A300050
G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1 + x*A(x)^n)^n / 2^(n+1).
Original entry on oeis.org
1, 1, 4, 28, 250, 2558, 28594, 340486, 4255344, 55300776, 742732646, 10267434138, 145692400018, 2118364506746, 31529605958892, 480186833802260, 7483472464151002, 119397596900634238, 1951747376021480874, 32721008993895160926, 563260078608381337148, 9967709437187109520736, 181544883799333028286098, 3406426523158387599683478, 65894531117591548919270114
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 250*x^4 + 2558*x^5 + 28594*x^6 + 340486*x^7 + 4255344*x^8 + 55300776*x^9 + 742732646*x^10 + ...
such that
A(x) = 1/2 + (1 + x*A(x))/2^2 + (1 + x*A(x)^2)^2/2^3 + (1 + x*A(x)^3)^3/2^4 + (1 + x*A(x)^4)^4/2^5 + (1 + x*A(x)^5)^5/2^6 + (1 + x*A(x)^6)^6/2^7 + ...
Also, due to a series identity,
A(x) = 1 + x*A(x)/(2 - A(x))^2 + x^2*A(x)^4/(2 - A(x)^2)^3 + x^3*A(x)^9/(2 - A(x)^3)^4 + x^4*A(x)^16/(2 - A(x)^4)^5 + x^5*A(x)^25/(2 - A(x)^5)^6 + x^6*A(x)^36/(2 - A(x)^6)^7 + ... + x^n * A(x)^(n^2) / (2 - A(x)^n)^(n+1) + ...
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{a(n) = my(A=1); for(i=0,n, A = sum(m=0,n, x^m * A^(m^2) / (2 - A^m + x*O(x^n))^(m+1) )); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
A302103
G.f. A(x) satisfies: A(x) = Sum_{n>=0} (2 + x*A(x)^n)^n / 3^(n+1).
Original entry on oeis.org
1, 1, 6, 63, 837, 12672, 208686, 3647568, 66697203, 1264307667, 24696153573, 495076265421, 10157438738790, 212900154037875, 4553735135491134, 99341289091151409, 2210262851488661562, 50173932628981325523, 1162965513498859292415, 27554435907912281877315, 668277970101220006626558, 16617278354076763108026795
Offset: 0
G.f.: A(x) = 1 + x + 6*x^2 + 63*x^3 + 837*x^4 + 12672*x^5 + 208686*x^6 + 3647568*x^7 + 66697203*x^8 + 1264307667*x^9 + 24696153573*x^10 + ...
such that
A(x) = 2/3 + (2 + x*A(x))/3^2 + (2 + x*A(x)^2)^2/3^3 + (2 + x*A(x)^3)^3/3^4 + (2 + x*A(x)^4)^4/3^5 + (2 + x*A(x)^5)^5/3^6 + (2 + x*A(x)^6)^6/3^7 + ...
Also, due to a series identity,
A(x) = 1 + x*A(x)/(3 - 2*A(x))^2 + x^2*A(x)^4/(3 - 2*A(x)^2)^3 + x^3*A(x)^9/(3 - 2*A(x)^3)^4 + x^4*A(x)^16/(3 - 2*A(x)^4)^5 + x^5*A(x)^25/(3 - 2*A(x)^5)^6 + x^6*A(x)^36/(3 - 2*A(x)^6)^7 + ... + x^n * A(x)^(n^2) / (3 - 2*A(x)^n)^(n+1) + ...
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{a(n) = my(A=1); for(i=0, n, A = sum(m=0, n, x^m * A^(m^2) / (3 - 2*A^m + x*O(x^n))^(m+1) )); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
A302104
G.f. A(x) satisfies: A(x) = Sum_{n>=0} (3 + x*A(x)^n)^n / 4^(n+1).
Original entry on oeis.org
1, 1, 8, 112, 1972, 39404, 853892, 19591692, 469250416, 11628163256, 296351290004, 7736140181364, 206273152705660, 5606990999026252, 155184267041459384, 4370129283473065984, 125189806731347999476, 3648813481714933367516, 108265665575110494127284, 3273367006162760350945260, 100977120404026793376264880, 3183255539561434435490787720
Offset: 0
G.f.: A(x) = 1 + x + 8*x^2 + 112*x^3 + 1972*x^4 + 39404*x^5 + 853892*x^6 + 19591692*x^7 + 469250416*x^8 + 11628163256*x^9 + 296351290004*x^10 + ...
such that
A(x) = 3/4 + (3 + x*A(x))/4^2 + (3 + x*A(x)^2)^2/4^3 + (3 + x*A(x)^3)^3/4^4 + (3 + x*A(x)^4)^4/4^5 + (3 + x*A(x)^5)^5/4^6 + (3 + x*A(x)^6)^6/4^7 + ...
Also, due to a series identity,
A(x) = 1 + x*A(x)/(4 - 3*A(x))^2 + x^2*A(x)^4/(4 - 3*A(x)^2)^3 + x^3*A(x)^9/(4 - 3*A(x)^3)^4 + x^4*A(x)^16/(4 - 3*A(x)^4)^5 + x^5*A(x)^25/(4 - 3*A(x)^5)^6 + x^6*A(x)^36/(4 - 3*A(x)^6)^7 + ... + x^n * A(x)^(n^2) / (4 - 3*A(x)^n)^(n+1) + ...
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{a(n) = my(A=1); for(i=0, n, A = sum(m=0, n, x^m * A^(m^2) / (4 - 3*A^m + x*O(x^n))^(m+1) )); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
Showing 1-3 of 3 results.
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