23015418712779922737206, 1274980039012724226987966, 74655326188457739033712062, 4607114081638141934903219532, 298862442692043953057588327202

Offset: 0

Paul D. Hanna, Nov 04 2018

nonn

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 82*x^3 + 928*x^4 + 13406*x^5 + 235690*x^6 + 4883702*x^7 + 116548222*x^8 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..375

{a(n) = polcoeff( sum(k=0, n, ((1+x +x*O(x^n))^(k+1) - 1)^k), n)}
for(n=0, 25, print1(a(n), ", "))

/* From e.g.f. infinite series: */
\p200 \\ set precision
{A = Vec(round( sum(n=0, 600, 1./(1 + (1+x +O(x^26))^(-n))^(n+1)) ))}
for(n=0, #A-1, print1(A[n+1], ", "))

Sum_{n>=0} (1+x)^(n*(n+1)) / (1 + (1+x)^n)^(n+1).

a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.1610886538654288138301722... and c = 0.8785394171057422507960514834733179025314463... - Vaclav Kotesovec, Oct 04 2020

cp's OEIS Frontend

A301585 G.f.: Sum_{n>=0} ((1+x)^(3*n) - 1)^n.

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A301586 G.f.: Sum_{n>=0} ((1+x)^(4*n) - 1)^n.

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PARI

Formula

A304642 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^(n+1) - A(x) )^n.

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PARI

Formula

A227619 G.f.: A(x) = 1+x + Sum_{n>=2} (A(x)^n - 1)^n.

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PARI

Formula

A303057 G.f. A(x) satisfies: A(x) = Sum_{n>=0} ((1+x)^n - 1)^n / A(x)^n.

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PARI

Formula

A121886 a(n) = (1/n!)* Sum_{k=0..n} |Stirling1(n,k)|*A122399(k).

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Mathematica

PARI

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Extensions

A232192 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^n - 1)^n.

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PARI

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A303654 G.f. A(x) satisfies: x = Sum_{n>=1} ((1+x)^n - 1)^n / A(x)^n.

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