A303056 -id:A303056 - cp's OEIS frontend

A317855 Decimal expansion of a constant related to the asymptotics of A122400.

3, 1, 6, 1, 0, 8, 8, 6, 5, 3, 8, 6, 5, 4, 2, 8, 8, 1, 3, 8, 3, 0, 1, 7, 2, 2, 0, 2, 5, 8, 8, 1, 3, 2, 4, 9, 1, 7, 2, 6, 3, 8, 2, 7, 7, 4, 1, 8, 8, 5, 5, 6, 3, 4, 1, 6, 2, 7, 2, 7, 8, 2, 0, 7, 5, 3, 7, 6, 9, 7, 0, 5, 9, 2, 1, 9, 3, 0, 4, 6, 1, 1, 2, 1, 9, 7, 5, 7, 4, 6, 8, 5, 4, 9, 7, 8, 4, 5, 9, 3, 2, 4, 2, 2, 7

Offset: 1

Vaclav Kotesovec, Aug 09 2018

			3.161088653865428813830172202588132491726382774188556341627278...

Cf. A121886, A122399, A122400, A122418, A122419, A122420, A227619, A232192, A243802, A244585, A248798, A317340, A326010.

Cf. A301584, A301585, A301586, A303056, A303057, A303654.

r = r /. FindRoot[E^(1/r)/r + (1 + E^(1/r)) * ProductLog[-E^(-1/r)/r] == 0, {r, 3/4}, WorkingPrecision -> 120]; RealDigits[(1 + Exp[1/r])*r^2][[1]]

r=solve(r=.8,1,exp(1/r)/r + (1+exp(1/r))*lambertw(-exp(-1/r)/r))
(1+exp(1/r))*r^2 \\ Charles R Greathouse IV, Jun 15 2021

Equals (1+exp(1/r))*r^2, where r = 0.873702433239668330496568304720719298213992... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0.

A304639 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n.

Original entry on oeis.org

1, 1, 2, 11, 117, 1735, 31853, 689043, 17079221, 476238926, 14742680162, 501584454703, 18605089712174, 747393133162471, 32332767332220442, 1498961537925543920, 74153115616699819304, 3899494667155151052688, 217246028175467702590241, 12783023090792392539557926, 792236994094236725330142276, 51585659784100723438219893047, 3520987513029712770759434038820

Offset: 0

Paul D. Hanna, May 16 2018

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 117*x^4 + 1735*x^5 + 31853*x^6 + 689043*x^7 + 17079221*x^8 + 476238926*x^9 + 14742680162*x^10 + 501584454703*x^11 + ...
is such that
1 = 1 + (1/(1-x) - A(x)) + (1/(1-x)^2 - A(x))^2  + (1/(1-x)^3 - A(x))^3 + (1/(1-x)^4 - A(x))^4 + (1/(1-x)^5 - A(x))^5 + (1/(1-x)^6 - A(x))^6 + (1/(1-x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1-x)/((1-x) + A(x))^2  +  (1-x)^2/((1-x)^2 + A(x))^3  +  (1-x)^3/((1-x)^3  +  A(x))^4 + (1-x)^4/((1-x)^4 + A(x))^5  +  (1-x)^5/((1-x)^5 + A(x))^6  +  (1-x)^6/((1-x)^6 + A(x))^7 + ...
PARTICULAR VALUES.
Although the power series A(x) diverges at x = -1, it may be evaluated formally.
Let t = A(-1) = 0.5452189736359494312349502450349441069576127988881794567242641...
then t satisfies
(1) 1 = Sum_{n>=0} ( 1/2^n - t )^n.
(2) 1 = Sum_{n>=0} ( 1 - 2^n*t )^n / 2^(n^2).
(3) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1).

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

Cf. A326262, A326263, A326264, A326265.

Cf. A303056.

{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, (1/(1-x +x^2*O(x^n))^m - 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/(1-x)^n - A(x) )^n.
(2) 1 = Sum_{n>=0} ( 1 - (1-x)^n*A(x) )^n / (1-x)^(n^2).
(3) 1 = Sum_{n>=0} (1-x)^n / ( (1-x)^n + A(x) )^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.16108865386542881383... and c = 0.16107844724485... - Vaclav Kotesovec, Oct 14 2020

A321602 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(2*n) - A(x))^n.

