A249936 - OEIS

%I #9 Nov 24 2014 22:21:11

%S 1,1,4,7,49,85,565,1016,7350,13195,95564,173088,1275166,2317299,

%T 17166633,31318180,233446920,427044441,3195627268,5859732238,

%U 43994595982,80828819606,608411734754,1119676074515,8445900399153,15565569301793,117619945650292,217042442850592,1642503844147711,3034229201413924

%N G.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} x^k * {[x^k] A(x)^(3*n)}.

%C Here [x^k] A(x)^(3*n) denotes the coefficient of x^k in A(x)^(3*n).

%F G.f. satisfies: A(x) = (1  + 3*x^2*G'(x^2)/G(x^2)) / (1 - x*G(x^2)^3), where A(x) = G(x/A(x)^3) and G(x) = A(x*G(x)^3) = ( (1/x)*Series_Reversion(x/A(x)^3) )^(1/3).

%e G.f.: A(x) = 1 + x + 4*x^2 + 7*x^3 + 49*x^4 + 85*x^5 + 565*x^6 + 1016*x^7 +...

%e Related expansions:

%e A(x)^3 = 1 + 3*x + 15*x^2 + 46*x^3 + 249*x^4 + 786*x^5 + 3907*x^6 +...

%e A(x)^6 = 1 + 6*x + 39*x^2 + 182*x^3 + 999*x^4 + 4446*x^5 + 22116*x^6 +...

%e A(x)^9 = 1 + 9*x + 72*x^2 + 435*x^3 + 2655*x^4 + 14247*x^5 + 77145*x^6 +...

%e A(x)^12 = 1 + 12*x + 114*x^2 + 832*x^3 + 5703*x^4 + 35076*x^5 + 208630*x^6 +...

%e GENERATING METHOD.

%e The initial terms, k=0..n, of the (3*n)-th power of g.f. A(x) begin:

%e n=0: [1];

%e n=3: [1, 3];

%e n=6: [1, 6, 39];

%e n=9: [1, 9, 72, 435];

%e n=12: [1, 12, 114, 832, 5703];

%e n=15: [1, 15, 165, 1400, 10710, 73683];

%e n=18: [1, 18, 225, 2166, 18324, 138924, 982281];

%e n=21: [1, 21, 294, 3157, 29274, 242004, 1847629, 13193967];

%e n=24: [1, 24, 372, 4400, 44370, 396720, 3251592, 24825192, 179183031]; ...

%e from which the antidiagonal sums form this sequence:

%e a(0) = 1;

%e a(1) = 1;

%e a(2) = 1 + 3 = 4;

%e a(3) = 1 + 6 = 7;

%e a(4) = 1 + 9 + 39 = 49;

%e a(5) = 1 + 12 + 72 = 85;

%e a(6) = 1 + 15 + 114 + 435 = 565;

%e a(7) = 1 + 18 + 165 + 832 = 1016; ...

%e ALTERNATE GENERATING METHOD.

%e Define G(x) such that G(x) = A(x*G(x)^3) = ( (1/x)*Series_Reversion(x/A(x)^3) )^(1/3):

%e G(x) = 1 + x + 7*x^2 + 55*x^3 + 548*x^4 + 5748*x^5 + 64357*x^6 + 745777*x^7 + 8903267*x^8 + 108597640*x^9 + 1348164588*x^10 +...

%e then A(x) = (1  + 3*x^2*G'(x^2)/G(x^2)) / (1 - x*G(x^2)^3).

%e Note that 1  + 3*x^2*G'(x^2)/G(x^2) begins:

%e 1 + 3*x^2 + 39*x^4 + 435*x^6 + 5703*x^8 + 73683*x^10 + 982281*x^12 +...

%e where the coefficients form the main diagonal of the above triangle.

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, x^k*polcoeff((A+x*O(x^m))^(3*m), k))+x*O(x^n))); polcoeff(A, n)}

%o for(n=0, 40, print1(a(n), ", "))

%o (PARI) /* ALTERNATE GENERATING METHOD (faster) */

%o {a(n)=local(A=1+x, G=1); for(i=0,#binary(n)+1, G=(1/x*serreverse(x/A^3 +x^2*O(x^n)))^(1/3); A=(1+3*x^2*subst(G'/G, x, x^2))/(1-x*subst(G^3, x, x^2))); polcoeff(A, n)}

%o for(n=0, 40, print1(a(n), ", "))

%Y Cf. A222658, A249935, A249937.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 24 2014