Paul D. Hanna - cp's OEIS frontend

A386659 G.f. A(x) satisfies A(x^3) = A(x)^3/(1 + 3*A(x)).

1, 1, 0, 0, 1, 0, -1, 1, 0, -2, 3, 0, -6, 10, 0, -22, 33, 0, -79, 122, 0, -299, 472, 0, -1179, 1871, 0, -4754, 7601, 0, -19553, 31449, 0, -81720, 132020, 0, -345949, 561034, 0, -1480475, 2408712, 0, -6394189, 10431950, 0, -27835400, 45521500, 0, -122008360, 199948108, 0, -538016031, 883331845, 0

Offset: 1

Paul D. Hanna, Aug 28 2025

Compare to: F(x^2) = F(x)^2/(1 + 2*F(x)) holds when F(x) = x/(1-x).

			G.f.: A(x) = x + x^2 + x^5 - x^7 + x^8 - 2*x^10 + 3*x^11 - 6*x^13 + 10*x^14 - 22*x^16 + 33*x^17 - 79*x^19 + 122*x^20 - 299*x^22 + 472*x^23 - 1179*x^25 + 1871*x^26 - 4754*x^28 + ...
where A(x^3) = A(x)^3/(1 + 3*A(x)).
RELATED SERIES.
The series trisections are A(x) = T1(x) + T2(x) + T3(x), with T3(x) = 0 and
T1(x) = x - x^7 - 2*x^10 - 6*x^13 - 22*x^16 - 79*x^19 - 299*x^22 - 1179*x^25 - 4754*x^28 - 19553*x^31 - 81720*x^34 - 345949*x^37 - 1480475*x^40 + ...
T2(x) = x^2 + x^5 + x^8 + 3*x^11 + 10*x^14 + 33*x^17 + 122*x^20 + 472*x^23 + 1871*x^26 + 7601*x^29 + 31449*x^32 + 132020*x^35 + 561034*x^38 + 2408712*x^41 + ...
where T1(x)*T2(x) = A(x^3) and
T2(x)/T1(x) = x + x^4 + 2*x^7 + 6*x^10 + 20*x^13 + 71*x^16 + 267*x^19 + 1041*x^22 + 4168*x^25 + 17047*x^28 + ... + A370446(n)*x^(3*n-2) + ...
The cube of A(x) also has interesting series trisections.
A(x)^3 = x^3 + 3*x^4 + 3*x^5 + x^6 + 3*x^7 + 6*x^8 - 3*x^10 + 6*x^11 - 9*x^13 + 12*x^14 + x^15 - 21*x^16 + 42*x^17 - 84*x^19 + 132*x^20 - x^21 - 309*x^22 + 465*x^23 + x^24 + ...
where cubic trisections, defined by A(x)^3 = C1(x) + C2(x) + C3(x), obey
C3(x) = A(x^3),
C1(x)*C2(x) = 9*A(x^3)^3,
C2(x)/C1(x) = T2(x)/T1(x) = x + x^4 + 2*x^7 + 6*x^10 + 20*x^13 + 71*x^16 + 267*x^19 + 1041*x^22 + ... + A370446(n)*x^(3*n-2) + ...
The cubic trisections begin
C1(x) = 3*x^4 + 3*x^7 - 3*x^10 - 9*x^13 - 21*x^16 - 84*x^19 - 309*x^22 - 1137*x^25 - 4449*x^28 - 17868*x^31 - 73137*x^34 - 304662*x^37 - 1286388*x^40 - ...
C2(x) = 3*x^5 + 6*x^8 + 6*x^11 + 12*x^14 + 42*x^17 + 132*x^20 + 465*x^23 + 1791*x^26 + 7059*x^29 + 28503*x^32 + 117498*x^35 + 491757*x^38 + 2084481*x^41 + ...
C3(x) = x^3 + x^6 + x^15 - x^21 + x^24 - 2*x^30 + 3*x^33 - 6*x^39 + 10*x^42 - 22*x^48 + 33*x^51 + ... + a(n)*x^(3*n) + ...
SPECIFIC VALUES.
A(r) = 1 and A(r^3) = 1/4 at r = 0.591403538949431343296352603332310036448543950513103383318429...
A(t) = 4/5 and A(t^3) = 64/425 at t = 0.510303761967726164722767738473741580674762344121899...
A(t) = 3/4 and A(t^3) = 27/208 at t = 0.488075704869119285515484767956113771965332978558674...
A(t) = 2/3 and A(t^3) = 8/81 at t = 0.4490656139430636435247188510711544862057647445925319...
A(t) = 1/2 and A(t^3) = 1/20 at t = 0.3627219904933172573963798296372201737748692616169519...
A(t) = 1/3 and A(t^3) = 1/54 at t = 0.2629820536068200748031820994203659473004640287705972...
A(t) = 1/4 and A(t^3) = 1/112 at t = r^3 = 0.206848205250953970652722994332475597057157203674066...
A(t) = 1/5 and A(t^3) = 1/200 at t = 0.170714946526968286919515308872119424149511936479752...
A(1/2) = 0.7765855959847885627987696942587081429921785817514493... where A(1/8) = A(1/2)^3/(1 + 3*A(1/2)).
A(1/3) = 0.4482359377100401660271468423571796863698018480508060... where A(1/27) = A(1/3)^3/(1 + 3*A(1/3)).
A(1/4) = 0.3134295384970268001359461486249333443235800254018265... where A(1/64) = A(1/4)^3/(1 + 3*A(1/4)).
A(1/8) = 0.1406550988235082384593126468031209848166962450443705...
A(1/27) = 0.038408848749171730717291402355749106248762924579924...
A(1/64) = 0.015869141556098751959628853939856842544839850661716...

