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

%I A322227 #10 Jan 05 2019 02:26:24
%S A322227 1,3,-12,-140,540,15456,-50932,-3176172,7343325,1053842295,
%T A322227 -1009469538,-515714090814,-374961500823,349796118587475,
%U A322227 949197425607720,-314320029983283752,-1565276549925545181,361569820089891813849,2715239099277372861920,-518323783521922446434520,-5333587428291215212424382,906157476001402934272328354,12062331313935951302447900940,-1897919702589547490476079347500,-31441371048822199544956413616625
%N A322227 a(n) = [x^(n-1)] Product_{k=1..n} (k + x - k*x^2), for n >= 1.
%C A322227 a(n) = n*(n+1)/2 * A322226(n) for n >= 1.
%H A322227 Paul D. Hanna, <a href="/A322227/b322227.txt">Table of n, a(n) for n = 1..301</a>
%e A322227 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 A322227 1;
%e A322227 1, 1, -1;
%e A322227 2, 3, -3, -3, 2;
%e A322227 6, 11, -12, -21, 12, 11, -6;
%e A322227 24, 50, -61, -140, 75, 140, -61, -50, 24;
%e A322227 120, 274, -375, -1011, 540, 1475, -540, -1011, 375, 274, -120;
%e A322227 720, 1764, -2696, -8085, 4479, 15456, -5005, -15456, 4479, 8085, -2696, -1764, 720;
%e A322227 5040, 13068, -22148, -71639, 42140, 169266, -50932, -221389, 50932, 169266, -42140, -71639, 22148, 13068, -5040; ...
%e A322227 in which this sequence forms a diagonal.
%e A322227 RELATED SEQUENCES.
%e A322227 Note that the terms in this sequence
%e A322227 [1, 3, -12, -140, 540, 15456, -50932, -3176172, 7343325, 1053842295, ...]
%e A322227 may be divided by triangular numbers n*(n+1)/2 to obtain A322226:
%e A322227 [1, 1, -2, -14, 36, 736, -1819, -88227, 163185, 19160769, -15294993, ...].
%t A322227 a[n_] := SeriesCoefficient[Product[k + x - k x^2, {k, 1, n}], {x, 0, n-1}];
%t A322227 Array[a, 25] (* _Jean-François Alcover_, Dec 29 2018 *)
%o A322227 (PARI) {T(n, k) = polcoeff( prod(m=1, n, m + x - m*x^2) +x*O(x^k), k)}
%o A322227 /* Print the irregular triangle */
%o A322227 for(n=0, 10, for(k=0, 2*n, print1( T(n, k), ", ")); print(""))
%o A322227 /* Print this sequence */
%o A322227 for(n=1, 30, print1( T(n, n-1), ", "))
%Y A322227 Cf. A322225, A322226.
%K A322227 sign
%O A322227 1,2
%A A322227 _Paul D. Hanna_, Dec 15 2018