Original entry on oeis.org

1, 2, 5, 68, 1521, 45328, 1660032, 71548008, 3533826841, 196432984748, 12128132342482, 823366216285428, 60966207548525287, 4890600994792550264, 422601696583826709492, 39142599000082019249968, 3869325702147169825040193, 406650337650126697706078146, 45281361448272561712508294157, 5325916931170845646048163850556, 659842223101960470758187538118437

Offset: 0

Paul D. Hanna, Nov 14 2018

nonn

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 68*x^3 + 1521*x^4 + 45328*x^5 + 1660032*x^6 + 71548008*x^7 + 3533826841*x^8 + 196432984748*x^9 + 12128132342482*x^10 + ...
such that
1 = 1  +  ((1+x)^2 - A(x))  +  ((1+x)^4 - A(x))^2  +  ((1+x)^6 - A(x))^3  +  ((1+x)^8 - A(x))^4  +  ((1+x)^10 - A(x))^5  +  ((1+x)^12 - A(x))^6  +  ((1+x)^14 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)^2/(1 + (1+x)^2*A(x))^2  +  (1+x)^8/(1 + (1+x)^4*A(x))^3  +  (1+x)^18/(1 + (1+x)^6*A(x))^4  +  (1+x)^32/(1 + (1+x)^8*A(x))^5  +  (1+x)^50/(1 + (1+x)^10*A(x))^6  + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 2 + 6*x + 182*x^2 + 5554*x^3 + 211172*x^4 + 9397920*x^5 + 476737830*x^6 + 27086036234*x^7 + 1702330030676*x^8 + ...
the coefficients of which are all even:
(1/2) * A'(x)/A(x) = 1 + 3*x + 91*x^2 + 2777*x^3 + 105586*x^4 + 4698960*x^5 + 238368915*x^6 + 13543018117*x^7 + 851165015338*x^8 + ...

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

Cf. A303056, A321603, A321604, A321605.

Cf. A326262.

{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(2*m) - Ser(A))^m ) )[#A] );H=A; A[n+1]}
for(n=0,30,print1(a(n),", "))

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

A321603 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(3*n) - A(x))^n.

Original entry on oeis.org

1, 3, 12, 235, 7872, 351924, 19340668, 1250971416, 92720438955, 7733929764167, 716488771114410, 72981787493017014, 8107675760704948748, 975749719762368998037, 126491959992115408069503, 17576241581408197850363955, 2606439876885873198662077692, 410925212330248782377865281826, 68641203626673300062880912740755, 12110976733338358608040713750036252

Offset: 0

Paul D. Hanna, Nov 14 2018

nonn

			G.f.: A(x) = 1 + 3*x + 12*x^2 + 235*x^3 + 7872*x^4 + 351924*x^5 + 19340668*x^6 + 1250971416*x^7 + 92720438955*x^8 + 7733929764167*x^9 + ...
such that
1 = 1  +  ((1+x)^3 - A(x))  +  ((1+x)^6 - A(x))^2  +  ((1+x)^9 - A(x))^3  +  ((1+x)^12 - A(x))^4  +  ((1+x)^15 - A(x))^5  +  ((1+x)^18 - A(x))^6  +  ((1+x)^21 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)^3/(1 + (1+x)^3*A(x))^2  +  (1+x)^12/(1 + (1+x)^6*A(x))^3  +  (1+x)^27/(1 + (1+x)^9*A(x))^4  +  (1+x)^48/(1 + (1+x)^12*A(x))^5  +  (1+x)^75/(1 + (1+x)^15*A(x))^6  + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 3 + 15*x + 624*x^2 + 28731*x^3 + 1638798*x^4 + 109462350*x^5 + 8333782509*x^6 + 710574703107*x^7 + 67015908514587*x^8 + ...
the coefficients of which are all divisible by 3:
(1/3) * A'(x)/A(x) = 1 + 5*x + 208*x^2 + 9577*x^3 + 546266*x^4 + 36487450*x^5 + 2777927503*x^6 + 236858234369*x^7 + 22338636171529*x^8 + ...

