A352236
G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 / (A(x) - 2*x*A'(x)).
Original entry on oeis.org
1, 1, 3, 19, 185, 2353, 36075, 638115, 12683761, 278485217, 6674259667, 173097575603, 4826128088489, 143896870347793, 4568544366818747, 153883892657000259, 5481761893234193889, 205939077652874352577, 8138639816942009694627, 337568614331296733526867
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 185*x^4 + 2353*x^5 + 36075*x^6 + 638115*x^7 + 12683761*x^8 + ...
such that A(x) = 1 + x*A(x)^2/(A(x) - 2*x*A'(x)).
Related table.
The table of coefficients of x^k in A(x)^(2*n+1) begins:
n=0: [1, 1, 3, 19, 185, 2353, 36075, ...];
n=1: [1, 3, 12, 76, 705, 8595, 127680, ...];
n=2: [1, 5, 25, 165, 1490, 17506, 252050, ...];
n=3: [1, 7, 42, 294, 2632, 30016, 419454, ...];
n=4: [1, 9, 63, 471, 4239, 47295, 643017, ...];
n=5: [1, 11, 88, 704, 6435, 70785, 939312, ...];
n=6: [1, 13, 117, 1001, 9360, 102232, 1329016, ...]; ...
in which the following pattern holds:
[x^n] A(x)^(2*n+1) = [x^(n-1)] (2*n+1) * A(x)^(2*n+1), n >= 1,
as illustrated by
[x^1] A(x)^3 = 3 = [x^0] 3*A(x)^3 = 3*1;
[x^2] A(x)^5 = 25 = [x^1] 5*A(x)^5 = 5*5;
[x^3] A(x)^7 = 294 = [x^2] 7*A(x)^7 = 7*42;
[x^4] A(x)^9 = 4239 = [x^3] 9*A(x)^9 = 9*471;
[x^5] A(x)^11 = 70785 = [x^4] 11*A(x)^11 = 11*6435;
[x^6] A(x)^13 = 1329016 = [x^5] 13*A(x)^13 = 13*102232; ...
Also, compare the above terms along the diagonal to the series
B(x) = A(x*B(x)^2) = 1 + x + 5*x^2 + 42*x^3 + 471*x^4 + 6435*x^5 + 102232*x^6 + 1837630*x^7 + ... + A317352(n)*x^n + ...
where B(x)^2 = (1/x) * Series_Reversion( x/A(x)^2 ).
-
/* Using A(x) = 1 + x*A(x)^2/(A(x) - 2*x*A'(x)) */
{a(n) = my(A=1); for(i=1,n, A = 1 + x*A^2/(A - 2*x*A' + x*O(x^n)) );
polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))
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/* Using [x^n] A(x)^(2*n+1) = [x^(n-1)] (2*n+1)*A(x)^(2*n+1) */
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff((x*Ser(A)^(2*(#A)-1) - Ser(A)^(2*(#A)-1)/(2*(#A)-1)),#A-1));A[n+1]}
for(n=0,30, print1(a(n),", "))
A352235
G.f. A(x) satisfies: A(x) = 1 + x*A(x) / (A(x) - 3*x*A'(x)).
Original entry on oeis.org
1, 1, 3, 24, 309, 5262, 108894, 2618718, 71246145, 2154788970, 71563126710, 2586270267600, 100995812044266, 4237522832234832, 190126298040192912, 9085093650185205498, 460711407231295513689, 24715373661154672634058, 1398648334415007990887454
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 24*x^3 + 309*x^4 + 5262*x^5 + 108894*x^6 + 2618718*x^7 + 71246145*x^8 + ...
such that A(x) = 1 + x*A(x)/(A(x) - 3*x*A'(x)).
Related table.
The table of coefficients of x^k in A(x)^(3*n+2) begins:
n=0: [1, 2, 7, 54, 675, 11286, 230742, ...];
n=1: [1, 5, 25, 190, 2210, 34981, 688635, ...];
n=2: [1, 8, 52, 416, 4642, 69872, 1322848, ...];
n=3: [1, 11, 88, 759, 8349, 120549, 2195886, ...];
n=4: [1, 14, 133, 1246, 13790, 193060, 3391017, ...];
n=5: [1, 17, 187, 1904, 21505, 295154, 5017618, ...];
n=6: [1, 20, 250, 2760, 32115, 436524, 7217250, ...]; ...
