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A322233 Coefficient of x^n in Product_{k=1..n} (k + x + (n+1-k)*x^2), for n >= 0. 3

%I

%S 1,1,6,27,350,2915,58156,714483,19279494,314352219,10702577028,

%T 217898279709,8961716977044,218984398436089,10559918611732824,

%U 301232711294645459,16665825788607694854,543748938748390962083,33949005044435560537540,1247096314671704716427281,86735021741507120417172516,3542923311990891744041871249,271628142382739329884065326824,12213792889354508458626454059325,1023380788552996903255162728503100

%N Coefficient of x^n in Product_{k=1..n} (k + x + (n+1-k)*x^2), for n >= 0.

%C Main diagonal of triangle A322229, where, for n >= 0:

%C (1) A322229(n,0) = A322229(n,2*n) = n!,

%C (2) Sum_{k=0..2*n} A322229(n,k) = (n+2)^n,

%C (3) Sum_{k=0..2*n} A322229(n,k)*(-1)^k = n^n.

%H Paul D. Hanna, <a href="/A322233/b322233.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) ~ (exp(2) + (-1)^n) * sqrt(3/Pi) * n^(n - 1/2) / 2. - _Vaclav Kotesovec_, Dec 27 2018

%e Triangle A322229, where row n gives coefficients in Product_{k=1..n} (k + x + (n+1-k)*x^2), begins

%e 1;

%e 1, 1, 1;

%e 2, 3, 6, 3, 2;

%e 6, 11, 32, 27, 32, 11, 6;

%e 24, 50, 189, 210, 350, 210, 189, 50, 24;

%e 120, 274, 1269, 1689, 3594, 2915, 3594, 1689, 1269, 274, 120;

%e 720, 1764, 9652, 14651, 37750, 37457, 58156, 37457, 37750, 14651, 9652, 1764, 720;

%e 5040, 13068, 82396, 138473, 417780, 481074, 896412, 714483, 896412, 481074, 417780, 138473, 82396, 13068, 5040; ...

%e in which the coefficient of x^n in row n yields this sequence.

%o (PARI) {A322229(n, k) = polcoeff( prod(m=1, n, m + x + (n+1-m)*x^2) +x*O(x^k), k)}

%o for(n=0, 30, print1( A322229(n, n), ", "))

%Y Cf. A322229, A322234.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 18 2018

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Last modified June 2 15:47 EDT 2020. Contains 334787 sequences. (Running on oeis4.)