A236310 - OEIS

%I #19 Feb 11 2024 02:34:00

%S 0,1,0,0,1,1,1,1,1,2,3,4,5,6,7,8,10,13,17,22,28,35,43,52,62,74,89,108,

%T 132,162,199,244,298,362,437,524,625,743,882,1047,1244,1480,1763,2102,

%U 2507,2989,3560,4233,5022,5943,7015,8261,9709,11393,13354,15641,18312,21435,25089,29365,34367,40213,47036,54985,64227,74950

%N Expansion of Sum_{k>=0} x^((k+1)^2)/(1-x)^k.

%C a(n) is the number of compositions of n such that the first part is equal to the number of parts and all parts are greater than or equal to the first part. - _John Tyler Rascoe_, Feb 10 2024

%H Alois P. Heinz, <a href="/A236310/b236310.txt">Table of n, a(n) for n = 0..10000</a>

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

%F G.f.: Sum_{k>0} A(x,k) where A(x,k) = (x^k)*(x^k/(1-x))^(k-1) is the g.f. for compositions of this kind with first part k. - _John Tyler Rascoe_, Feb 10 2024

%e From _John Tyler Rascoe_, Feb 10 2024: (Start)

%e The compositions for n = 9..11 are:

%e 9:  [3,3,3], [2,7];

%e 10: [3,4,3], [3,3,4], [2,8];

%e 11: [3,4,4], [3,3,5], [3,5,3], [2,9].

%e (End)

%o (PARI) N=66; q='q+O('q^N);

%o gf=sum(n=0, N, q^((n+1)^2) / (1-q)^n );

%o concat([0],Vec(gf))

%Y Cf. A098131 (g.f. Sum_{k>=0} x^(k^2)/(1-x)^k).

%Y Cf. A219282 (g.f. Sum_{k>=0} x^(k*(k+1)/2)/(1-x)^k).

%Y Cf. A063978 (g.f. Sum_{k>=0} x^(k^2)/(1-x)^(k+1)).

%K nonn

%O 0,10

%A _Joerg Arndt_, Apr 22 2014