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URL: https://oeis.org/A107882

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A107882
Column 1 of triangle A107880.
4
1, 2, 5, 19, 104, 766, 7197, 82910, 1136923, 18141867, 330940109, 6803936050, 155839142185, 3938383850350, 108934529005948, 3275059508166297, 106388204134734785, 3714826559490125850, 138796913898027894261
OFFSET
0,2
FORMULA
G.f.: 1 = Sum_{k>=0} a(k)*x^k*(1-x)^(2 + k*(k+1)/2).
From Benedict W. J. Irwin, Nov 26 2016: (Start)
Conjecture: a(n) can be expressed with a series of nested sums,
a(2) = Sum_{i=1..2} i+1,
a(3) = Sum_{i=1..2}Sum_{j=1..i+1} j+2,
a(4) = Sum_{i=1..2}Sum_{j=1..i+1}Sum_{k=1..j+2} k+3,
a(5) = Sum_{i=1..2}Sum_{j=1..i+1}Sum_{k=1..j+2}Sum_{l=1..k+3} l+4. (End)
EXAMPLE
G.f. = 1 + 2*x + 5*x^2 + 19*x^3 + 104*x^4 + 766*x^5 + 7197*x^6 + 82910*x^7 + ...
1 = 1*(1-x)^2 + 2*x*(1-x)^3 + 5*x^2*(1-x)^5 +
19*x^3*(1-x)^8 + 104*x^4*(1-x)^12 + 766*x^5*(1-x)^17 +...
MATHEMATICA
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 *)
PROG
(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)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 04 2005
STATUS
approved