A107882 - OEIS

%I #12 Nov 26 2016 18:19:15

%S 1,2,5,19,104,766,7197,82910,1136923,18141867,330940109,6803936050,

%T 155839142185,3938383850350,108934529005948,3275059508166297,

%U 106388204134734785,3714826559490125850,138796913898027894261

%N Column 1 of triangle A107880.

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

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

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

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

%e G.f. = 1 + 2*x + 5*x^2 + 19*x^3 + 104*x^4 + 766*x^5 + 7197*x^6 + 82910*x^7 + ...

%e 1 = 1*(1-x)^2 + 2*x*(1-x)^3 + 5*x^2*(1-x)^5 +

%e 19*x^3*(1-x)^8 + 104*x^4*(1-x)^12 + 766*x^5*(1-x)^17 +...

%t a[ n_, k_: 2, 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))^(2+k*(k+1)/2)),n)}

%Y Cf. A107880, A107881, A107883.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 04 2005