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A229711 G.f.: Sum_{n>=0} a(n)*x^n / (1+x)^(n^3) = x.

%I A229711 #11 Sep 27 2013 21:44:13
%S A229711 1,1,7,154,7329,621054,83287785,16339143828,4433073578739,
%T A229711 1595084475573057,736780843688600494,425703341782263982836,
%U A229711 301237142332910524156150,256518292539312393631293756,259004327874862610288497260501,306183323229810278424153632807196
%N A229711 G.f.: Sum_{n>=0} a(n)*x^n / (1+x)^(n^3) = x.
%e A229711 G.f.: x = 1*x/(1+x) + 1*x^2/(1+x)^8 + 7*x^3/(1+x)^27 + 154*x^4/(1+x)^64 + 7329*x^5/(1+x)^125 + 621054*x^6/(1+x)^216 + 83287785*x^7/(1+x)^343 +...
%e A229711 ALTERNATE GENERATING METHOD.
%e A229711 Also forms the final terms in rows of the triangle where row n+1 equals the partial sums of row n with the final term repeated 3*n*(n-1)+1 times, starting with a '1' in row 1, as illustrated by:
%e A229711 1;
%e A229711 1, 1, 1, 1, 1, 1, 1;
%e A229711 1, 2, 3, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7;
%e A229711 1, 3, 6, 10, 15, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154; ...
%e A229711 MATRIX GENERATING METHOD.
%e A229711 Given triangle T(n, k) = binomial(k^3+n-k-1, n-k), such that the g.f. of column k equals 1/(1-x)^(k^3) for k>=1, which begins:
%e A229711 1;
%e A229711 1, 1;
%e A229711 1, 8, 1;
%e A229711 1, 36, 27, 1;
%e A229711 1, 120, 378, 64, 1;
%e A229711 1, 330, 3654, 2080, 125, 1;
%e A229711 1, 792, 27405, 45760, 7875, 216, 1;
%e A229711 1, 1716, 169911, 766480, 333375, 23436, 343, 1; ...
%e A229711 then this sequence forms column 1 (ignoring signs) of the matrix inverse:
%e A229711 1;
%e A229711 -1, 1;
%e A229711 7, -8, 1;
%e A229711 -154, 180, -27, 1;
%e A229711 7329, -8616, 1350, -64, 1;
%e A229711 -621054, 731502, -116244, 5920, -125, 1;
%e A229711 83287785, -98171784, 15685569, -820480, 19125, -216, 1;
%e A229711 -16339143828, 19265191212, -3085386984, 163253040, -3963750, 50652, -343, 1; ...
%o A229711 (PARI) /* GENERATING FUNCTION: */
%o A229711 {a(n)=local(F=1/(1+x+x*O(x^n))); polcoeff(x-sum(k=1, n-1, a(k)*x^k*F^(k^3)), n)}
%o A229711 for(n=1,20,print1(a(n),", "))
%o A229711 (PARI) /* SUMMATION METHOD: */
%o A229711 {A=[1, 1]; for(i=1, 20, A=concat(A, -Vec(sum(n=0, #A-1, A[n+1]*x^n/(1+x+x*O(x^#A))^(n^3)))[#A+1])); for(n=1, #A-1, print1(A[n+1], ", "))}
%o A229711 (PARI) /* MATRIX METHOD: */
%o A229711 {a(n)=local(M=matrix(n,n,r,c,if(r>=c,binomial(c^3+r-c-1, r-c))));-(-1)^n*(M^-1)[n,1]}
%o A229711 for(n=1,20,print1(a(n),", "))
%Y A229711 Cf. A177447.
%K A229711 nonn
%O A229711 1,3
%A A229711 _Paul D. Hanna_, Sep 27 2013