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

Cf. A303056, A321602, A321604, A321605.

Cf. A326263.

{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(3*m) - Ser(A))^m ) )[#A] );H=A; A[n+1]}
for(n=0,30,print1(a(n),", "))

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

A321604 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(4*n) - A(x))^n.

Original entry on oeis.org

1, 4, 22, 564, 25157, 1499576, 109904860, 9480509576, 937113401201, 104240673195936, 12878161899791760, 1749261564410844864, 259132759251207789056, 41585481940418457992816, 7188476201158569394613976, 1331880173688346226092103696, 263358773243148578509342224153, 55363099822436514905885084770968, 12330972024423209530808891225876436, 2900976547500300324930009436969260936

Offset: 0

Paul D. Hanna, Nov 14 2018

nonn

			G.f.: A(x) = 1 + 4*x + 22*x^2 + 564*x^3 + 25157*x^4 + 1499576*x^5 + 109904860*x^6 + 9480509576*x^7 + 937113401201*x^8 + 104240673195936*x^9 + ...
such that
1 = 1  +  ((1+x)^4 - A(x))  +  ((1+x)^8 - A(x))^2  +  ((1+x)^12 - A(x))^3  +  ((1+x)^16 - A(x))^4  +  ((1+x)^20 - A(x))^5  +  ((1+x)^24 - A(x))^6  +  ((1+x)^28 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)^4/(1 + (1+x)^4*A(x))^2  +  (1+x)^16/(1 + (1+x)^8*A(x))^3  +  (1+x)^36/(1 + (1+x)^12*A(x))^4  +  (1+x)^64/(1 + (1+x)^16*A(x))^5  +  (1+x)^100/(1 + (1+x)^20*A(x))^6  + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 4 + 28*x + 1492*x^2 + 91788*x^3 + 6981484*x^4 + 621939700*x^5 + 63151305340*x^6 + 7181135905380*x^7 + 903210250234696*x^8 + ...
the coefficients of which are all divisible by 4:
(1/4) * A'(x)/A(x) = 1 + 7*x + 373*x^2 + 22947*x^3 + 1745371*x^4 + 155484925*x^5 + 15787826335*x^6 + 1795283976345*x^7 + 225802562558674*x^8 + ...

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

Cf. A303056, A321602, A321603, A321605.

Cf. A326264.

{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(4*m) - Ser(A))^m ) )[#A] );H=A; A[n+1]}
for(n=0,30,print1(a(n),", "))

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

A321605 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(5*n) - A(x))^n.

Original entry on oeis.org

1, 5, 35, 1110, 61830, 4607001, 422112085, 45521033720, 5625206604320, 782244114339935, 120812011501389376, 20514224767917807795, 3798925417133114909240, 762102329400356260363990, 164678708686403817727101920, 38140958485665617437764886383, 9427520984195812306085385378080, 2477372683628569966077893189614835, 689743886246438120027048924784220410

Offset: 0

Paul D. Hanna, Nov 14 2018

nonn

			G.f.: A(x) = 1 + 5*x + 35*x^2 + 1110*x^3 + 61830*x^4 + 4607001*x^5 + 422112085*x^6 + 45521033720*x^7 + 5625206604320*x^8 + 782244114339935*x^9 + ...
such that
1 = 1  +  ((1+x)^5 - A(x))  +  ((1+x)^10 - A(x))^2  +  ((1+x)^15 - A(x))^3  +  ((1+x)^20 - A(x))^4  +  ((1+x)^25 - A(x))^5  +  ((1+x)^30 - A(x))^6  +  ((1+x)^35 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)^5/(1 + (1+x)^5*A(x))^2  +  (1+x)^20/(1 + (1+x)^10*A(x))^3  +  (1+x)^45/(1 + (1+x)^15*A(x))^4  +  (1+x)^80/(1 + (1+x)^20*A(x))^5  +  (1+x)^125/(1 + (1+x)^25*A(x))^6  + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 5 + 45*x + 2930*x^2 + 225545*x^3 + 21445630*x^4 + 2388480630*x^5 + 303204843520*x^6 + 43104182972905*x^7 + 6777636393880895*x^8 + ...
the coefficients of which are all divisible by 5:
(1/5) * A'(x)/A(x) = 1 + 9*x + 586*x^2 + 45109*x^3 + 4289126*x^4 + 477696126*x^5 + 60640968704*x^6 + 8620836594581*x^7 + 1355527278776179*x^8 + ...