in which the following pattern holds:
[x^n] A(x)^(3*n+2) = [x^(n-1)] (3*n+2) * A(x)^(3*n+2), n >= 1,
as illustrated by
[x^1] A(x)^2 = 2 = [x^0] 2*A(x)^2 = 2*1;
[x^2] A(x)^5 = 25 = [x^1] 5*A(x)^5 = 5*5;
[x^3] A(x)^8 = 416 = [x^2] 8*A(x)^8 = 8*52;
[x^4] A(x)^11 = 8349 = [x^3] 11*A(x)^11 = 11*759;
[x^5] A(x)^14 = 193060 = [x^4] 14*A(x)^14 = 14*13790;
[x^6] A(x)^17 = 5017618 = [x^5] 17*A(x)^17 = 17*295154; ...
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/* Using A(x) = 1 + x*A(x)/(A(x) - 3*x*A'(x)) */
{a(n) = my(A=1); for(i=1,n, A = 1 + x*A/(A - 3*x*A' + x*O(x^n)) );
polcoeff(A,n)}
for(n=0,20, print1(a(n),", "))
-
/* Using [x^n] A(x)^(3*n+2) = [x^(n-1)] (3*n+2)*A(x)^(3*n+2) */
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff((x*Ser(A)^(3*(#A-2)+2) - Ser(A)^(3*(#A-2)+2)/(3*(#A-2)+2)),#A-1));A[n+1]}
for(n=0,20, print1(a(n),", "))
A352237
G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 / (A(x) - 3*x*A'(x)).
Original entry on oeis.org
1, 1, 4, 37, 532, 9994, 226252, 5910445, 173581060, 5634589906, 199792389160, 7671942375898, 316936631324368, 14011781050744984, 660054967923455212, 33008607551445324157, 1746771084107236755604, 97536010045722766992778, 5731874036042145864368824
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 37*x^3 + 532*x^4 + 9994*x^5 + 226252*x^6 + 5910445*x^7 + 173581060*x^8 + ...
such that A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)).
Related table.
The table of coefficients of x^k in A(x)^(3*n+1) begins:
n=0: [1, 1, 4, 37, 532, 9994, 226252, ...];
n=1: [1, 4, 22, 200, 2717, 48788, 1069122, ...];
n=2: [1, 7, 49, 462, 6069, 104664, 2219784, ...];
n=3: [1, 10, 85, 850, 11020, 183832, 3777355, ...];
n=4: [1, 13, 130, 1391, 18083, 294203, 5869734, ...];
n=5: [1, 16, 184, 2112, 27852, 445632, 8659920, ...];
n=6: [1, 19, 247, 3040, 41002, 650161, 12353059, ...]; ...
in which the following pattern holds:
[x^n] A(x)^(3*n+1) = [x^(n-1)] (3*n+1) * A(x)^(3*n+1), n >= 1,
as illustrated by
[x^1] A(x)^4 = 4 = [x^0] 4*A(x)^4 = 4*1;
[x^2] A(x)^7 = 49 = [x^1] 7*A(x)^7 = 7*7;
[x^3] A(x)^10 = 850 = [x^2] 10*A(x)^10 = 10*85;
[x^4] A(x)^13 = 18083 = [x^3] 13*A(x)^13 = 13*1391;
[x^5] A(x)^16 = 445632 = [x^4] 16*A(x)^16 = 16*27852;
[x^6] A(x)^19 = 12353059 = [x^5] 19*A(x)^19 = 19*650161; ...
Also, compare the above terms along the diagonal to the series
B(x) = A(x*B(x)^3) = 1 + x + 7*x^2 + 85*x^3 + 1391*x^4 + 27852*x^5 + 650161*x^6 + 17204220*x^7 + ...
where B(x)^3 = (1/x) * Series_Reversion( x/A(x)^3 ).
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/* Using A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)) */
{a(n) = my(A=1); for(i=1,n, A = 1 + x*A^2/(A - 3*x*A' + x*O(x^n)) );
polcoeff(A,n)}
for(n=0,20, print1(a(n),", "))
-
/* Using [x^n] A(x)^(3*n+1) = [x^(n-1)] (3*n+1)*A(x)^(3*n+1) */
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff((x*Ser(A)^(3*(#A)-2) - Ser(A)^(3*(#A)-2)/(3*(#A)-2)),#A-1));A[n+1]}
for(n=0,20, print1(a(n),", "))
Showing 1-3 of 3 results.