A107886 Column 1 of triangle A107884.

1, 3, 9, 37, 210, 1575, 14943, 173109, 2381814, 38087355, 695745075, 14317460370, 328142173159, 8296618775100, 229557238129530, 6903176055689085, 224285333475911340, 7832574292981396104, 292678312428437482293

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

G.f.: 1 = Sum_{k>=0} a(k)*x^k*(1-x)^(3 + 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..3} i+1,

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

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

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

Example

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

Mathematica

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 *)

Prog

(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)

Crossrefs

Cf. A107884, A107885, A107887, A107888.

Sequence in context: A300170 A155159 A222518 * A220452 A130407 A137031

Adjacent sequences: A107883 A107884 A107885 * A107887 A107888 A107889

Keyword

nonn

Author

Paul D. Hanna, Jun 04 2005

Status

approved