A038464 -id:A038464 - cp's OEIS frontend

A038465 Sums of 3 distinct powers of 3.

13, 31, 37, 39, 85, 91, 93, 109, 111, 117, 247, 253, 255, 271, 273, 279, 325, 327, 333, 351, 733, 739, 741, 757, 759, 765, 811, 813, 819, 837, 973, 975, 981, 999, 1053, 2191, 2197, 2199, 2215, 2217, 2223, 2269, 2271, 2277, 2295, 2431, 2433, 2439, 2457, 2511, 2917

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Base 3 interpretation of A038445.

Cf. A000244, A038464, A038466, A038467, A038468, A038469.

Total/@Subsets[3^Range[0,10],{3}]//Union (* Harvey P. Dale, Jul 10 2017 *)

from math import comb, isqrt
from sympy import integer_nthroot
def A038465(n): return 3**((r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(t:=(n>comb(m+2,3)))+1,3))-comb((k:=isqrt(b:=r+1<<1))+(b>k*(k+1)),2))+3**((a:=isqrt(s:=n-comb(m-(t^1)+2,3)<<1))+((s<<2)>(a<<2)*(a+1)+1))+3**(m+t+1) # Chai Wah Wu, Apr 04 2025

Offset corrected by Amiram Eldar, Jul 13 2022

A038466 Sums of 4 distinct powers of 3.

Original entry on oeis.org

40, 94, 112, 118, 120, 256, 274, 280, 282, 328, 334, 336, 352, 354, 360, 742, 760, 766, 768, 814, 820, 822, 838, 840, 846, 976, 982, 984, 1000, 1002, 1008, 1054, 1056, 1062, 1080, 2200, 2218, 2224, 2226, 2272, 2278, 2280, 2296, 2298, 2304, 2434, 2440, 2442, 2458

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Base 3 interpretation of A038446.

Cf. A000244, A038464, A038465, A038467, A038468, A038469.

Sort[Plus @@@ Subsets[3^Range[0, 7], {4}]] (* Amiram Eldar, Jul 13 2022 *)

from itertools import islice
def A038466_gen(): # generator of terms
    yield int(bin(n:=15)[2:],3)
    while True: yield int(bin((n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b))[2:],3)
A038466_list = list(islice(A038466_gen(),30)) # Chai Wah Wu, Apr 04 2025

Offset corrected by Amiram Eldar, Jul 13 2022

A038467 Sums of 5 distinct powers of 3.

Original entry on oeis.org

121, 283, 337, 355, 361, 363, 769, 823, 841, 847, 849, 985, 1003, 1009, 1011, 1057, 1063, 1065, 1081, 1083, 1089, 2227, 2281, 2299, 2305, 2307, 2443, 2461, 2467, 2469, 2515, 2521, 2523, 2539, 2541, 2547, 2929, 2947, 2953, 2955, 3001, 3007, 3009, 3025, 3027, 3033

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Base 3 interpretation of A038447.

Cf. A000244, A038464, A038465, A038466, A038468, A038469.

Union[Total/@Subsets[3^Range[0,7],{5}]]  (* Harvey P. Dale, Feb 23 2011 *)

A038468 Sums of 6 distinct powers of 3.

Original entry on oeis.org

364, 850, 1012, 1066, 1084, 1090, 1092, 2308, 2470, 2524, 2542, 2548, 2550, 2956, 3010, 3028, 3034, 3036, 3172, 3190, 3196, 3198, 3244, 3250, 3252, 3268, 3270, 3276, 6682, 6844, 6898, 6916, 6922, 6924, 7330, 7384, 7402, 7408, 7410, 7546, 7564, 7570, 7572, 7618

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000244, A038464, A038465, A038466, A038467, A038469.

Base 3 interpretation of A038448.

