A374446
Positions of zeros in the expansion of Product_{k>=0} (1 - x^(2^k))^3; A373308(a(n)) = 0 for n >= 1.
Original entry on oeis.org
2, 11, 14, 47, 50, 59, 62, 191, 194, 203, 206, 239, 242, 251, 254, 767, 770, 779, 782, 815, 818, 827, 830, 959, 962, 971, 974, 1007, 1010, 1019, 1022, 3071, 3074, 3083, 3086, 3119, 3122, 3131, 3134, 3263, 3266, 3275, 3278, 3311, 3314, 3323, 3326, 3839, 3842, 3851, 3854, 3887, 3890, 3899
Offset: 1
Product_{k>=0} (1 - x^(2^k))^3 = 1 - 3*x + 0*x^2 + 8*x^3 - 9*x^4 + 3*x^5 + 8*x^6 - 24*x^7 + 15*x^8 + 19*x^9 - 24*x^10 + 0*x^11 + 17*x^12 - 27*x^13 + 0*x^14 + 64*x^15 + ... + A373308(n)*x^n + ...
in which the coefficients of {x^2, x^11, x^14, x^47, ..., x^a(n), ...} are zero.
Compare to numbers with only even length runs in their binary expansion: A001196 = [3, 12, 15, 48, 51, 60, 63, 192, 195, 204, 207, 240, 243, 252, 255, 768, ...]; it appears that a(n) = A001196(n) - 1 for n >= 1.
A373309
Number of binary partitions of n into three kinds of parts.
Original entry on oeis.org
1, 3, 9, 19, 42, 78, 146, 246, 420, 668, 1068, 1620, 2470, 3618, 5310, 7546, 10746, 14910, 20706, 28134, 38262, 51090, 68238, 89706, 117964, 153012, 198468, 254332, 325914, 413214, 523778, 657606, 825444, 1027292, 1278060, 1577748, 1947062, 2386002, 2922702, 3557162, 4327644
Offset: 0
G.f.: A(x) = 1 + 3*x + 9*x^2 + 19*x^3 + 42*x^4 + 78*x^5 + 146*x^6 + 246*x^7 + 420*x^8 + 668*x^9 + 1068*x^10 + 1620*x^11 + 2470*x^12 + ...
where A(x) = 1/((1-x)^3*(1-x^2)^3*(1-x^4)^3* ... * (1 - x^(2^k))^3 * ...).
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nmax = 40; CoefficientList[Series[1/Product[(1 - x^(2^k))^3, {k, 0, Log[2, nmax] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 25 2024 *)
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{a(n) = my(A = 1/prod(k=0,#binary(n), (1 - x^(2^k) +x*O(x^n))^3 )); polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))
A373313
Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x)/(1 - A(x))^3 ).
Original entry on oeis.org
1, 3, 18, 127, 999, 8376, 73400, 664143, 6157467, 58190531, 558478098, 5428532148, 53331912158, 528721992000, 5282688183600, 53140908294191, 537760961917833, 5470638540940401, 55914705172750446, 573908634206898951, 5913010265931479289, 61132102068652970100, 634002859944973526904
Offset: 1
G.f.: A(x) = x + 3*x^2 + 18*x^3 + 127*x^4 + 999*x^5 + 8376*x^6 + 73400*x^7 + 664143*x^8 + 6157467*x^9 + 58190531*x^10 + ...
where A( x*A(x)/(1 - A(x))^3 ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 45*x^4 + 362*x^5 + 3084*x^6 + 27318*x^7 + 249149*x^8 + 2323968*x^9 + 22067697*x^10 + ...
Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then
R(x) = x * Product_{n>=0} (1 - x^(2^n))^3 = x - 3*x^2 + 8*x^4 - 9*x^5 + 3*x^6 + 8*x^7 - 24*x^8 + 15*x^9 + 19*x^10 + ... + A373308(n-1) * x^n + ...
thus,
x = A(x) * (1 - A(x))^3 * (1 - A(x)^2)^3 * (1 - A(x)^4)^3 * (1 - A(x)^8)^3 * (1 - A(x)^16)^3 * ... * (1 - A(x)^(2^n))^3 * ...
Also, notice that the cube root of A(x)/x is an integral series
(A(x)/x)^(1/3) = 1 + x + 5*x^2 + 32*x^3 + 239*x^4 + 1937*x^5 + 16578*x^6 + 147408*x^7 + 1348465*x^8 + 12608851*x^9 + 119972595*x^10 + ...
SPECIFIC VALUES.
A(t) = 1/6 at t = (1/6) * Product_{n>=0} (1 - 1/6^(2^n))^3 = 0.0884290923082561345726735004152032422138677544...
A(t) = 1/7 at t = (1/7) * Product_{n>=0} (1 - 1/7^(2^n))^3 = 0.0844605844040460521136280418653467784637497846...
A(t) = 1/8 at t = (1/8) * Product_{n>=0} (1 - 1/8^(2^n))^3 = 0.0798174217593180496284155971364088109289815675...
A(1/12) = 0.1379538716718371951653031812720929490038524971492263...
A(1/13) = 0.1160657279647048938238673646663527089747582497393475...
A(1/14) = 0.1016889922856297159061963243507242491941351481713051...
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{a(n) = my(A = serreverse(x*prod(k=0,#binary(n), (1 - x^(2^k) + x*O(x^n))^3))); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))
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{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( subst(Ser(A),x, x*Ser(A)/(1 - Ser(A))^3 ) - Ser(A)^2,#A)); A[n+1]}
for(n=1,30,print1(a(n),", "))
Showing 1-3 of 3 results.
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