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A322238 a(n) = [x^n] Product_{k=1..n} (k + x + k*x^2), for n >= 0.

%I A322238 #19 Dec 30 2018 00:41:38
%S A322238 1,1,5,23,215,1815,24495,302619,5320667,86715299,1876495799,
%T A322238 38014052089,976259270857,23653053031933,702814658564889,
%U A322238 19822951593203139,668803323061123779,21526439460249188211,812828119716458951775,29400983906741712373461,1228278466826435935830261,49325712126290139872176221,2258694878457443286997591293,99715752513094342744003434597,4966247793605869355035248188325
%N A322238 a(n) = [x^n] Product_{k=1..n} (k + x + k*x^2), for n >= 0.
%H A322238 Paul D. Hanna, <a href="/A322238/b322238.txt">Table of n, a(n) for n = 0..300</a>
%F A322238 a(n+1) = 2*(n+1) * A322237(n) + a(n), for n >= 1.
%F A322238 a(n+1) = n*(n+1)^2 * A322236(n) + a(n), for n >= 1.
%e A322238 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 A322238 1;
%e A322238 1, 1, 1;
%e A322238 2, 3, 5, 3, 2;
%e A322238 6, 11, 24, 23, 24, 11, 6;
%e A322238 24, 50, 131, 160, 215, 160, 131, 50, 24;
%e A322238 120, 274, 825, 1181, 1890, 1815, 1890, 1181, 825, 274, 120;
%e A322238 720, 1764, 5944, 9555, 17471, 19866, 24495, 19866, 17471, 9555, 5944, 1764, 720;
%e A322238 5040, 13068, 48412, 85177, 173460, 223418, 313628, 302619, 313628, 223418, 173460, 85177, 48412, 13068, 5040;
%e A322238 40320, 109584, 440684, 834372, 1860153, 2642220, 4120122, 4521924, 5320667, 4521924, 4120122, 2642220, 1860153, 834372, 440684, 109584, 40320; ...
%e A322238 in which the central terms equal this sequence.
%e A322238 RELATED SEQUENCES.
%e A322238 Note that the terms in the secondary diagonal A322237 in the above triangle
%e A322238 [1, 3, 24, 160, 1890, 19866, 313628, 4521924, 89489025, 1642616195, ...]
%e A322238 may be divided by triangular numbers to obtain A322236:
%e A322238 [1, 1, 4, 16, 126, 946, 11201, 125609, 1988645, 29865749, 592326527, ...].
%o A322238 (PARI) {T(n,k) = polcoeff( prod(m=1,n, m + x + m*x^2) +x*O(x^k), k)}
%o A322238 /* Print the irregular triangle */
%o A322238 for(n=0,10, for(k=0,2*n, print1( T(n,k),", "));print(""))
%o A322238 /* Print this sequence */
%o A322238 for(n=0,30, print1( T(n,n),", "))
%Y A322238 cF. A322235, A322236, A322237.
%Y A322238 Cf. A322228 (variant), A322894 (variant).
%K A322238 nonn
%O A322238 0,3
%A A322238 _Paul D. Hanna_, Dec 12 2018