|
%I #10 Apr 06 2019 01:14:41
%S 1,3,10,41,190,973,5413,32351,205966,1387807,9845083,73215780,
%T 568757151,4601092084,38660287934,336623442207,3031260737552,
%U 28178974826871,269995107206317,2662508737568260,26987695641386128,280844928307623929,2997258604356945337,32772404387384205040,366794262989293809151,4198563078511314225148,49113768374394883013208,586698015175211371037407,7152213983896219165256687
%N G.f.: Sum_{n>=0} x^n * (1 + (1+x)^n)^n / (1 - x*(1+x)^n)^(n+1).
%H Paul D. Hanna, <a href="/A325059/b325059.txt">Table of n, a(n) for n = 0..500</a>
%F G.f.: Sum_{n>=0} x^n * (1 + (1+x)^n)^n / (1 - x*(1+x)^n)^(n+1).
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * ( (1+x)^n + (1+x)^k )^(n-k).
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * Sum_{j=0..n-k} binomial(n-k,j) * (1 + x)^((n-j)*(n-k)).
%F FORMULAS INVOLVING TERMS.
%F a(n) = Sum_{i=0..n} Sum_{j=0..n-i} Sum_{k=0..n-i-j} binomial(n-i,j) * binomial(n-i-j,k) * binomial((n-i-j)*(n-i-k),i).
%F a(n) = Sum_{i=0..n} Sum_{j=0..n-i} Sum_{k=0..n-i-j} binomial((n-i-j)*(n-i-k),i) * (n-i)! / ((n-i-j-k)!*j!*k!).
%e G.f.: A(x) = 1 + 3*x + 10*x^2 + 41*x^3 + 190*x^4 + 973*x^5 + 5413*x^6 + 32351*x^7 + 205966*x^8 + 1387807*x^9 + 9845083*x^10 + 73215780*x^11 + 568757151*x^12 + ...
%e such that
%e A(x) = 1/(1-x) + x*(1 + (1+x))/(1 - x*(1+x))^2 + x^2*(1 + (1+x)^2)^2/(1 - x*(1+x)^2)^3 + x^3*(1 + (1+x)^3)^3/(1 - x*(1+x)^3)^4 + x^4*(1 + (1+x)^4)^4/(1 - x*(1+x)^4)^5 + x^5*(1 + (1+x)^5)^5/(1 - x*(1+x)^5)^6 + x^6*(1 + (1+x)^6)^6/(1 - x*(1+x)^6)^7 + x^7*(1 + (1+x)^7)^7/(1 - x*(1+x)^7)^8 + ...
%o (PARI) {a(n) = my(A = sum(m=0, n+1, x^m*((1+x +x*O(x^n) )^m + 1)^m/(1 - x*(1+x +x*O(x^n) )^m )^(m+1) )); polcoeff(A, n)}
%o for(n=0, 35, print1(a(n), ", "))
%o (PARI) {a(n) = sum(i=0, n, sum(k=0, n-i, sum(j=0, n-i-k, binomial(n-i, k) * binomial(n-i-k, j) * binomial((n-i-k)*(n-i-j), i) )))}
%o for(n=0, 35, print1(a(n), ", "))
%o (PARI) {a(n) = sum(i=0, n, sum(j=0, n-i, sum(k=0, n-i-j, binomial((n-i-j)*(n-i-k), i) * (n-i)! / ((n-i-j-k)!*j!*k!) )))}
%o for(n=0, 35, print1(a(n), ", "))
%Y Cf. A323680.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 28 2019
|
