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A322237 a(n) = [x^(n-1)] Product_{k=1..n} (k + x + k*x^2), for n >= 1.

%I A322237 #12 Dec 28 2018 09:31:32
%S A322237 1,3,24,160,1890,19866,313628,4521924,89489025,1642616195,39093550782,
%T A322237 872184375426,24255771626177,637337897821275,20280636608372520,
%U A322237 613459886387884248,21980602229339160099,752319889132243511097,29971937072992355586420,1145176991891992950865380,50212935598435298798304888,2118631687709497814282735724,101386084189432812756067599036
%N A322237 a(n) = [x^(n-1)] Product_{k=1..n} (k + x + k*x^2), for n >= 1.
%C A322237 a(n) = n*(n+1)/2 * A322236(n) for n >= 1.
%H A322237 Paul D. Hanna, <a href="/A322237/b322237.txt">Table of n, a(n) for n = 1..301</a>
%e A322237 The irregular triangle A322235 formed from coefficients of x^k in Product_{m=1..n} (m + x + m*x^2), for n >= 0, k = 0..2*n, begins
%e A322237 1;
%e A322237 1, 1, 1;
%e A322237 2, 3, 5, 3, 2;
%e A322237 6, 11, 24, 23, 24, 11, 6;
%e A322237 24, 50, 131, 160, 215, 160, 131, 50, 24;
%e A322237 120, 274, 825, 1181, 1890, 1815, 1890, 1181, 825, 274, 120;
%e A322237 720, 1764, 5944, 9555, 17471, 19866, 24495, 19866, 17471, 9555, 5944, 1764, 720;
%e A322237 5040, 13068, 48412, 85177, 173460, 223418, 313628, 302619, 313628, 223418, 173460, 85177, 48412, 13068, 5040;
%e A322237 40320, 109584, 440684, 834372, 1860153, 2642220, 4120122, 4521924, 5320667, 4521924, 4120122, 2642220, 1860153, 834372, 440684, 109584, 40320; ...
%e A322237 in which this sequence forms a diagonal.
%t A322237 a[n_] := SeriesCoefficient[Product[k + x + k x^2, {k, 1, n}], {x, 0, n-1}];
%t A322237 Array[a, 23] (* _Jean-François Alcover_, Dec 28 2018 *)
%o A322237 (PARI) {T(n,k) = polcoeff( prod(m=1,n, m + x + m*x^2) +x*O(x^k), k)}
%o A322237 /* Print the irregular triangle */
%o A322237 for(n=0,10, for(k=0,2*n, print1( T(n,k),", "));print(""))
%o A322237 /* Print this sequence */
%o A322237 for(n=1,30, print1( T(n,n-1),", "))
%Y A322237 Cf. A322235, A322238, A322236.
%K A322237 nonn
%O A322237 1,2
%A A322237 _Paul D. Hanna_, Dec 15 2018