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A322226 a(n) = A322227(n) / (n*(n+1)/2), where A322227(n) = [x^(n-1)] Product_{k=1..n} (k + x - k*x^2), for n >= 1.

%I A322226 #14 Jan 05 2019 02:28:20
%S A322226 1,1,-2,-14,36,736,-1819,-88227,163185,19160769,-15294993,-6611719113,
%T A322226 -4120456053,3331391605595,7909978546731,-2311176691053557,
%U A322226 -10230565685787877,2114443392338548619,14290732101459857168,-2468208492961535459212,-23089123066195736850322,3581650102772343613724618,43704098963536055443651815,-6326399008631824968253597825,-96742680150222152446019734205
%N A322226 a(n) = A322227(n) / (n*(n+1)/2), where A322227(n) = [x^(n-1)] Product_{k=1..n} (k + x - k*x^2), for n >= 1.
%H A322226 Paul D. Hanna, <a href="/A322226/b322226.txt">Table of n, a(n) for n = 1..301</a>
%e A322226 The irregular triangle A322225 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 A322226 1;
%e A322226 1, 1, -1;
%e A322226 2, 3, -3, -3, 2;
%e A322226 6, 11, -12, -21, 12, 11, -6;
%e A322226 24, 50, -61, -140, 75, 140, -61, -50, 24;
%e A322226 120, 274, -375, -1011, 540, 1475, -540, -1011, 375, 274, -120;
%e A322226 720, 1764, -2696, -8085, 4479, 15456, -5005, -15456, 4479, 8085, -2696, -1764, 720;
%e A322226 5040, 13068, -22148, -71639, 42140, 169266, -50932, -221389, 50932, 169266, -42140, -71639, 22148, 13068, -5040; ...
%e A322226 in which the central terms equal A322228.
%e A322226 RELATED SEQUENCES.
%e A322226 Note that the terms in the secondary diagonal A322227 in the above triangle
%e A322226 [1, 3, -12, -140, 540, 15456, -50932, -3176172, 7343325, 1053842295, ...]
%e A322226 may be divided by triangular numbers to obtain this sequence
%e A322226 [1, 1, -2, -14, 36, 736, -1819, -88227, 163185, 19160769, -15294993, ...].
%t A322226 a[n_] := SeriesCoefficient[Product[k + x - k x^2, {k, 1, n}], {x, 0, n - 1}]/(n (n + 1)/2);
%t A322226 Array[a, 25] (* _Jean-François Alcover_, Dec 29 2018 *)
%o A322226 (PARI) {A322225(n, k) = polcoeff( prod(m=1, n, m + x - m*x^2) +x*O(x^k), k)}
%o A322226 /* Print the irregular triangle */
%o A322226 for(n=0, 10, for(k=0, 2*n, print1( A322225(n, k), ", ")); print(""))
%o A322226 /* Print this sequence */
%o A322226 for(n=1, 30, print1( A322225(n, n-1)/(n*(n+1)/2), ", "))
%Y A322226 Cf. A322227, A322228.
%Y A322226 Cf. A322236 (variant), A322894 (variant).
%K A322226 sign
%O A322226 1,3
%A A322226 _Paul D. Hanna_, Dec 15 2018