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A322236 a(n) = A322237(n) / (n*(n+1)/2), where A322237(n) = [x^(n-1)] Product_{k=1..n} (k + x + k*x^2), for n >= 1.

%I A322236 #15 Dec 28 2018 04:58:22
%S A322236 1,1,4,16,126,946,11201,125609,1988645,29865749,592326527,11181850967,
%T A322236 266546940947,6069884741155,169005305069771,4510734458734443,
%U A322236 143664066858425883,4399531515393236907,157747037226275555718,5453223770914252146978,217372015577641986139848,8374038291341888594002908,367340884744321785348071011,15606634300050239405862650475
%N A322236 a(n) = A322237(n) / (n*(n+1)/2), where A322237(n) = [x^(n-1)] Product_{k=1..n} (k + x + k*x^2), for n >= 1.
%H A322236 Paul D. Hanna, <a href="/A322236/b322236.txt">Table of n, a(n) for n = 1..301</a>
%e A322236 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 A322236 1;
%e A322236 1, 1, 1;
%e A322236 2, 3, 5, 3, 2;
%e A322236 6, 11, 24, 23, 24, 11, 6;
%e A322236 24, 50, 131, 160, 215, 160, 131, 50, 24;
%e A322236 120, 274, 825, 1181, 1890, 1815, 1890, 1181, 825, 274, 120;
%e A322236 720, 1764, 5944, 9555, 17471, 19866, 24495, 19866, 17471, 9555, 5944, 1764, 720;
%e A322236 5040, 13068, 48412, 85177, 173460, 223418, 313628, 302619, 313628, 223418, 173460, 85177, 48412, 13068, 5040;
%e A322236 40320, 109584, 440684, 834372, 1860153, 2642220, 4120122, 4521924, 5320667, 4521924, 4120122, 2642220, 1860153, 834372, 440684, 109584, 40320; ...
%e A322236 in which the central terms equal A322238.
%e A322236 RELATED SEQUENCES.
%e A322236 Note that the terms in the secondary diagonal (A322237), beginning
%e A322236 [1, 3, 24, 160, 1890, 19866, 313628, 4521924, 89489025, 1642616195, ...]
%e A322236 may be divided by triangular numbers to obtain this sequence
%e A322236 [1, 1, 4, 16, 126, 946, 11201, 125609, 1988645, 29865749, 592326527, ...].
%t A322236 a[n_] := SeriesCoefficient[Product[k + x + k x^2, {k, 1, n}], {x, 0, n-1}]/ (n(n+1)/2);
%t A322236 Array[a, 24] (* _Jean-François Alcover_, Dec 28 2018 *)
%o A322236 (PARI) {T(n, k) = polcoeff( prod(m=1, n, m + x + m*x^2) +x*O(x^k), k)}
%o A322236 /* Print the irregular triangle */
%o A322236 for(n=0, 10, for(k=0, 2*n, print1( T(n, k), ", ")); print(""))
%o A322236 /* Print this sequence */
%o A322236 for(n=1, 30, print1( T(n, n-1)/(n*(n+1)/2), ", "))
%Y A322236 Cf. A322237, A322238.
%K A322236 nonn
%O A322236 1,3
%A A322236 _Paul D. Hanna_, Dec 15 2018