|
%I #36 Jul 02 2019 21:53:37
%S 1,1,2,3,10,27,109,427,1958,9467,49459,274712,1614199,9996580,
%T 64940226,441179351,3125044744,23021059143,175976694409,1393077001768,
%U 11400165893604,96286628620151,838123560744653,7508677200329118,69152466448641019,653972815019717914,6344196087718370108,63073829812214409363,642093553544993640915,6687618467901426663337,71209887695115322487153,774636418450000537370791
%N G.f.: Sum_{n>=0} x^n * ((1+x)^n + 1)^n / (1 + x*(1+x)^n)^(n+1).
%C More generally, the following sums are equal:
%C (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (q^n + p)^n / (1 + p*q^n*r)^(n+k),
%C (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * (q^n - p)^n / (1 - p*q^n*r)^(n+k),
%C for any fixed integer k; here, k = 1 and r = x, p = 1, q = (1+x). See other examples for k = 2 (A326006), k = 3 (A326007), k = 4 (A326008).
%H Paul D. Hanna, <a href="/A323680/b323680.txt">Table of n, a(n) for n = 0..500</a>
%F G.f.: Sum_{n>=0} x^n * ((1+x)^n + 1)^n / (1 + x*(1+x)^n)^(n+1).
%F G.f.: Sum_{n>=0} x^n * ((1+x)^n - 1)^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) * ( (1+x)^n + (1+x)^k )^(n-k) * (-1)^k.
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * Sum_{j=0..n-k} (-1)^j * 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} (-1)^k * 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) * (-1)^j * (n-i)! / ((n-i-j-k)!*j!*k!).
%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 10*x^4 + 27*x^5 + 109*x^6 + 427*x^7 + 1958*x^8 + 9467*x^9 + 49459*x^10 + 274712*x^11 + 1614199*x^12 + ...
%e such that
%e A(x) = 1/(1+x) + x*((1+x) + 1)/(1 + x*(1+x))^2 + x^2*((1+x)^2 + 1)^2/(1 + x*(1+x)^2)^3 + x^3*((1+x)^3 + 1)^3/(1 + x*(1+x)^3)^4 + x^4*((1+x)^4 + 1)^4/(1 + x*(1+x)^4)^5 + x^5*((1+x)^5 + 1)^5/(1 + x*(1+x)^5)^6 + x^6*((1+x)^6 + 1)^6/(1 + x*(1+x)^6)^7 + x^7*((1+x)^7 + 1)^7/(1 + x*(1+x)^7)^8 + ...
%e also,
%e A(x) = 1/(1-x) + x*((1+x) - 1)/(1 - x*(1+x))^2 + x^2*((1+x)^2 - 1)^2/(1 - x*(1+x)^2)^3 + x^3*((1+x)^3 - 1)^3/(1 - x*(1+x)^3)^4 + x^4*((1+x)^4 - 1)^4/(1 - x*(1+x)^4)^5 + x^5*((1+x)^5 - 1)^5/(1 - x*(1+x)^5)^6 + x^6*((1+x)^6 - 1)^6/(1 - x*(1+x)^6)^7 + x^7*((1+x)^7 - 1)^7/(1 - x*(1+x)^7)^8 + ...
%e RELATED INFINITE SERIES.
%e At x = -1/2, the g.f. as a power series in x diverges, but the related series converges:
%e S = Sum_{n>=0} (-1/2)^n*(1/2^n + 1)^n / (1 - 1/2^(n+1))^(n+1), and
%e S = Sum_{n>=0} (-1/2)^n*(1/2^n - 1)^n / (1 + 1/2^(n+1))^(n+1).
%e Equivalently,
%e S = Sum_{n>=0} (-2)^n * (2^n + 1)^n / (2^(n+1) - 1)^(n+1), and
%e S = Sum_{n>=0} 2^n * (2^n - 1)^n / (2^(n+1) + 1)^(n+1) ;
%e written explicitly,
%e S = 1/(2-1) - 2*(2+1)/(2^2-1)^2 + 2^2*(2^2+1)^2/(2^3-1)^3 - 2^3*(2^3+1)^3/(2^4-1)^4 + 2^4*(2^4+1)^4/(2^5-1)^5 - 2^5*(2^5+1)^5/(2^6-1)^6 + 2^6*(2^6+1)^6/(2^7-1)^7 + ...
%e also,
%e S = 1/(2+1) + 2*(2-1)/(2^2+1)^2 + 2^2*(2^2-1)^2/(2^3+1)^3 + 2^3*(2^3-1)^3/(2^4+1)^4 + 2^4*(2^4-1)^4/(2^5+1)^5 + 2^5*(2^5-1)^5/(2^6+1)^6 + 2^6*(2^6-1)^6/(2^7+1)^7 + ...
%e where
%e S = 0.54250659711853510199583159448775795614278675261848614946772936514239222...
%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(j=0,n-i, sum(k=0,n-i-j, (-1)^k * binomial(n-i, j) * binomial(n-i-j, k) * binomial((n-i-j)*(n-i-k), 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, (-1)^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. A323681, A323695, A325059, A326006, A326007, A326008.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 11 2019
| 