A052853
A simple grammar.
Original entry on oeis.org
0, 1, 2, 5, 14, 42, 138, 466, 1643, 5919, 21773, 81279, 307483, 1175352, 4534161, 17626999, 68992703, 271641249, 1075144364, 4275274867, 17071822275, 68428152475, 275217386092, 1110375948303, 4492641333003, 18225081419544, 74111194585752, 302040709982249
Offset: 0
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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spec := [S,{C=Prod(Z,B),S=Cycle(C),B=Set(S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
A205773
G.f. satisfies: A(x) = 1/Product_{n>=1} (1 - 3*x^n*A(x^n)).
Original entry on oeis.org
1, 3, 21, 156, 1335, 12153, 116778, 1160715, 11848530, 123420648, 1306709841, 14019657771, 152092615971, 1665531792021, 18386262679557, 204393214435791, 2286101345820933, 25708109998131381, 290490321604346535, 3296566844230833750, 37555644504960139647
Offset: 0
G.f.: A(x) = 1 + 3*x + 21*x^2 + 156*x^3 + 1335*x^4 + 12153*x^5 +...
where
A(x) = 1/((1 - 3*x*A(x)) * (1 - 3*x^2*A(x^2)) * (1 - 3*x^3*A(x^3)) *...).
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{a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-3*x^k*subst(A, x, x^k+x*O(x^n))))); polcoeff(A, n)}
A205774
G.f. satisfies: A(x) = 1/Product_{n>=1} (1 - x^n*A(x^n)^3).
Original entry on oeis.org
1, 1, 5, 27, 177, 1245, 9399, 73659, 595510, 4923724, 41451675, 354071010, 3061018302, 26732084764, 235476740731, 2089770720125, 18666863392846, 167697751329817, 1514206777182411, 13734387733516323, 125083419013852945, 1143367086845429280
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 27*x^3 + 177*x^4 + 1245*x^5 +...
where
A(x) = 1/((1 - x*A(x)^3) * (1 - x^2*A(x^2)^3) * (1 - x^3*A(x^3)^3) *...).
A205775
G.f. satisfies: A(x) = 1/Product_{n>=1} (1 - x^n*A(x^n)^n).
Original entry on oeis.org
1, 1, 3, 8, 26, 79, 276, 936, 3376, 12259, 45648, 171739, 655664, 2524835, 9813259, 38410167, 151332137, 599541153, 2387199083, 9547195445, 38335338712, 154484001619, 624579964260, 2532713370789, 10298393401623, 41979975505800, 171522040764060
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 26*x^4 + 79*x^5 + 276*x^6 + 936*x^7 +...
where
A(x) = 1/((1 - x*A(x)) * (1 - x^2*A(x^2)^2) * (1 - x^3*A(x^3)^3) *...).
A308379
E.g.f. A(x) satisfies: A(x) = x * Product_{k>=1} 1/(1 - A(x^k))^(1/k).
Original entry on oeis.org
1, 2, 15, 152, 2255, 40944, 938161, 25026896, 777966129, 27346727600, 1077001807871, 46870231698168, 2235954785893231, 115950345421719704, 6496012991027031585, 390935629387700612384, 25153144712405994085409, 1722934940168892344912928, 125180348349211811174365615
Offset: 1
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terms = 19; A[] = 0; Do[A[x] = x Product[1/(1 - A[x^k])^(1/k), {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] Range[0, terms]! // Rest
A329802
G.f. A(x) satisfies: A(x) = 1 / (1 - x * Product_{k>=1} A(x^k)).
Original entry on oeis.org
1, 1, 2, 6, 19, 64, 219, 777, 2803, 10315, 38496, 145516, 555764, 2142060, 8320207, 32538518, 128012533, 506300507, 2011932479, 8028941336, 32163411045, 129291553211, 521372223648, 2108522273338, 8549844313915, 34753397386201, 141584261960345
Offset: 0
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nmax = 26; A[] = 0; Do[A[x] = 1/(1 - x Product[A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]