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a(n) = [y^(n+1)] (n + y) * Product_{k=1..2*n} (k + y), for n >= 0.
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%I #12 Jan 26 2019 11:17:57

%S 1,4,55,1260,40593,1690920,86550035,5260335080,370410456273,

%T 29664913887180,2663386839535695,265000164136279572,

%U 28945346029081686865,3443628513369917505360,443271719760096505911675,61385459345641259759898000,9100387546322497725789848865,1438068852777042379374392377620,241308826278118770656171323634855,42852242077203438281471161279058300

%N a(n) = [y^(n+1)] (n + y) * Product_{k=1..2*n} (k + y), for n >= 0.

%C A diagonal in triangle A268647: a(n) = A268647(n, n+1) for n >= 0.

%H Paul D. Hanna, <a href="/A322627/b322627.txt">Table of n, a(n) for n = 0..300</a>

%e The coefficients of y^k in (n + y) * Product_{j=1..2*n} (j + y), for k=0..2*n+1, yields row n of triangle A268647, which begins:

%e 0, 1;

%e 2, 5, 4, 1;

%e 48, 124, 120, 55, 12, 1;

%e 2160, 6012, 6636, 3829, 1260, 238, 24, 1;

%e 161280, 478656, 582080, 387260, 157080, 40593, 6720, 690, 40, 1;

%e 18144000, 56772000, 74396520, 54801076, 25494150, 7927205, 1690920, 248523, 24750, 1595, 60, 1; ...

%e this sequence is the diagonal a(n) = A268647(n, n+1) for n >= 0.

%o (PARI) /* a(n) = [y^(n+1)] (n + y)*Product_{k=1..2*n} (k + y) */

%o {A268647(n, k) = polcoeff((n + y)*prod(k=1, 2*n, k + y), k, y)}

%o {a(n) = A268647(n, n+1)}

%o for(n=0, 25, print1(a(n),", "));

%Y Cf. A268647.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 26 2019