Paul D. Hanna, Table of n, a(n) for n = 1..2200

Cf. A370446, A264228.

{a(n) = my(V=[0,1]); for(i=1,n, V = concat(V,0); A = Ser(V);
V[#V] = polcoef( subst(A,x, x^3) - A^3/(1 + 3*A), #V+1)/3; ); V[n+1] }
for(n=1,54,print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n along with series trisections T1(x) = Sum_{n>=0} a(3*n+1)*x^(3*n+1) and T2(x) = Sum_{n>=0} a(3*n+2)*x^(3*n+2) satisfy the following formulas.
(1) A(x^3) = A(x)^3/(1 + 3*A(x)).
(2) a(3*n) = 0 for n >= 1.
(3) T1(x)*T2(x) = A(x^3).
(4) T2(x)/T1(x) = G(x^3)/x^2 where g.f. G(x) of A370446 satisfies G(x)^3 + x^4/G(x)^3 = G(x^3) + x^4/G(x^3) - 3*x^2.
(5) A(-F(-x)) = x where g.f. F(x) of A264228 satisfies F(x)^3 = F( x^3/(1-3*x) ).

A386665 G.f. satisfies: A(x) = Sum_{n>=0} ( (1+x)^n - A(x)^(-1/2) )^n / ( 2 - (1+x)^n * A(x)^(-1/2) )^(n+1).

Original entry on oeis.org

1, 1, 8, 90, 1336, 24406, 530234, 13410942, 388841734, 12762735148, 469004980720, 19105730068460, 855146084504046, 41724450644602328, 2204075802189470532, 125300401263988607716, 7626356269363721248332, 494723229572772238087966, 34070289390944902842701094, 2482276670026891882801017812

Offset: 0

Paul D. Hanna, Aug 29 2025

It appears that lim_{n->oo} ( a(n+1)/a(n) )/(n+1) exists and is near 4.

			G.f.: A(x) = 1 + x + 8*x^2 + 90*x^3 + 1336*x^4 + 24406*x^5 + 530234*x^6 + 13410942*x^7 + 388841734*x^8 + 12762735148*x^9 + 469004980720*x^10 + ...
RELATED SERIES.
A(x)^(1/2) = 1 + 2*(x/4) + 62*(x/4)^2 + 2756*(x/4)^3 + 163574*(x/4)^4 + 11997852*(x/4)^5 + 1047984172*(x/4)^6 + 106571791752*(x/4)^7 + 12417003030694*(x/4)^8 + ...
A(x)^(-1/2) = 1 - 2*(x/4) - 58*(x/4)^2 - 2516*(x/4)^3 - 149434*(x/4)^4 - 11055996*(x/4)^5 - 976190180*(x/4)^6 - 100318703592*(x/4)^7 - 11796814729146*(x/4)^8 - ...