Sort[Plus @@@ Subsets[3^Range[0, 8], {6}]] (* Amiram Eldar, Jul 12 2022 *)

from itertools import islice
def A038468_gen(): # generator of terms
    yield int(bin(n:=63)[2:],3)
    while True: yield int(bin((n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b))[2:],3)
A038468_list = list(islice(A038468_gen(),30)) # Chai Wah Wu, Apr 05 2025

Offset corrected by Amiram Eldar, Jul 12 2022

A038469 Sums of 7 distinct powers of 3.

Original entry on oeis.org

1093, 2551, 3037, 3199, 3253, 3271, 3277, 3279, 6925, 7411, 7573, 7627, 7645, 7651, 7653, 8869, 9031, 9085, 9103, 9109, 9111, 9517, 9571, 9589, 9595, 9597, 9733, 9751, 9757, 9759, 9805, 9811, 9813, 9829, 9831, 9837, 20047, 20533, 20695, 20749, 20767, 20773, 20775

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Base 3 interpretation of A038449.

Cf. A000244, A038464, A038465, A038466, A038467, A038468.

Sort[Plus @@@ Subsets[3^Range[0, 8], {7}]] (* Amiram Eldar, Jul 13 2022 *)

Offset corrected by Amiram Eldar, Jul 13 2022

A309758 Numbers that are sums of consecutive powers of 3.

Original entry on oeis.org

1, 3, 4, 9, 12, 13, 27, 36, 39, 40, 81, 108, 117, 120, 121, 243, 324, 351, 360, 363, 364, 729, 972, 1053, 1080, 1089, 1092, 1093, 2187, 2916, 3159, 3240, 3267, 3276, 3279, 3280, 6561, 8748, 9477, 9720, 9801, 9828, 9837, 9840, 9841, 19683, 26244, 28431

Offset: 1

Ilya Gutkovskiy, Aug 15 2019

nonn

Numbers of the form (3^i - 3^j)/2 with i > j.

			1080 = 3^3 + 3^4 + 3^5 + 3^6, so 1080 is in the sequence.
+------+--------+
| a(n) | base 3*|
+------+--------+
|   1  |     1  |
|   3  |    10  |
|   4  |    11  |
|   9  |   100  |
|  12  |   110  |
|  13  |   111  |
|  27  |  1000  |
|  36  |  1100  |
|  39  |  1110  |
|  40  |  1111  |
+------+--------+
* - a(n) written in base 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000244, A005836, A023758, A038464, A309759, A309761.

[seq(seq((3^i-3^j)/2,j=i-1..0,-1),i=1..20)]; # Robert Israel, Aug 19 2019

from math import isqrt
def A309758(n): return 3**(m:=isqrt(n<<3)+1>>1)-3**(m*(m+1)-(n<<1)>>1)>>1 # Chai Wah Wu, Apr 04 2025

A371709 Expansion of g.f. A(x) satisfying A( xA(x)^2 + xA(x)^3 ) = A(x)^3.

Original entry on oeis.org

1, 1, 1, 2, 6, 16, 39, 99, 271, 764, 2157, 6128, 17658, 51534, 151635, 448962, 1337493, 4008040, 12072594, 36524898, 110943633, 338218626, 1034509917, 3173811240, 9763898994, 30113782641, 93094164244, 288415278638, 895332445053, 2784580242557, 8675408291598, 27072326322939

Offset: 1

Paul D. Hanna, May 02 2024

nonn

Compare to the following identities of the Catalan function C(x) = x + C(x)^2 (A000108):
(1) C(x)^2 = C( x*C(x)*(1 + C(x)) ),
(2) C(x)^4 = C( x*C(x)^3*(1 + C(x))*(1 + C(x)^2) ),
(3) C(x)^8 = C( x*C(x)^7*(1 + C(x))*(1 + C(x)^2)*(1 + C(x)^4) ),
(4) C(x)^(2^n) = C( x*C(x)^(2^n-1)*Product_{k=0..n-1} (1 + C(x)^(2^k)) ) for n > 0.
a(3^n) == 1 (mod 3) for n >= 0.
a(2*3^n) == 1 (mod 3) for n >= 0.
a(n) == 2 (mod 3) iff n is the sum of 2 distinct powers of 3 (A038464).

