A325997 - OEIS

%I #19 Jul 07 2019 09:48:00

%S 1,4,3,10,8,18,7,40,9,44,41,54,13,150,15,88,127,168,19,298,21,324,275,

%T 180,25,854,132,238,524,774,31,1286,33,1180,893,378,674,2998,39,460,

%U 1406,3744,43,2790,45,3458,4397,648,49,8420,303,4714,2960,6270,55,7060,6492,9120,4049,990,61,30748,63,1120,7697,13788,15082,17626,69,16834,6971,28788,73,48088,75,1558,39792,25578,12091,41578,81,77874,11540,1890,85,121650,58227

%N G.f.: Sum_{n>=0} (n+1) * (x + x^n)^n.

%C More generally, the following sums are equal:

%C (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,

%C (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),

%C for any fixed integer k; here, k = 2 and p = x, q = x, r = 1.

%H Robert Israel, <a href="/A325997/b325997.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: Sum_{n>=0} (n+1) * (x + x^n)^n.

%F G.f.: Sum_{n>=0} (n+1) * x^(n^2) / (1 - x^(n+1))^(n+2).

%e G.f.: A(x) = 1 + 4*x + 3*x^2 + 10*x^3 + 8*x^4 + 18*x^5 + 7*x^6 + 40*x^7 + 9*x^8 + 44*x^9 + 41*x^10 + 54*x^11 + 13*x^12 + 150*x^13 + 15*x^14 + 88*x^15 +...

%e where

%e A(x) = 1 + 2*(x + x) + 3*(x + x^2)^2 + 4*(x + x^3)^3 + 5*(x + x^4)^4 + 6*(x + x^5)^5 + 7*(x + x^6)^6 + 8*(x + x^7)^7 + 9*(x + x^8)^8 + 10*(x + x^9)^9 + ...

%e Also

%e A(x) = 1/(1-x)^2 + 2*x/(1 - x^2)^3 + 3*x^4/(1 - x^3)^4 + 4*x^9/(1 - x^4)^5 + 5*x^16/(1 - x^5)^6 + 6*x^25/(1 - x^6)^7 + 7*x^36/(1 - x^7)^8 + 8*x^49/(1 - x^8)^9 + 9*x^64/(1 - x^9)^10 + 10*x^81/(1 - x^10)^11 + ...

%p G:= add((n+1)*(x+x^n)^n,n=0..100):

%p S:= series(G,x,101):

%p seq(coeff(S,x,n),n=0..100); # _Robert Israel_, Jun 02 2019

%o (PARI) {a(n)=polcoeff(sum(m=0, n, (m+1) * (x + x^m +x*O(x^n))^m), n)}

%o for(n=0, 100, print1(a(n), ", "))

%o (PARI) {a(n)=polcoeff(sum(m=0, n, (m+1) * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+2)), n)}

%o for(n=0, 100, print1(a(n), ", "))

%Y Cf. A217669, A325998, A325999.

%K nonn,look

%O 0,2

%A _Paul D. Hanna_, Jun 02 2019