cp's OEIS Frontend

A245374 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 3*(n+1)*x) ).

%I A245374 #8 Apr 01 2025 18:00:56
%S A245374 1,3,12,54,288,1782,12474,96714,819882,7536402,74610234,790692354,
%T A245374 8921660922,106687646802,1346863560714,17890362862434,249297686894682,
%U A245374 3634756665823602,55317506662094634,876911386062810114,14451743847813157242,247171758180997987602,4380263376360686471754
%N A245374 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 3*(n+1)*x) ).
%F A245374 G.f.: Sum_{n>=0} n! * (3*x)^n * (1+x)^n / Product_{k=1..n} (1 + 3*k*x).
%F A245374 a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..k} (-1)^i * binomial(k,i) * 3^(n-k) * (k-i+1)^(n-k).
%e A245374 G.f.: A(x) = 1 + 3*x + 12*x^2 + 54*x^3 + 288*x^4 + 1782*x^5 + 12474*x^6 +...
%e A245374 where we have the following series identity:
%e A245374 A(x) = 1/((1+x)*(1-3*x)) + x/((1+x)^2*(1-6*x)) + x^2/((1+x)^3*(1-9*x))+ x^3/((1+x)^4*(1-12*x))+ x^4/((1+x)^5*(1-15*x)) + x^5/((1+x)^6*(1-18*x)) +...
%e A245374 is equal to
%e A245374 A(x) = 1 + 3*x*(1+x)/(1+3*x) + 2!*(3*x)^2*(1+x)^2/((1+3*x)*(1+6*x)) + 3!*(3*x)^3*(1+x)^3/((1+3*x)*(1+6*x)*(1+9*x)) + 4!*(3*x)^4*(1+x)^4/((1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)) + 5!*(3*x)^5*(1+x)^5/((1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)*(1+15*x)) +...
%o A245374 (PARI) {a(n)=polcoeff( sum(m=0, n, x^m/((1+x)^(m+1)*(1 - 3*(m+1)*x) +x*O(x^n))), n)}
%o A245374 for(n=0, 30, print1(a(n), ", "))
%o A245374 (PARI) {a(n)=polcoeff( sum(m=0, n, 3^m*m!*x^m*(1+x)^m/prod(k=1, m, 1+3*k*x +x*O(x^n))), n)}
%o A245374 for(n=0, 30, print1(a(n), ", "))
%o A245374 (PARI) {a(n)=sum(k=0, floor(n/2), sum(i=0, k, (-1)^i*binomial(k, i)*(k-i+1)^(n-k)*3^(n-k)))}
%o A245374 for(n=0, 30, print1(a(n), ", "))
%Y A245374 Cf. A229046, A245373, A245375, A245376.
%K A245374 nonn
%O A245374 0,2
%A A245374 _Paul D. Hanna_, Jul 19 2014