Paul D. Hanna, Table of n, a(n) for n = 0..250

Cf. A322735, A317350, A322737.

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

G.f. A(x) = Sum_{n>=0} a(n)*x^n along with B(x) = A(x)^(1/2) satisfies the following formulas.
(1) A(x) = Sum_{n>=0} ( (1+x)^n - 1/B(x) )^n / ( 2 - (1+x)^n/B(x) )^(n+1).
(2) A(x) = Sum_{n>=0} ( (1+x)^n + 1/B(x) )^n / ( 2 + (1+x)^n/B(x) )^(n+1).
(3) B(x) = Sum_{n>=0} ( (1+x)^n*B(x) - 1 )^n / ( 2*B(x) - (1+x)^n )^(n+1).
(4) B(x) = Sum_{n>=0} ( (1+x)^n*B(x) + 1 )^n / ( 2*B(x) + (1+x)^n )^(n+1).

A386656 E.g.f.: Sum_{n>=0} (3^n*x + LambertW(x))^n / n!.

Original entry on oeis.org

1, 4, 98, 21901, 45203076, 864855654349, 151334120052647134, 240066304912259832915171, 3437872829353908000927273009224, 443629285010311848968435132228644809721, 515464807017361539745514781011221080738833641050

Offset: 0

Paul D. Hanna, Aug 23 2025

Conjecture: for n >= 6, a(n) (mod 6) equals [4, 3, 2, 1, 0, 1] repeating.
In general, the following sums are equal:
(C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
(C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = 3 with p = LambertW(x)/x, r = x.

			E.g.f.: A(x) = 1 + 4*x + 98*x^2/2! + 21901*x^3/3! + 45203076*x^4/4! + 864855654349*x^5/5! + 151334120052647134*x^6/6! + ...
where A(x) = Sum_{n>=0} (3^n*x + LambertW(x))^n / n!.
RELATED SERIES.
LambertW(x) = x - 2*x^2/2! + 3^2*x^3/3! - 4^3*x^4/4! + 5^4*x^5/5! - 6^5*x^6/6! + 7^6*x^7/7! + ... + (-1)^(n-1) * n^(n-1)*x^n/n! + ...
where exp(LambertW(x)) = x/LambertW(x);
also, (x/LambertW(x))^y = Sum_{k>=0} y*(y - k)^(k-1) * x^k/k!.

Cf. A386655 (q=2), A386657 (q=4), A386658 (q=5), A386648.

{a(n) = sum(k=0,n, binomial(n,k) * 3^(k*(k+1)) * (3^k - (n-k))^(n-k-1) )}
for(n=0, 12, print1(a(n), ", "))

{a(n) = my(A = sum(m=0, n, (3^m + lambertw(x +x^3*O(x^n))/x)^m *x^m/m! )+x*O(x^n)); n! * polcoeff(A, n)}
for(n=0, 12, print1(a(n), ", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = Sum_{n>=0} (3^n*x + LambertW(x))^n / n!.
(2) A(x) = Sum_{n>=0} 3^(n^2) * exp( LambertW(x) * 3^n ) * x^n / n!.
(3) A(x) = Sum_{n>=0} 3^(n^2) * (x/LambertW(x))^(3^n) * x^n / n!.
(4) A(x) = Sum_{n>=0} 3^(n*(n+1)) * x^n/n! * Sum_{k>=0} (3^n - k)^(k-1) * x^k/k!.
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(k*(k+1)) * (3^k - (n-k))^(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n*k) * (1 - (n-k)/3^k)^(n-k-1).

A386657 E.g.f.: Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.

Original entry on oeis.org

1, 5, 287, 274532, 4362420261, 1131407873777920, 4729288202285254702123, 317048074495318899943286044736, 340323907513179399929311813628104334217, 5846207259092593125133941613189798019292422881280

Offset: 0

Paul D. Hanna, Aug 23 2025

Conjecture: for n >= 6, a(n) (mod 6) equals [1, 2, 3, 2, 3, 4] repeating.
In general, the following sums are equal:
(C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
(C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = 4 with p = LambertW(x)/x, r = x.