			G.f. A(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 16*x^6 + 39*x^7 + 99*x^8 + 271*x^9 + 764*x^10 + 2157*x^11 + 6128*x^12 + 17658*x^13 + 51534*x^14 + 151635*x^15 + 448962*x^16 + ...
where A( x*A(x)^2*(1 + A(x)) ) = A(x)^3.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 17*x^6 + 48*x^7 + 126*x^8 + 332*x^9 + 918*x^10 + 2616*x^11 + 7504*x^12 + ...
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 36*x^7 + 105*x^8 + 292*x^9 + 801*x^10 + 2256*x^11 + 6515*x^12 + 18981*x^13 + ...
A(x)^2 + A(x)^3 = x^2 + 3*x^3 + 6*x^4 + 12*x^5 + 30*x^6 + 84*x^7 + 231*x^8 + 624*x^9 + 1719*x^10 + 4872*x^11 + 14019*x^12 + 40599*x^13 + ...
Let B(x) be the series reversion of g.f. A(x), B(A(x)) = x, then
B(x) * (1+x)/(1+x^3) = x - 2*x^4 + 3*x^7 - 5*x^10 + 7*x^13 - 9*x^16 + 12*x^19 - 15*x^22 + 18*x^25 - 23*x^28 + ... + (-1)^n*A005704(n)*x^(3*n+1) + ...
where A005704 is the number of partitions of 3*n into powers of 3.
We can show that g.f. A(x) = A( x*A(x)^2*(1 + A(x)) )^(1/3) satisfies
(4) A(x) = x * Product_{n>=0} (1 + A(x)^(3^n))
by substituting x*A(x)^2*(1 + A(x)) for x in (4) to obtain
A(x)^3 = x * A(x)^2*(1 + A(x)) * Product_{n>=1} (1 + A(x)^(3^n))
which is equivalent to formula (4).
SPECIFIC VALUES.
A(3/10) = 0.526165645044542830201162330432965674027415264612114520...
A(1/4) = 0.353259384374080248921564026412797625837830114153200664...
A(1/5) = 0.255218141344695821239609680309162895225297482063273545...
A(t) = 1/2 and A(t*3/8) = 1/8 at t = (1/2)/Product_{n>=0} (1 + 1/2^(3^n)) = 0.295718718466711580562679377308518930409875701753934072...
A(t) = 1/3 and A(t*4/27) = 1/27 at t = (1/3)/Product_{n>=0} (1 + 1/3^(3^n)) = 0.241059181496179959557718992589733756750585121455883861...
A(t) = 1/4 and A(t*5/64) = 1/64 at t = (1/4)/Product_{n>=0} (1 + 1/4^(3^n)) = 0.196922325724019432212969925740117827612003158137366017...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A371716, A370440, A370441, A370446, A264228, A198951.

Cf. A005704, A038464.

/* Using series reversion of x/Product_{n>=0} (1 + x^(3^n)) */
{a(n) = my(A); A = serreverse( x/prod(k=0,ceil(log(n)/log(3)), (1 + x^(3^k) +x*O(x^n)) ) ); polcoeff(A,n)}
for(n=1,35, print1(a(n),", "))

/* Using A(x)^3 = A( x*A(x)^2 + x*A(x)^3 ) */
{a(n) = my(A=[1],F); for(i=1,n, A = concat(A,0); F = x*Ser(A);
A[#A] = polcoeff( subst(F,x, x*F^2 + x*F^3 ) - F^3, #A+2) ); A[n]}
for(n=1,35, print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = A( x*A(x)^2*(1 + A(x)) ).
(2) A(x)^9 = A( x*A(x)^8*(1 + A(x))*(1 + A(x)^3) ).
(3) A(x)^27 = A( x*A(x)^26*(1 + A(x))*(1 + A(x)^3)*(1 + A(x)^9) ).
(4) A(x) = x * Product_{n>=0} (1 + A(x)^(3^n)).
(5) A(x) = Series_Reversion( x / Product_{n>=0} (1 + x^(3^n)) ).
a(n) ~ c * d^n / n^(3/2), where d = 3.2753449994351908157330968510747739... and c = 0.1559869008021616116037651076359... - Vaclav Kotesovec, May 03 2024
The radius of convergence r of g.f. A(x) and A(r) satisfy 1 = Sum_{n>=0} 3^n * A(r)^(3^n) / (1 + A(r)^(3^n)) and r = A(r) / Product_{n>=0} (1 + A(r)^(3^n)), where r = 0.30531134893345362211... = 1/d (d is given above) and A(r) = 0.600582105427215700175254768411726892599... - Paul D. Hanna, May 03 2024

