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),", "))
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/* 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),", "))
A352238
G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 / (A(x) - 4*x*A'(x)).
Original entry on oeis.org
1, 1, 5, 61, 1161, 28857, 864141, 29861749, 1160382737, 49854838897, 2340623599381, 119051103325613, 6516915195123097, 381912592990453545, 23856225840952434333, 1582482450123627473637, 111113139625779846025761, 8234335766045466358238433
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 61*x^3 + 1161*x^4 + 28857*x^5 + 864141*x^6 + 29861749*x^7 + 1160382737*x^8 + ...
such that A(x) = 1 + x*A(x)^2/(A(x) - 4*x*A'(x)).
Related table.
The table of coefficients of x^k in A(x)^(4*n+1) begins:
n=0: [1, 1, 5, 61, 1161, 28857, 864141, ...];
n=1: [1, 5, 35, 415, 7430, 176286, 5107530, ...];
n=2: [1, 9, 81, 993, 17127, 389583, 10916559, ...];
n=3: [1, 13, 143, 1859, 31564, 693212, 18802212, ...];
n=4: [1, 17, 221, 3077, 52309, 1118549, 29427153, ...];
n=5: [1, 21, 315, 4711, 81186, 1704906, 43640030, ...];
n=6: [1, 25, 425, 6825, 120275, 2500555, 62513875, ...]; ...
in which the following pattern holds:
[x^n] A(x)^(4*n+1) = [x^(n-1)] (4*n+1) * A(x)^(4*n+1), n >= 1,
as illustrated by
[x^1] A(x)^5 = 5 = [x^0] 5*A(x)^5 = 5*1;
[x^2] A(x)^9 = 81 = [x^1] 9*A(x)^9 = 9*9;
[x^3] A(x)^13 = 1859 = [x^2] 13*A(x)^13 = 13*143;
[x^4] A(x)^17 = 52309 = [x^3] 17*A(x)^17 = 17*3077;
[x^5] A(x)^21 = 1704906 = [x^4] 21*A(x)^21 = 21*81186;
[x^6] A(x)^25 = 62513875 = [x^5] 25*A(x)^25 = 25*2500555; ...
Also, compare the above terms along the diagonal to the series
B(x) = A(x*B(x)^4) = 1 + x + 9*x^2 + 143*x^3 + 3077*x^4 + 81186*x^5 + 2500555*x^6 + 87388600*x^7 + ...
where B(x)^4 = (1/x) * Series_Reversion( x/A(x)^4 ).
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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 - 4*x*A' + x*O(x^n)) );
polcoeff(A,n)}
for(n=0,20, print1(a(n),", "))
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/* Using [x^n] A(x)^(4*n+1) = [x^(n-1)] (4*n+1)*A(x)^(4*n+1) */
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff((x*Ser(A)^(4*(#A)-3) - Ser(A)^(4*(#A)-3)/(4*(#A)-3)),#A-1));A[n+1]}
for(n=0,20, print1(a(n),", "))
Showing 1-3 of 3 results.