A375439
Expansion of g.f. A(x) satisfying A(x) = x + x^2 + (2*A(x)^3 + A(x^3))/3.
Original entry on oeis.org
1, 1, 1, 2, 4, 9, 20, 48, 117, 290, 734, 1880, 4868, 12730, 33556, 89072, 237904, 638873, 1723930, 4672008, 12710904, 34703894, 95054188, 261116816, 719223064, 1985934212, 5496123033, 15242821108, 42357113994, 117918233704, 328833828334, 918470764376, 2569238134248, 7197046596440
Offset: 1
G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 20*x^7 + 48*x^8 + 117*x^9 + 290*x^10 + 734*x^11 + 1880*x^12 + 4868*x^13 + 12730*x^14 + 33556*x^15 + ...
where A(x) = x + x^2 + (2*A(x)^3 + A(x^3))/3.
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 30*x^7 + 72*x^8 + 175*x^9 + 435*x^10 + 1101*x^11 + 2819*x^12 + 7302*x^13 + 19095*x^14 + 50332*x^15 + ...
Let B(x) be the series reversion of A(x), B(A(x)) = x, then B(x) begins
B(x) = x - x^2 + x^3 - 2*x^4 + 4*x^5 - 9*x^6 + 22*x^7 - 57*x^8 + 152*x^9 - 411*x^10 + 1119*x^11 - 3063*x^12 + 8436*x^13 - 23405*x^14 + 65452*x^15 + ...
SPECIFIC VALUES.
A(1/3) = 0.6046115975458048490061476622502250915528261368314569825...
where A(1/3) = 4/9 + (2*A(1/3)^3 + A(1/27))/3.
A(1/4) = 0.345218924086872316546119663994502755734706567000751...
A(1/5) = 0.253555647303827972834265469178971877524548605418192...
A(1/6) = 0.201444567662949882659512632012060178593075505771758...
A(1/7) = 0.167365364255434800795732539120237367470157092655512...
A(1/8) = 0.143236474390624253781858259379882809014038308155736...
A(1/27) = 0.03846365186207481603806452459437536518999937182129...
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{a(n) = my(A=[0,1],Ax=x); for(i=1,n, A = concat(A,0); Ax=Ser(A);
A[#A] = polcoeff( x + x^2 + ( 2*Ax^3 + subst(Ax,x,x^3) )/3 - Ax,#A-1) );A[n+1]}
for(n=1,40,print1(a(n),", "))
A287211
The number of plane rooted complete ternary trees with 2n+1 unlabeled leaves (hence n internal nodes including the root where n starts at 0) satisfying these two conditions: (1) if one of the three children of any internal node is the greatest in deglex order then that child is not the leftmost child; (2) if one of the three children of any internal node is the smallest in deglex order then that child is not the rightmost child. Deglex order refers to degree-lexicographical order defined inductively on the number of leaves (see details under Comments).
Original entry on oeis.org
1, 1, 2, 6, 21, 78, 308, 1264, 5332, 22994, 100896, 449004
Offset: 0
Association types for arities 1, 3, 5, 7 are as follows in deglex order. See Links for a-file with association types for arities up to 11.
Arity 1, number of types 1:
a.
Arity 3, number of types 1:
[abc].
Arity 5, number of types 2:
[ab[cde]],
[a[bcd]e].
Arity 7, number of types 6:
[ab[cd[efg]]],
[ab[c[def]g]],
[a[bcd][efg]],
[a[bc[def]]g],
[a[b[cde]f]g],
[[abc]d[efg]].
A375438
Expansion of g.f. A(x) satisfying A(x) = x + x^2 + (A(x)^3 + 2*A(x^3))/3.
Original entry on oeis.org
1, 1, 1, 1, 2, 4, 6, 12, 24, 46, 93, 191, 393, 819, 1724, 3648, 7772, 16654, 35850, 77520, 168289, 366629, 801328, 1756620, 3861157, 8508247, 18791480, 41591566, 92237598, 204931918, 456096480, 1016720162, 2269865456, 5074732028, 11360680664, 25464831258, 57146836290
Offset: 1
G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 6*x^7 + 12*x^8 + 24*x^9 + 46*x^10 + 93*x^11 + 191*x^12 + 393*x^13 + 819*x^14 + 1724*x^15 + ...
where A(x) = x + x^2 + (A(x)^3 + 2*A(x^3))/3.
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 18*x^7 + 36*x^8 + 70*x^9 + 138*x^10 + 279*x^11 + 571*x^12 + 1179*x^13 + 2457*x^14 + 5168*x^15 + ...
Let B(x) be the series reversion, B(A(x)) = x, then B(x) begins
B(x) = x - x^2 + x^3 - x^4 + 3*x^6 - 8*x^7 + 9*x^8 + 18*x^9 - 134*x^10 + 442*x^11 - 997*x^12 + 1428*x^13 - 10*x^14 - 7640*x^15 + ...
SPECIFIC VALUES.
A(2/5) = 0.741461459188681119672668058998130332678610537393868...
A(1/3) = 0.515838591521774544528452689654484632143493145820237...
A(1/4) = 0.335691575266570204286454430830296229544471248787335...
A(1/5) = 0.250623759777806277498640241328479184446996870792060...
A(1/6) = 0.200220725056085320106333620370114891484630832364434...
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{a(n) = my(A=[0,1],Ax=x); for(i=1,n, A = concat(A,0); Ax=Ser(A);
A[#A] = polcoeff( x + x^2 + ( Ax^3 + 2*subst(Ax,x,x^3))/3 - Ax,#A-1) );A[n+1]}
for(n=1,40,print1(a(n),", "))
Showing 1-3 of 3 results.
Comments