			E.g.f.: A(x) = 1 + 5*x + 287*x^2/2! + 274532*x^3/3! + 4362420261*x^4/4! + 1131407873777920*x^5/5! + 4729288202285254702123*x^6/6! + ...
where A(x) = Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.
RELATED SERIES.
LambertW(x) = x - 2*x^2/2! + 3^2*x^3/3! - 4^3*x^4/4! + 5^4*x^5/5! - 6^5*x^6/6! + 7^6*x^7/7! + ... + (-1)^(n-1) * n^(n-1)*x^n/n! + ...
where exp(LambertW(x)) = x/LambertW(x);
also, (x/LambertW(x))^y = Sum_{k>=0} y*(y - k)^(k-1) * x^k/k!.

Cf. A386655 (q=2), A386656 (q=3), A386658 (q=5), A386648.

{a(n) = sum(k=0,n, binomial(n,k) * 4^(k*(k+1)) * (4^k - (n-k))^(n-k-1) )}
for(n=0, 12, print1(a(n), ", "))

{a(n) = my(A = sum(m=0, n, (4^m + lambertw(x +x^3*O(x^n))/x)^m *x^m/m! )+x*O(x^n)); n! * polcoeff(A, n)}
for(n=0, 12, print1(a(n), ", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.
(2) A(x) = Sum_{n>=0} 4^(n^2) * exp( LambertW(x) * 4^n ) * x^n / n!.
(3) A(x) = Sum_{n>=0} 4^(n^2) * (x/LambertW(x))^(4^n) * x^n / n!.
(4) A(x) = Sum_{n>=0} 4^(n*(n+1)) * x^n/n! * Sum_{k>=0} (4^n - k)^(k-1) * x^k/k!.
a(n) = Sum_{k=0..n} binomial(n,k) * 4^(k*(k+1)) * (4^k - (n-k))^(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * 4^(n*k) * (1 - (n-k)/4^k)^(n-k-1).

A386658 E.g.f.: Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.

Original entry on oeis.org

1, 6, 674, 2000229, 153566609748, 298500361403750381, 14557504055095871311168750, 17765160070810827062009088144577731, 542112188572462226990932242595876785196798632, 413592212104548192173492724488185195719396124921931347641

Offset: 0

Paul D. Hanna, Aug 23 2025

Conjecture: for n >= 6, a(n) (mod 6) equals [4, 3, 0, 3, 0, 5] repeating.
In general, the following sums are equal:
(C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
(C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = 5 with p = LambertW(x)/x, r = x.

			E.g.f.: A(x) = 1 + 6*x + 674*x^2/2! + 2000229*x^3/3! + 153566609748*x^4/4! + 298500361403750381*x^5/5! + 14557504055095871311168750*x^6/6! + ...
where A(x) = Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.
RELATED SERIES.
LambertW(x) = x - 2*x^2/2! + 3^2*x^3/3! - 4^3*x^4/4! + 5^4*x^5/5! - 6^5*x^6/6! + 7^6*x^7/7! + ... + (-1)^(n-1) * n^(n-1)*x^n/n! + ...
where exp(LambertW(x)) = x/LambertW(x);
also, (x/LambertW(x))^y = Sum_{k>=0} y*(y - k)^(k-1) * x^k/k!.

Cf. A386655 (q=2), A386656 (q=3), A386657 (q=4), A386648.

{a(n,q=5) = sum(k=0,n, binomial(n,k) * q^(k*(k+1)) * (q^k - (n-k))^(n-k-1) )}
for(n=0, 12, print1(a(n), ", "))

{a(n,q=5) = my(A = sum(m=0, n, (q^m + lambertw(x +x^3*O(x^n))/x)^m *x^m/m! )+x*O(x^n)); n! * polcoeff(A, n)}
for(n=0, 12, print1(a(n), ", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.
(2) A(x) = Sum_{n>=0} 5^(n^2) * exp( LambertW(x) * 5^n ) * x^n / n!.
(3) A(x) = Sum_{n>=0} 5^(n^2) * (x/LambertW(x))^(5^n) * x^n / n!.
(4) A(x) = Sum_{n>=0} 5^(n*(n+1)) * x^n/n! * Sum_{k>=0} (5^n - k)^(k-1) * x^k/k!.
a(n) = Sum_{k=0..n} binomial(n,k) * 5^(k*(k+1)) * (5^k - (n-k))^(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * 5^(n*k) * (1 - (n-k)/5^k)^(n-k-1).