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

Cf. A303056, A321602, A321603, A321604.

Cf. A326265.

{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(5*m) - Ser(A))^m ) )[#A] );H=A; A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^(5*n) - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(5*n^2) / (1 + (1+x)^(5*n)*A(x))^(n+1).

A303926 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1 + x*A(x))^n - A(x) )^n.

Original entry on oeis.org

1, 1, 2, 12, 130, 1912, 34715, 743217, 18255118, 505070221, 15532353184, 525533183871, 19403298048040, 776437898905606, 33479679336072541, 1547841068340501230, 76390272348430998076, 4008960603544297652028, 222949077434693015546579, 13098226217965693342007714, 810657425687536689904281842

Offset: 0

Paul D. Hanna, May 03 2018

nonn

Compare to: 1 = Sum_{n>=0} ( (1 + x*G(x))^k - G(x) )^n holds trivially for fixed k>0 when G(x) = (1 + x*G(x))^k ; this sequence explores the case when k varies with n.

			G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 130*x^4 + 1912*x^5 + 34715*x^6 + 743217*x^7 + 18255118*x^8 + 505070221*x^9 + 15532353184*x^10 + ...
such that
1 = 1 + ((1 + x*A(x)) - A(x)) + ((1 + x*A(x))^2 - A(x))^2 + ((1 + x*A(x))^3 - A(x))^3 + ((1 + x*A(x))^4 - A(x))^4 + ((1 + x*A(x))^5 - A(x))^5 + ...

Cf. A303927, A303928, A303923, A303056.

{a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ( (1 + x*Ser(A))^m - 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*A(x))^n - A(x) )^n.
(2) 1 = Sum_{n>=0} (1 + x*A(x))^(n^2) / (1 + A(x)*(1 + x*A(x))^n)^(n+1). - Paul D. Hanna, Dec 06 2018
G.f.: x/Series_Reversion( x*F(x) ) such that 1 = Sum_{n>=0} ((1 + x*F(x)^2)^n - F(x))^n, where F(x) is the g.f. of A303927.
G.f.: sqrt( x/Series_Reversion( x*G(x)^2 ) ) such that 1 = Sum_{n>=0} ((1 + x*G(x)^3)^n - G(x))^n, where G(x) is the g.f. of A303928.

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

A326282 G.f. A(x) satisfies: 1 = Sum_{n>=0} 2^n * ((1+x)^n - A(x))^n.

Original entry on oeis.org

1, 1, 2, 28, 586, 16336, 559164, 22519620, 1039209116, 53968031108, 3112841732920, 197413519635632, 13654508980460736, 1023144120035225664, 82581014079320743504, 7144332294806845079568, 659630258631919908187784, 64748755209330058463666656, 6733915902264715745675338784, 739732094650896407811045989408, 85594689069528757090534336595600

Offset: 0

Paul D. Hanna, Jun 22 2019

nonn

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),
for any fixed integer k; here, k = 1 with r = 2, p = -A(x), q = (1+x).

			G.f.: A(x) = 1 + x + 2*x^2 + 28*x^3 + 586*x^4 + 16336*x^5 + 559164*x^6 + 22519620*x^7 + 1039209116*x^8 + 53968031108*x^9 + 3112841732920*x^10 + ...
such that
1 = 1  +  2*((1+x) - A(x))  +  2^2*((1+x)^2 - A(x))^2  +  2^3*((1+x)^3 - A(x))^3  +  2^4*((1+x)^4 - A(x))^4  +  2^5*((1+x)^5 - A(x))^5  +  2^6*((1+x)^6 - A(x))^6  +  2^7*((1+x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + 2*A(x))  +  2*(1+x)/(1 + 2*(1+x)*A(x))^2  +  2^2*(1+x)^4/(1 + 2*(1+x)^2*A(x))^3  +  2^3*(1+x)^9/(1 + 2*(1+x)^3*A(x))^4  +  2^4*(1+x)^16/(1 + 2*(1+x)^4*A(x))^5  +  2^5*(1+x)^25/(1 + 2*(1+x)^5*A(x))^6  +  2^6*(1+x)^36/(1 + 2*(1+x)^6*A(x))^7 + ...

