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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
 1, 1, 6, 27, 350, 2915, 58156, 714483, 19279494, 314352219, 10702577028, 217898279709, 8961716977044, 218984398436089, 10559918611732824, 301232711294645459, 16665825788607694854, 543748938748390962083, 33949005044435560537540, 1247096314671704716427281, 86735021741507120417172516, 3542923311990891744041871249, 271628142382739329884065326824, 12213792889354508458626454059325, 1023380788552996903255162728503100 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Main diagonal of triangle A322229, where, for n >= 0: (1) A322229(n,0) = A322229(n,2*n) = n!, (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 Paul D. Hanna, Table of n, a(n) for n = 0..300 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 in row n yields this sequence. PROG (PARI) {A322229(n, k) = polcoeff( prod(m=1, n, m + x + (n+1-m)*x^2) +x*O(x^k), k)} for(n=0, 30, print1( A322229(n, n), ", ")) CROSSREFS Cf. A322229, A322234. Sequence in context: A267630 A047778 A290802 * A164985 A048436 A276412 Adjacent sequences:  A322230 A322231 A322232 * A322234 A322235 A322236 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 18 2018 STATUS approved

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Last modified March 29 20:20 EDT 2020. Contains 333117 sequences. (Running on oeis4.)