A376226 G.f. satisfies A(x) = A(x^3 + 6xA(x)^3) / A(x^2 + 4xA(x)^2).

Original entry on oeis.org

1, 2, 10, 66, 518, 4484, 41424, 399900, 3983698, 40622502, 421780380, 4442833776, 47353725678, 509717438804, 5532808254500, 60492412303032, 665570138005230, 7363717939202660, 81872879608989990, 914314572022052508, 10251126194392776384, 115346231108018654736, 1302114832694059544892

Offset: 1

Paul D. Hanna, Oct 04 2024

nonn

Compare to C(x) = C(x^3 + 3*x*C(x)^3) / C(x^2 + 2*x*C(x)^2), where C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).
Conjectures:
(C1) a(n) == 1 (mod 3) iff n = 3^k for some k >= 0.
(C2) a(n) == 2 (mod 3) iff n = A038464(k)/2 for some k >= 1, where A038464 lists the sums of 2 distinct powers of 3.

			G.f.: A(x) = x + 2*x^2 + 10*x^3 + 66*x^4 + 518*x^5 + 4484*x^6 + 41424*x^7 + 399900*x^8 + 3983698*x^9 + 40622502*x^10 + 421780380*x^11 + 4442833776*x^12 + ...
where A(x) = A(x^3 + 6*x*A(x)^3) / A(x^2 + 4*x*A(x)^2).
RELATED SERIES.
A(x^2 + 4*x*A(x)^2) = x^2 + 4*x^3 + 18*x^4 + 112*x^5 + 794*x^6 + 6360*x^7 + 55266*x^8 + 509968*x^9 + 4914150*x^10 + 48889752*x^11 + 498234420*x^12 + ...
A(x^3 + 6*x*A(x)^3) = x^3 + 6*x^4 + 36*x^5 + 254*x^6 + 1980*x^7 + 16812*x^8 + 152002*x^9 + 1440828*x^10 + 14148936*x^11 + 142715046*x^12 + ...
A(x)^2 = x^2 + 4*x^3 + 24*x^4 + 172*x^5 + 1400*x^6 + 12360*x^7 + 115500*x^8 + 1123552*x^9 + 11255688*x^10 + 115291188*x^11 + 1201533048*x^12 + ...
A(x)^3 = x^3 + 6*x^4 + 42*x^5 + 326*x^6 + 2766*x^7 + 25020*x^8 + 237364*x^9 + 2332860*x^10 + 23547474*x^11 + 242620986*x^12 + ...
A(x)^2 / A(x^2 + 4*x*A(x)^2) = 1 + 6*x^2 + 36*x^3 + 354*x^4 + 3264*x^5 + 32010*x^6 + 320400*x^7 + 3276558*x^8 + 34050444*x^9 + 358651116*x^10 + 3820385664*x^11 + 41087069040*x^12 + ...
which also equals A(x)^3 / A(x^3 + 6*x*A(x)^3).

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

{a(n) = my(A=[0,1],Ax=x); for(i=1,n, A=concat(A,0); Ax=Ser(A);
A[#A] = polcoeff( subst(Ax,x, x^3 + 6*x*Ax^3 ) - Ax*subst(Ax,x, x^2 + 4*x*Ax^2 ),#A+1)); A[n+1]}
for(n=1,25,print1(a(n),", "))

a(n) ~ c * d^n / n^(3/2), where d = 12.086418637032871629430806055580752... and c = 0.01774947449130389477598279659776... - Vaclav Kotesovec, Oct 10 2024

A384830 G.f. satisfies A(x) = A(x^3 - 3xA(x)^3) / A(x^2 - 2xA(x)^2).

