A301465 -id:A301465 - cp's OEIS frontend

A301306 G.f.: Sum_{n>=0} (1 + (1+x)^n)^n * x^n.

1, 2, 5, 16, 60, 254, 1188, 6043, 33080, 193249, 1197001, 7819995, 53648847, 385090323, 2883045424, 22451716833, 181437812058, 1518374146260, 13133970646948, 117235109969112, 1078235776311405, 10204120439288725, 99244762587719585, 990878067150790140, 10145281310155565842, 106420501631411705747, 1142671059786354295966, 12548652816798990883431, 140839029768184796119004

Offset: 0

Paul D. Hanna, Mar 21 2018

nonn

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 16*x^3 + 60*x^4 + 254*x^5 + 1188*x^6 + 6043*x^7 + 33080*x^8 + 193249*x^9 + 1197001*x^10 + ...
such that
A(x) = 1 + (1 + (1+x))*x + (1 + (1+x)^2)^2*x^2 + (1 + (1+x)^3)^3*x^3 + (1 + (1+x)^4)^4*x^4 + (1 + (1+x)^5)^5*x^5 + (1 + (1+x)^6)^6*x^6 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A301465.

{a(n) = my(A=1); A = sum(k=0,n, (1 + (1+x)^k +x*O(x^n))^k * x^k ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

{a(n) = sum(j=0,n, sum(k=0,n-j, binomial(n-j,k) * binomial((n-j)*k,j) ))}
for(n=0,30,print1(a(n),", "))

G.f.: Sum_{n>=0} x^n * (1+x)^(n^2) / (1 - x*(1+x)^n)^(n+1).
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (1+x)^(n*k).
a(n) = Sum_{j=0..n} Sum_{k=0..n-j} binomial(n-j, k) * binomial((n-j)*k, j).

A301436 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 2*(1+x)^n - A(x) )^n / 2^(n+1).

Original entry on oeis.org

1, 6, 50, 1582, 82722, 5842550, 511261682, 52903385886, 6290859281538, 843328959011622, 125706002934030898, 20617322695573745742, 3689811206934015405474, 715633021826704924420758, 149544785675949258192968178, 33502338836970792659941911358, 8011296279710787237594088464898, 2036927238948023349890031708437830, 548778491694092921577420334962662962, 156179940994829385561873698156273034606

Offset: 0

Paul D. Hanna, Mar 24 2018

nonn

			G.f.: A(x) = 1 + 6*x + 50*x^2 + 1582*x^3 + 82722*x^4 + 5842550*x^5 + 511261682*x^6 + 52903385886*x^7 + 6290859281538*x^8 + ...
such that
1 = 1/2  +  (2*(1+x) - A(x))/2^2  +  (2*(1+x)^2 - A(x))^2/2^3  +  (2*(1+x)^3 - A(x))^3/2^4  +  (2*(1+x)^4 - A(x))^4/2^5  +  (2*(1+x)^5 - A(x))^5/2^6 + ...
Also,
1 = 1/(2 + A(x))  +  2*(1+x)/(2 + (1+x)*A(x))^2  +  2^2*(1+x)^4/(2 + (1+x)^2*A(x))^3  +  2^3*(1+x)^9/(2 + (1+x)^3*A(x))^4  +  2^4*(1+x)^16/(2 + (1+x)^4*A(x))^5  +  2^5*(1+x)^25/(2 + (1+x)^5*A(x))^6  + ...

Paul D. Hanna, Table of n, a(n) for n = 0..50

Cf. A303653, A301465.

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

cp's OEIS Frontend

A301306 G.f.: Sum_{n>=0} (1 + (1+x)^n)^n * x^n.

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