A386655 E.g.f.: Sum_{n>=0} (2^n*x + LambertW(x))^n / n!.

Original entry on oeis.org

1, 3, 23, 708, 82677, 39043808, 75384175459, 594418947869568, 19030890530555146281, 2460681168464503636482816, 1280084112577610436036966382815, 2672769582069469500760580570122074560, 22366167041278673568399013569832022272725469

Offset: 0

Paul D. Hanna, Aug 23 2025

Conjecture: for n >= 6, a(n) (mod 6) equals [1, 0, 3, 0, 3, 2] repeating.
In general, the following sums are equal:
(C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
(C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = 2 with p = LambertW(x)/x, r = x.

			E.g.f.: A(x) = 1 + 3*x + 23*x^2/2! + 708*x^3/3! + 82677*x^4/4! + 39043808*x^5/5! + 75384175459*x^6/6! + 594418947869568*x^7/7! + ...
where A(x) = Sum_{n>=0} (2^n*x + LambertW(x))^n / n!.
RELATED SERIES.
LambertW(x) = x - 2*x^2/2! + 3^2*x^3/3! - 4^3*x^4/4! + 5^4*x^5/5! - 6^5*x^6/6! + 7^6*x^7/7! + ... + (-1)^(n-1) * n^(n-1)*x^n/n! + ...
where exp(LambertW(x)) = x/LambertW(x);
also, (x/LambertW(x))^y = Sum_{k>=0} y*(y - k)^(k-1) * x^k/k!.

Cf. A386656 (q=3), A386657 (q=4), A386658 (q=5), A386648.

nmax = 15; Join[{1}, Rest[CoefficientList[Series[Sum[(2^k*x + LambertW[x])^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]] (* Vaclav Kotesovec, Aug 23 2025 *)

{a(n) = sum(k=0,n, binomial(n,k) * 2^(k*(k+1)) * (2^k - (n-k))^(n-k-1) )}
for(n=0, 12, print1(a(n), ", "))

{a(n) = my(A = sum(m=0, n, (2^m + lambertw(x +x^3*O(x^n))/x)^m *x^m/m! )+x*O(x^n)); n! * polcoeff(A, n)}
for(n=0, 12, print1(a(n), ", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = Sum_{n>=0} (2^n*x + LambertW(x))^n / n!.
(2) A(x) = Sum_{n>=0} 2^(n^2) * exp( LambertW(x) * 2^n ) * x^n / n!.
(3) A(x) = Sum_{n>=0} 2^(n^2) * (x/LambertW(x))^(2^n) * x^n / n!.
(4) A(x) = Sum_{n>=0} 2^(n*(n+1)) * x^n/n! * Sum_{k>=0} (2^n - k)^(k-1) * x^k/k!.
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(k*(k+1)) * (2^k - (n-k))^(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n*k) * (1 - (n-k)/2^k)^(n-k-1).
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Aug 23 2025

A385915 G.f. satisfies A(x) = -(A(x^3) + A(x^4)) / A(-x^2).

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 2, 2, 4, 7, 4, 11, 4, 12, 8, 19, 14, 29, 23, 47, 32, 74, 44, 110, 83, 164, 135, 276, 196, 439, 304, 663, 489, 1051, 768, 1656, 1192, 2581, 1856, 4046, 2888, 6317, 4547, 9848, 7130, 15440, 11106, 24186, 17377, 37729, 27231, 59062, 42614, 92484, 66682, 144664, 104328, 226371, 163174, 354230

Offset: 1

Paul D. Hanna, Aug 21 2025

a(n) ~ c*d^n, where d = 1.25088706673007476636567388431275493535326837186841972110953..., and c = 0.52020182784673907154955829274303103782236665499908622597516... if n is even, or c = 0.29984063427406787235208627225075145443391314443990234248956... if n is odd.

