A107886 - OEIS

%I #8 Nov 26 2016 19:14:23

%S 1,3,9,37,210,1575,14943,173109,2381814,38087355,695745075,

%T 14317460370,328142173159,8296618775100,229557238129530,

%U 6903176055689085,224285333475911340,7832574292981396104,292678312428437482293

%N Column 1 of triangle A107884.

%F G.f.: 1 = Sum_{k>=0} a(k)*x^k*(1-x)^(3 + k*(k+1)/2).

%F From _Benedict W. J. Irwin_, Nov 26 2016: (Start)

%F Conjecture: a(n) can be expressed with a series of nested sums,

%F a(2) = Sum_{i=1..3} i+1,

%F a(3) = Sum_{i=1..3}Sum_{j=1..i+1} j+2,

%F a(4) = Sum_{i=1..3}Sum_{j=1..i+1}Sum_{k=1..j+2} k+3,

%F a(5) = Sum_{i=1..3}Sum_{j=1..i+1}Sum_{k=1..j+2}Sum_{l=1..k+3} l+4. (End)

%e G.f. = 1 + 3*x + 9*x^2 + 37*x^3 + 210*x^4 + 1575*x^5 + 14943*x^6 + ...

%e 1 = 1*(1-x)^3 + 3*x*(1-x)^4 + 9*x^2*(1-x)^6 + 37*x^3*(1-x)^9 + 210*x^4*(1-x)^13 + 1575*x^5*(1-x)^18 + ...

%t a[ n_, k_: 3, j_: 0] := If[ n < 1, Boole[n >= 0], a[ n, k, j] = Sum[ a[ n - 1, i, j + 1], {i, k + j}]]; (* _Michael Somos_, Nov 26 2016 *)

%o (PARI) a(n)=polcoeff(1-sum(k=0,n-1,a(k)*x^k*(1-x+x*O(x^n))^(3+k*(k+1)/2)),n)

%Y Cf. A107884, A107885, A107887, A107888.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 04 2005