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A326269 G.f.: Sum_{n>=0} (1+x + x^n)^n * x^n. 1

%I

%S 1,1,3,3,7,10,17,27,41,70,109,168,276,439,688,1099,1774,2820,4488,

%T 7219,11596,18574,29844,48040,77302,124515,200756,323695,522168,

%U 843020,1361409,2198679,3552094,5740668,9279009,14999925,24252057,39216310,63419775,102569373,165898349,268344639,434076911,702197193,1135967897,1837747824,2973155053,4810149922,7782281092,12591037633,20371441356

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

%C More generally, the following sums are equal:

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

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

%C for any fixed integer k; this sequence results when k=1, p = 1+x, q = x, r = x.

%H Paul D. Hanna, <a href="/A326269/b326269.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%F a(n) ~ (5 + sqrt(5))/10 * Phi^n, where Phi = (1 + sqrt(5))/2.

%e G.f.: A(x) = 1 + x + 3*x^2 + 3*x^3 + 7*x^4 + 10*x^5 + 17*x^6 + 27*x^7 + 41*x^8 + 70*x^9 + 109*x^10 + 168*x^11 + 276*x^12 + 439*x^13 + 688*x^14 + 1099*x^15 + ...

%e such that

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

%e also

%e A(x) = 1/(1-x-x^2) + x^2/(1-x^2-x^3)^2 + x^6/(1-x^3-x^4)^3 + x^12/(1-x^4-x^5)^4 + x^20/(1-x^5-x^6)^5 + x^30/(1-x^6-x^7)^6 + x^42/(1-x^7-x^8)^7 + ...

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

%o for(n=0,50, print1(a(n),", "))

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

%o for(n=0,50, print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 02 2019

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Last modified January 19 20:41 EST 2020. Contains 331066 sequences. (Running on oeis4.)