Radius of convergence r of g.f. A(x) satisfies A(-r^2) = 0 and A(r^8) = -A(-r^6) where r = 0.79943267989338565357086513413379878916201504254400772696808...

			G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 2*x^6 + 2*x^7 + 2*x^8 + 4*x^9 + 7*x^10 + 4*x^11 + 11*x^12 + 4*x^13 + 12*x^14 + 8*x^15 + 19*x^16 + 14*x^17 + ...
where A(x) = -(A(x^3) + A(x^4)) / A(-x^2).
RELATED SERIES.
A(x^3) + A(x^4) = x^3 + x^4 + x^6 + x^8 + x^9 + 3*x^12 + 2*x^16 + 2*x^18 + 2*x^21 + 4*x^24 + 4*x^27 + 2*x^28 + 7*x^30 + 2*x^32 + 4*x^33 + ...
where A(x^3) + A(x^4) = -A(x)*A(-x^2).
Let B(x) satisfy A(B(x)) = x then
B(x) = x - x^2 + x^3 - 2*x^4 + 8*x^5 - 30*x^6 + 96*x^7 - 293*x^8 + 945*x^9 - 3274*x^10 + 11679*x^11 - 41637*x^12 + 148232*x^13 - 531931*x^14 + 1932116*x^15 + ...
where -x*A(-B(x)^2) = A(B(x)^3) + A(B(x)^4).
SPECIFIC VALUES.
The radius of convergence r satisfies A(-r^2) = 0 and A(r^8) = -A(-r^6) (see values given below).
Also, -A(r^2)*A(-r^4) = A(r^6) - A(-r^6) (verify using values given below).
Pertinent values of the form A(+-r^n) are as follows.
A(r^2) = 2.25568357277545334879615953002039818218094300680461...
A(r^3) = 1.13146138072432881698300088743197078335065492774082...
A(r^4) = 0.71641352894347364220461475600253964520851864903599...
A(r^6) = 0.35711522301431469636315144464034804067241658443788...
A(r^8) = 0.20089620831732266634625988585698697217669751459458...
A(r^9) = 0.15416703344989606171813799820914068395259643759003...
A(r^12) = 0.07313926779510423315813760034344399834214660198599...
A(-r^3) = -0.21985397321719823661610779049573983213878785101569...
A(-r^4) = -0.24738019022989345323378959578383752763617413768043...
A(-r^6) = -0.20089620831732266634625988585698697217669751459458...
A(-r^8) = -0.14204971726158381650811456142497985591356688167501...
A(-r^9) = -0.11730709739793756826670096033821084565818297459544...
A(-r^12) = -0.06376735751196287607391908207491894730221727143981...
...
A(1/2) = 1.076088418495368709480408335347599544142613316488183... where A(1/2) = -(A(1/8) + A(1/16)) / A(-1/4).
A(1/3) = 0.510506935002162613788658566406685686749792809989670... where A(1/3) = -(A(1/27) + A(1/81)) / A(-1/9).
A(1/4) = 0.336602030533756099597633504337162753515731187520108... where A(1/4) = -(A(1/64) + A(1/256)) / A(-1/16).
A(1/8) = 0.143075145485757320815392125895338831436230491529773...
A(1/16) = 0.066681035393498075099284704625662933959506411882465...
A(1/27) = 0.038463353126979968975084607088546188041484601759398...
A(1/64) = 0.015873074561121087188918170893728977022238322582751...
A(1/81) = 0.0125000229473609050663549233903594151185002805142699...
A(1/256) = 0.003921568859375695990570527967100959643728275680027...
A(-1/4) = -0.194924670941580390863804372991458359617278204823944...
A(-1/9) = -0.099828959373735016072583725604101146855506666073829...
A(-1/16) = -0.058807260874534978256299853660358029251607884417895...
...

Paul D. Hanna, Table of n, a(n) for n = 1..4100

Cf. A385908.

{a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1, #binary(n+1), A = -(subst(A, x, x^3) + subst(A, x, x^4))/subst(A, x, -x^2) +x*O(x^n); ); polcoef(H=A, n)}
for(n=1,100, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = -(A(x^3) + A(x^4)) / A(-x^2).
(2) -A(-x^2) = (A(x^3) - A(-x^3)) / (A(x) - A(-x)).

