A300279
G.f.: Sum_{n>=0} (1 + x*(1+x)^n)^n / 2^(n+1).
Original entry on oeis.org
1, 1, 4, 16, 86, 544, 3904, 31328, 276798, 2660564, 27576614, 306051500, 3615559236, 45241980928, 597141146374, 8283583741588, 120393776421550, 1828261719906800, 28937578248560784, 476355010859517352, 8139464481630136242, 144109168217154747856, 2639508261422244889106, 49940898467864797567140, 974790619672853340925800
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 86*x^4 + 544*x^5 + 3904*x^6 + 31328*x^7 + 276798*x^8 + 2660564*x^9 + 27576614*x^10 + ...
such that
A(x) = 1/2 + (1 + x*(1+x))/2^2 + (1 + x*(1+x)^2)^2/2^3 + (1 + x*(1+x)^3)^3/2^4 + (1 + x*(1+x)^4)^4/2^5 + (1 + x*(1+x)^5)^5/2^6 + (1 + x*(1+x)^6)^6/2^7 + ...
Also, due to a series identity,
A(x) = 1 + x*(1+x)/(2 - (1+x))^2 + x^2*(1+x)^4/(2 - (1+x)^2)^3 + x^3*(1+x)^9/(2 - (1+x)^3)^4 + x^4*(1+x)^16/(2 - (1+x)^4)^5 + x^5*(1+x)^25/(2 - (1+x)^5)^6 + x^6*(1+x)^36/(2 - (1+x)^6)^7 + ... + x^n * (1+x)^(n^2) / (2 - (1+x)^n)^(n+1) + ...
Triangle A300280 is defined by
T(n,k) = Sum_{j>=0} C(j+k, k) * C((j+k)*k, n-k) / 2^(j+k+1), begins:
1;
0, 1;
0, 3, 1;
0, 5, 10, 1;
0, 7, 57, 21, 1;
0, 9, 252, 246, 36, 1;
0, 11, 969, 2158, 710, 55, 1;
0, 13, 3414, 15927, 10260, 1635, 78, 1; ...
the row sums of which form this sequence.
RELATED INFINITE SERIES.
At x = -1/2: the following sums are equal
S1 = Sum_{n>=1} (2^n - 1)^(n-1) / 2^(n^2),
S1 = Sum_{n>=1} (-1)^(n-1) / (2^n - 1)^n.
Explicitly,
S1 = 1/2 + 3/2^4 + 7^2/2^9 + 15^3/2^16 + 31^4/2^25 + 63^5/2^36 + 127^6/2^49 + 255^7/2^64 + 511^8/2^81 + 1023^9/2^100 + 2047^10/2^121 + 4095^11/2^144 + ...
S1 = 1 - 1/3^2 + 1/7^3 - 1/15^4 + 1/31^5 - 1/63^6 + 1/127^7 - 1/255^8 + 1/511^9 - 1/1023^10 + 1/2047^11 - 1/4095^12 + 1/8191^13 - 1/16383^14 + ...
where S1 = 0.891784622610953349715890136060239421022216970366139189336822360...
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{a(n) = my(A = sum(m=0, n, x^m * (1+x)^(m^2) / (2 - (1 + x + x*O(x^n))^m )^(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),", "))
A302105
G.f. A(x) satisfies: A(x) = Sum_{n>=0} (4 + x*A(x)^n)^n / 5^(n+1).
Original entry on oeis.org
1, 1, 10, 175, 3835, 95090, 2551480, 72360700, 2139052845, 65329175385, 2049247480265, 65752776679275, 2151923601749290, 71691421965972905, 2428004656549037580, 83523871228996755395, 2917260885363111908770, 103451501815230690971935, 3726040763307222530311125, 136400452641372633368206185, 5080478361492407723101242440
Offset: 0
G.f.: A(x) = 1 + x + 10*x^2 + 175*x^3 + 3835*x^4 + 95090*x^5 + 2551480*x^6 + 72360700*x^7 + 2139052845*x^8 + 65329175385*x^9 + 2049247480265*x^10 + ...
such that
A(x) = 4/5 + (4 + x*A(x))/5^2 + (4 + x*A(x)^2)^2/5^3 + (4 + x*A(x)^3)^3/5^4 + (4 + x*A(x)^4)^4/5^5 + (4 + x*A(x)^5)^5/5^6 + (4 + x*A(x)^6)^6/5^7 + ...
Also, due to a series identity,
A(x) = 1 + x*A(x)/(5 - 4*A(x))^2 + x^2*A(x)^4/(5 - 4*A(x)^2)^3 + x^3*A(x)^9/(5 - 4*A(x)^3)^4 + x^4*A(x)^16/(5 - 4*A(x)^4)^5 + x^5*A(x)^25/(5 - 4*A(x)^5)^6 + x^6*A(x)^36/(5 - 4*A(x)^6)^7 + ... + x^n * A(x)^(n^2) / (5 - 4*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) / (5 - 4*A^m + x*O(x^n))^(m+1) )); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
Showing 1-4 of 4 results.
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