Original entry on oeis.org

1, -1, 4, -21, 110, -616, 3738, -23619, 152470, -1003776, 6726702, -45720504, 314307018, -2181641134, 15269811260, -107651952999, 763745165826, -5448656285938, 39063995033178, -281309141648214, 2033846965665156, -14757571862304042, 107431429198117338, -784411267743868602, 5743068864740600214

Offset: 1

Paul D. Hanna, Jul 11 2025

sign

Compare to C(x) = C(x^3 + 3*x*C(x)^3) / C(x^2 + 2*x*C(x)^2), where C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).
Conjectures:
(C.1) a(n) is odd iff n = 2^k for k >= 0.
(C.2) a(n) == 1 (mod 3) iff n = 3^k for k >= 0.
(C.3) a(n) == 2 (mod 3) iff n = A038464(k)/2 for k >= 1, where A038464 lists sums of 2 distinct powers of 3.

			G.f.: A(x)  = x - x^2 + 4*x^3 - 21*x^4 + 110*x^5 - 616*x^6 + 3738*x^7 - 23619*x^8 + 152470*x^9 - 1003776*x^10 + 6726702*x^11 - 45720504*x^12 + 314307018*x^13 - 2181641134*x^14 + 15269811260*x^15 - 107651952999*x^16 +- ...
where A(x) = A(x^3 - 3*x*A(x)^3) / A(x^2 - 2*x*A(x)^2).
RELATED SERIES.
A(x)^2 = x^2 - 2*x^3 + 9*x^4 - 50*x^5 + 278*x^6 - 1620*x^7 + 10029*x^8 - 64262*x^9 + 420054*x^10 - 2793960*x^11 + 18874530*x^12 + ...
A(x)^3 = x^3 - 3*x^4 + 15*x^5 - 88*x^6 + 516*x^7 - 3123*x^8 + 19771*x^9 - 128748*x^10 + 853182*x^11 - 5739306*x^12 + ...
A(x^3 - 3*x*A(x)^3) = x^3 - 3*x^4 + 9*x^5 - 46*x^6 + 270*x^7 - 1575*x^8 + 9517*x^9 - 60228*x^10 + 391950*x^11 - 2595666*x^12 + ...
A(x^2 - 2*x*A(x)^2) = x^2 - 2*x^3 + 3*x^4 - 14*x^5 + 92*x^6 - 528*x^7 + 3027*x^8 - 18674*x^9 + 120414*x^10 - 790332*x^11 + 5264400*x^12 + ...
A(x)^2 / A(x^2 - 2*x*A(x)^2) = 1 + 6*x^2 - 24*x^3 + 120*x^4 - 696*x^5 + 4362*x^6 - 27720*x^7 + 179496*x^8 - 1188324*x^9 + 8004222*x^10 - 54609300*x^11 + 376571358*x^12 + ...
which also equals A(x)^3 / A(x^3 - 3*x*A(x)^3).
SPECIFIC VALUES.
A(t) = 1/9 at t = 0.121516943263807312205895948801335726496880098390997...
A(t) = 1/10 at t = 0.10853522544585482043493483988448061537900985360239...
  where 1/10 = A(t^3 - 3*t/10^3) / A(t^2 - 2*t/10^2).
A(1/8) = 0.11406786932603073004140288621992624859772243547241...
  where A(1/8) = A(1/8^3 - 3/8*A(1/8)^3) / A(1/8^2 - 2/8*A(1/8)^2).
A(1/9) = 0.10221632839303036250437950418981106643715860663579...
A(1/10) = 0.09261382926552257152263444179118404183078004746053...
A(1/11) = 0.08467305103076088485212166668503171895257965884840...
A(1/12) = 0.07799525781645435314575205510854905593664831977089...
A(-1/8) = -0.1721552141574965794714379396217931153561352397408...
A(-1/9) = -0.1381403570271418152616530595087372513703838778026...
A(-1/10) = -0.1188984739221037989978618004289786780166116550840...

