A202474 Coefficients of y^(n-1) in Product_{k=1..n} (1 + k*y + y^2) for n >= 1.
1, 3, 14, 80, 539, 4179, 36630, 358056, 3860922, 45519870, 582466235, 8038684290, 119018991779, 1881685721265, 31638175704546, 563703015007056, 10609073237333432, 210305960538762456, 4379808881917047898, 95604092878386437940, 2182706554812339958778
Offset: 1
Keywords
Examples
E.g.f.: A(x) = x + 3*x^2/2! + 14*x^3/3! + 80*x^4/4! + 539*x^5/5! + 4179*x^6/6! + 36630*x^7/7! + 358056*x^8/8! + 3860922*x^9/9! + 45519870*x^10/10! + ...
The coefficients in Product_{k=1..n} (1+k*x+x^2), n>=0, form the triangle:
[1];
[(1), 1, 1];
[1,(3), 4, 3, 1];
[1, 6, (14), 18, 14, 6, 1];
[1, 10, 39, (80), 100, 80, 39, 10, 1];
[1, 15, 90, 285, (539), 660, 539, 285, 90, 15, 1];
[1, 21, 181, 840, 2339, (4179), 5038, 4179, 2339, 840, 181, 21, 1];
[1, 28, 329, 2128, 8400, 21392, (36630), 43624, 36630, 21392, 8400, 2128, 329, 28, 1]; ...
the coefficients in parenthesis form the initial terms of this sequence.
Programs
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Mathematica
Flatten[{1,Table[Coefficient[Expand[Product[1 + k*x + x^2,{k,1,n+1}]],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Feb 10 2015 *) -
PARI
{a(n) = polcoeff(prod(k=1,n,1 + k*x + x^2 +x*O(x^n)),n-1)} for(n=1,30,print1(a(n),", "))
Formula
E.g.f.: 1/(1-x) * Sum_{n>=0} log(1 - x)^(2*n+1) / (n!*(n+1)!).
Extensions
Changed offset to 1 to agree with e.g.f. - Paul D. Hanna, Mar 02 2019