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

Cf. A303056, A326283, A326284.

{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, 2^m*((1+x)^m - Ser(A))^m ) )[#A]/2 ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} 2^n * ((1+x)^n - A(x))^n.
(2) 1 = Sum_{n>=0} 2^n * (1+x)^(n^2) / (1 + 2*(1+x)^n*A(x))^(n+1).
a(n) ~ c * (1 + 2*exp(1/r))^n * r^(2*n) * n! / sqrt(n), where r = 0.925556278640887084941460444526398190071550948416... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/2 and c = 0.06492129653731... - Vaclav Kotesovec, Oct 13 2020

A326283 G.f. A(x) satisfies: 1 = Sum_{n>=0} 3^n * ((1+x)^n - A(x))^n.

Original entry on oeis.org

1, 1, 3, 60, 1839, 75006, 3756681, 221373108, 14946242445, 1135512643905, 95807726020377, 8887382719047009, 899076658872459384, 98527185014966618211, 11629941100277822632476, 1471364340776928876041874, 198658352816171170123764339, 28515062757712810453709971653, 4336489052405032386983807856510, 696570875522773511632334350536870

Offset: 0

Paul D. Hanna, Jun 22 2019

nonn

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),
for any fixed integer k; here, k = 1 with r = 3, p = -A(x), q = (1+x).

			G.f.: A(x) = 1 + x + 3*x^2 + 60*x^3 + 1839*x^4 + 75006*x^5 + 3756681*x^6 + 221373108*x^7 + 14946242445*x^8 + 1135512643905*x^9 + 95807726020377*x^10 + ...
such that
1 = 1  +  3*((1+x) - A(x))  +  3^2*((1+x)^2 - A(x))^2  +  3^3*((1+x)^3 - A(x))^3  +  3^4*((1+x)^4 - A(x))^4  +  3^5*((1+x)^5 - A(x))^5  +  3^6*((1+x)^6 - A(x))^6  +  3^7*((1+x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + 3*A(x))  +  3*(1+x)/(1 + 3*(1+x)*A(x))^2  +  3^2*(1+x)^4/(1 + 3*(1+x)^2*A(x))^3  +  3^3*(1+x)^9/(1 + 3*(1+x)^3*A(x))^4  +  3^4*(1+x)^16/(1 + 3*(1+x)^4*A(x))^5  +  3^5*(1+x)^25/(1 + 3*(1+x)^5*A(x))^6  +  3^6*(1+x)^36/(1 + 3*(1+x)^6*A(x))^7 + ...

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

Cf. A303056, A326282, A326284.

{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, 3^m*((1+x)^m - Ser(A))^m ) )[#A]/3 ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} 3^n * ((1+x)^n - A(x))^n.
(2) 1 = Sum_{n>=0} 3^n * (1+x)^(n^2) / (1 + 3*(1+x)^n*A(x))^(n+1).
a(n) ~ c * (1 + 3*exp(1/r))^n * r^(2*n) * n! / sqrt(n), where r = 0.947093169766093813913446822751643203941993193936... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/3 and c = 0.04495001143628... - Vaclav Kotesovec, Oct 13 2020

cp's OEIS Frontend

A317855 Decimal expansion of a constant related to the asymptotics of A122400.

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A304639 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n.

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A321602 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(2*n) - A(x))^n.

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A321603 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(3*n) - A(x))^n.

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A321604 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(4*n) - A(x))^n.

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A321605 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(5*n) - A(x))^n.

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A303926 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1 + x*A(x))^n - A(x) )^n.

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

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