Paul D. Hanna, Table of n, a(n) for n = 1..731

Cf. A038464, A376226, A384270.

{a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A] = polcoeff( subst(Ax, x, x^3 - 3*x*Ax^3 ) - Ax*subst(Ax, x, x^2 - 2*x*Ax^2 ), #A+1)); A[n+1]}
for(n=1, 25, print1(a(n), ", "))

a(n) ~ (-1)^(n+1) * c * d^n / n^(3/2), where d = 7.79529565596481925683030276174290122793451926009119... and c = 0.0350717712305315691918705178165312896756485548321... - Vaclav Kotesovec, Jul 16 2025

A385912 A diagonal (unsigned) of triangle A385910; a(n) = -A385910(n+2,n) for n >= 1.

Original entry on oeis.org

1, 8, 49, 296, 1815, 11284, 70924, 449616, 2869779, 18418400, 118749345, 768537120, 4990021764, 32489701776, 212048505160, 1386886206112, 9087724409547, 59646983740680, 392071446052195, 2580601721867400, 17005938279649935, 112190574812699460, 740878216459158960, 4897062582469861440, 32395964187696107700

Offset: 1

Paul D. Hanna, Jul 14 2025

nonn

Triangle A385910 has g.f. A(x,y) where A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y).
The main diagonal of A385910 has g.f. G(x) = 1 + x*G(x)^3 (A001764).
Conjectures:
(C.1) a(n) is odd iff n = 2*A003714(k) + 1 for k >= 0, where A003714 is the Fibbinary numbers.
(C.2) a(n) == 1 (mod 3) iff n = 3^k + 1 for k >= 0.
(C.3) a(n) == 2 (mod 3) iff n = A038464(k)/2 + 1 for k >= 1 or n = 1, where A038464 lists sums of 2 distinct powers of 3.

			G.f. A(x) = x + 8*x^2 + 49*x^3 + 296*x^4 + 1815*x^5 + 11284*x^6 + 70924*x^7 + 449616*x^8 + 2869779*x^9 + 18418400*x^10 + 118749345*x^11 + 768537120*x^12 + ...

Paul D. Hanna, Table of n, a(n) for n = 1..180

Cf. A385910, A003714, A118113, A038464.

\\ a(n) = -A385910(n+2,n) for n >= 1
{A385910(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A] = polcoeff( subst(Ax, x, x^3 + 3*y*x*Ax^3 ) - Ax*subst(Ax, x, x^2 + 2*y*x*Ax^2 ), #A+1)); A[n+1]}
for(n=1, 25, print1(polcoef(-A385910(n+2),n),", "))

cp's OEIS Frontend

A038465 Sums of 3 distinct powers of 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A038466 Sums of 4 distinct powers of 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A038467 Sums of 5 distinct powers of 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A038468 Sums of 6 distinct powers of 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A038469 Sums of 7 distinct powers of 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A309758 Numbers that are sums of consecutive powers of 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

A371709 Expansion of g.f. A(x) satisfying A( x*A(x)^2 + x*A(x)^3 ) = A(x)^3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A376226 G.f. satisfies A(x) = A(x^3 + 6*x*A(x)^3) / A(x^2 + 4*x*A(x)^2).

Views

Author

Keywords

Comments

Examples

Links

Programs

PARI

A371709 Expansion of g.f. A(x) satisfying A( xA(x)^2 + xA(x)^3 ) = A(x)^3.

A376226 G.f. satisfies A(x) = A(x^3 + 6xA(x)^3) / A(x^2 + 4xA(x)^2).

A384830 G.f. satisfies A(x) = A(x^3 - 3xA(x)^3) / A(x^2 - 2xA(x)^2).