|
|
A322234
|
|
Coefficient of x^(n-1) in Product_{k=1..n} (k + x + (n+1-k)*x^2), for n >= 1.
|
|
3
|
|
|
1, 3, 32, 210, 3594, 37457, 896412, 12899520, 398293065, 7333821066, 277832923686, 6242376401170, 280450554344037, 7444127625398565, 387040500799449256, 11856077367530958400, 700385265181462012083, 24325550963621400151002, 1609526297187139688483020, 62510661428443938909212244, 4579921704394565828797270344, 196706096648058100149910675962, 15809764480004991957201666377340
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
A diagonal of triangle A322229, where, for n >= 0:
(2) Sum_{k=0..2*n} A322229(n,k) = (n+2)^n,
(3) Sum_{k=0..2*n} A322229(n,k)*(-1)^k = n^n.
|
|
LINKS
|
|
|
FORMULA
|
a(n) ~ (exp(2) - (-1)^n) * sqrt(3/Pi) * n^(n - 1/2) / 2. - Vaclav Kotesovec, Dec 27 2018
|
|
EXAMPLE
|
Triangle A322229, where row n gives coefficients in Product_{k=1..n} (k + x + (n+1-k)*x^2), begins
1;
1, 1, 1;
2, 3, 6, 3, 2;
6, 11, 32, 27, 32, 11, 6;
24, 50, 189, 210, 350, 210, 189, 50, 24;
120, 274, 1269, 1689, 3594, 2915, 3594, 1689, 1269, 274, 120;
720, 1764, 9652, 14651, 37750, 37457, 58156, 37457, 37750, 14651, 9652, 1764, 720;
5040, 13068, 82396, 138473, 417780, 481074, 896412, 714483, 896412, 481074, 417780, 138473, 82396, 13068, 5040; ...
in which the coefficient of x^(n-1) in row n for n>=1 yields this sequence.
|
|
MATHEMATICA
|
a[n_] := SeriesCoefficient[Product[k+x+(n-k+1)x^2, {k, 1, n}], {x, 0, n-1}];
|
|
PROG
|
(PARI) {A322229(n, k) = polcoeff( prod(m=1, n, m + x + (n+1-m)*x^2) +x*O(x^k), k)}
for(n=1, 30, print1( A322229(n, n-1), ", "));
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|