A385057 E.g.f. satisfies A(x) = exp( Sum_{n>=1} (Integral A(x)^n dx)^n / n ).

Original entry on oeis.org

1, 1, 3, 18, 173, 2368, 43025, 991070, 28030227, 950818494, 37995695979, 1763496545502, 93967776822477, 5692538342703978, 388772833646583213, 29711642817587338986, 2524166742181661207511, 236956380718244960455206, 24446253183753019240769463, 2757979540962272093582650734, 338712272097534292284500861745

Offset: 0

Paul D. Hanna, Aug 19 2025

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 18*x^3/3! + 173*x^4/4! + 2368*x^5/5! + 43025*x^6/6! + 991070*x^7/7! + 28030227*x^8/8! + 950818494*x^9/9! + 37995695979*x^10/10! + ...
where
A(x) = exp( (Integral A(x) dx) + (Integral A(x)^2 dx)^2/2 + (Integral A(x)^3 dx)^3/3 + (Integral A(x)^4 dx)^4/4 + ... ).
Also,
A'(x) = A(x)^2 + A(x)^3*(Integral A(x)^2 dx) + A(x)^4*(Integral A(x)^3 dx)^2 + A(x)^5*(Integral A(x)^4 dx)^3 + ...
RELATED SERIES.
log(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 104*x^4/4! + 1437*x^5/5! + 26642*x^6/6! + 629127*x^7/7! + ... + A268294(n)*x^n/n! + ...

Paul D. Hanna, Table of n, a(n) for n = 0..201

Cf. A268294 (log).

{a(n) = my(A = 1 + x +x*O(x^n)); for(i=0, n+1, A = exp( sum(m=1, n+1, intformal(A^m)^m/m ) ) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = exp( Sum_{n>=1} (Integral A(x)^n dx)^n / n ).
(2) A'(x) = Sum_{n>=1} A(x)^(n+1) * (Integral A(x)^n dx)^(n-1).
(3) A(x) = exp(B(x)), where B(x) is the e.g.f. of A268294.

A387041 G.f. A(x) satisfies (A(x) - x^2) o (x - A(x)^2) = x, where operator 'o' denotes composition.

Original entry on oeis.org

1, 2, 6, 41, 348, 3360, 35632, 406104, 4904914, 62180918, 821752456, 11263836924, 159523476148, 2327336091732, 34894961587312, 536671299862721, 8453184479505430, 136188177741639378, 2241801065131393700, 37670062720274627960, 645649822816127973456, 11279877783091509190416

Offset: 1

Paul D. Hanna, Aug 14 2025

nonn

			G.f.: A(x) = x + 2*x^2 + 6*x^3 + 41*x^4 + 348*x^5 + 3360*x^6 + 35632*x^7 + 406104*x^8 + 4904914*x^9 + 62180918*x^10 + ...
where A(x) - x^2 = x + A(A(x) - x^2)^2;
also, A(x - A(x)^2) = x + (x - A(x)^2)^2 = x + x^2 - 2*x*A(x)^2 + A(x)^4.
RELATED SERIES.
A(A(x) - x^2) = x + 3*x^2 + 16*x^3 + 126*x^4 + 1174*x^5 + 12278*x^6 + 139496*x^7 + 1689597*x^8 + 21553566*x^9 + 287191110*x^10 + ...
A(x - A(x)^2) = x + x^2 - 2*x^3 - 7*x^4 - 24*x^5 - 164*x^6 - 1452*x^7 - 14312*x^8 - 153354*x^9 - 1757322*x^10 - ...
A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 106*x^5 + 896*x^6 + 8604*x^7 + 90561*x^8 + 1023592*x^9 + 12258452*x^10 + ...
A(x)^4 = x^4 + 8*x^5 + 48*x^6 + 340*x^7 + 2896*x^8 + 27768*x^9 + 289862*x^10 + ...

