A132330
G.f.: A(x) = 1 + x*(A_2)^3; A_2 = 1 + x^2*(A_3)^3; A_3 = 1 + x^3*(A_4)^3; ... A_n = 1 + x^n*(A_{n+1})^3 for n>=1 with A_1 = A(x).
Original entry on oeis.org
1, 1, 0, 3, 0, 3, 9, 1, 18, 9, 36, 45, 57, 90, 114, 351, 165, 558, 738, 1044, 1791, 1908, 3915, 4926, 8568, 8553, 17217, 26271, 30474, 50967, 68526, 113319, 144324, 219195, 299359, 473454, 665424, 860733, 1396350, 1895913, 2762550, 3790935, 5695974
Offset: 0
A171793
Triangle read by rows: T(n,k) is the number of ternary trees with n edges and path length k; 0<=k<=n(n-1)/2.
Original entry on oeis.org
1, 1, 0, 3, 0, 0, 3, 9, 0, 0, 0, 1, 18, 9, 27, 0, 0, 0, 0, 0, 9, 45, 57, 54, 27, 81, 0, 0, 0, 0, 0, 0, 0, 36, 87, 270, 81, 297, 171, 162, 81, 243, 0, 0, 0, 0, 0, 0, 0, 0, 0, 84, 261, 567, 756, 936, 585, 972, 729, 891, 513, 486, 243, 729, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 126, 774, 1080
Offset: 0
G.f.: A(x,q) = 1 + x + (3*q)*x^2 + (3*q^2 + 9*q^3)*x^3 + (q^3 + 18*q^4 + 9*q^5 + 27*q^6)*x^4 +...
A(x,q)^3 = 1 + 3*x + (3 + 9*q)*x^2 + (1 + 18*q + 9*q^2 + 27*q^3)*x^3 +...
Triangle begins:
1;
1;
0,3;
0,0,3,9;
0,0,0,1,18,9,27;
0,0,0,0,0,9,45,57,54,27,81;
0,0,0,0,0,0,0,36,87,270,81,297,171,162,81,243;
0,0,0,0,0,0,0,0,0,84,261,567,756,936,585,972,729,891,513,486,243,729;
0,0,0,0,0,0,0,0,0,0,0,126,774,1080,2817,2682,4383,1998,4941,3294,3780,2241,4374,2187,2673,1539,1458,729,2187; ...
-
{T(n,k)=local(A=1+x);for(i=1,n,A=1+x*subst(A,x,q*x+x*O(x^n))^3); polcoeff(polcoeff(A,n,x)+O(q^(n*(n-1)/2+1)),k,q)}
A275690
G.f. A(x) satisfies: 1 = ...(((((A(x) - x)^(1/3) - x^2)^(1/3) - x^3)^(1/3) - x^4)^(1/3) - x^5)^(1/3) -...- x^n)^(1/3) -..., an infinite series of nested cube roots.
Original entry on oeis.org
1, 1, 3, 9, 30, 99, 334, 1116, 3744, 12504, 41724, 138840, 461187, 1528554, 5057028, 16699293, 55051065, 181184337, 595400772, 1953715239, 6401926227, 20950064478, 68472011889, 223521012585, 728827015536, 2373846887673, 7723658267667, 25104640758607, 81519763177575, 264463605423009, 857192148657477, 2775964660002954, 8982278557410627, 29040795844301862, 93819208534071840, 302863860771034455, 976981070712962919, 3149327670664845204
Offset: 0
-
{a(n) = my(A=1 +x*O(x^n)); for(k=0, n, A = A^3 + x^(n+1-k)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
Showing 1-3 of 3 results.