Paul D. Hanna, Table of n, a(n) for n = 1..520

{a(n) = my(V=[0,1]); for(i=1,n, V = concat(V,0); A = Ser(V);
V[#V] = polcoef(x + (x - A^2)^2 - subst(A,x, x - A^2),#V-1)); polcoef(A,n)}
for(n=1,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = x + x^2 + A(A(x) - x^2)^2.
(2) A(x - A(x)^2) = x + (x - A(x)^2)^2.
(3) A(A(x) - x^2) = sqrt( A(x) - x^2 - x ).
(4) A(x)^2 = x - sqrt( A(x - A(x)^2) - x ).
(5) A(x) = x^2 + Series_Reversion(x - A(x)^2).
(6) A(x) = sqrt( x - Series_Reversion(A(x) - x^2) ).
(7) A(x) = x + x^2 + Sum_{n>=0} d^n/dx^n A(x)^(2*n+2) / (n+1)!.
(8) A(x) = x^2 + x*exp( Sum_{n>=0} d^n/dx^n (A(x)^(2*n+2)/x) / (n+1)! ).

A386648 E.g.f. A(x) satisfies 1 = Sum_{n>=0} ( A(x)^n + LambertW(-x) )^n / n!.

Original entry on oeis.org

1, 1, 10, 63, 806, 9485, 161540, 2752155, 59021506, 1310350929, 34127883032, 934203381287, 28851653188814, 942891341070237, 33962521081521076, 1297690184864525043, 53814377103189792794, 2366017294084046632937, 111607600081334119137488, 5565256312162642805357247

Offset: 1

Paul D. Hanna, Aug 12 2025

nonn

Conjecture: for n > 6, a(n) (mod 6) equals [2,3,4,3,2,5] repeating.
In general, the following sums are equal:
(C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
(C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = A(x) with p = LambertW(-x), r = 1.

			E.g.f.: A(x) = x + x^2/2! + 10*x^3/3! + 63*x^4/4! + 806*x^5/5! + 9485*x^6/6! + 161540*x^7/7! + 2752155*x^8/8! + 59021506*x^9/9! + 1310350929*x^10/10! + ...
where 1 = Sum_{n>=0} ( A(x)^n + LambertW(-x) )^n / n!.
RELATED SERIES.
-LambertW(-x) = x + 2*x^2/2! + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! + 6^5*x^6/6! + 7^6*x^7/7! + ... + n^(n-1)*x^n/n! + ...
where exp(LambertW(-x)) = -x/LambertW(-x);
also, (-x/LambertW(-x))^y = Sum_{k>=0} y*(y - k)^(k-1) * (-x)^k/k!.

Paul D. Hanna, Table of n, a(n) for n = 1..102

Cf. A386645.

{a(n) = my(A=[0,1]); for(i=0, n, A=concat(A, 0);
A[#A] = polcoeff(1 - sum(m=0, #A, (Ser(A)^m + lambertw(-x +x^3*O(x^n)))^m /m! ), #A-1) ); n!*A[n+1]}
for(n=1, 20, print1(a(n), ", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) 1 = Sum_{n>=0} ( A(x)^n + LambertW(-x) )^n / n!.
(2) 1 = Sum_{n>=0} A(x)^(n^2) * exp( LambertW(-x) * A(x)^n ) / n!.
(3) 1 = Sum_{n>=0} A(x)^(n^2) * (-x/LambertW(-x))^(A(x)^n) / n!.
(4) 1 = Sum_{n>=0} A(x)^(n*(n+1))/n! * Sum_{k>=0} (A(x)^n - k)^(k-1) * (-x)^k/k!.

cp's OEIS Frontend

User: Paul D. Hanna

A386659 G.f. A(x) satisfies A(x^3) = A(x)^3/(1 + 3*A(x)).

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A386665 G.f. satisfies: A(x) = Sum_{n>=0} ( (1+x)^n - A(x)^(-1/2) )^n / ( 2 - (1+x)^n * A(x)^(-1/2) )^(n+1).

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A386656 E.g.f.: Sum_{n>=0} (3^n*x + LambertW(x))^n / n!.

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A386657 E.g.f.: Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.

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A386658 E.g.f.: Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.

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A386655 E.g.f.: Sum_{n>=0} (2^n*x + LambertW(x))^n / n!.

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A385915 G.f. satisfies A(x) = -(A(x^3) + A(x^4)) / A(-x^2).

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