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A155755 Triangle T(n, k) = A143491(n+2, k+2) + A143491(n+2, n-k+2), read by rows.

%I A155755 #9 Jun 06 2021 06:53:46
%S A155755 2,3,3,7,10,7,25,35,35,25,121,168,142,168,121,721,1064,735,735,1064,
%T A155755 721,5041,8055,5399,3330,5399,8055,5041,40321,69299,49371,22449,22449,
%U A155755 49371,69299,40321,362881,663740,509830,223300,109298,223300,509830,663740,362881
%N A155755 Triangle T(n, k) = A143491(n+2, k+2) + A143491(n+2, n-k+2), read by rows.
%C A155755 This symmetric summation of the triangle A143491 is equivalent to the coefficient [x^m] (p_n(x) + x^n*p_n(1/x)) of the polynomials defined in A143491 plus their reverses.
%H A155755 G. C. Greubel, <a href="/A155755/b155755.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A155755 T(n, k) = A143491(n+2, k+2) + A143491(n+2, n-k+2).
%F A155755 Sum_{k=0..n} T(n, k) = (n+2)!.
%e A155755 Triangle begins as:
%e A155755        2;
%e A155755        3,      3;,
%e A155755        7,     10,      7;
%e A155755       25,     35,     35,     25;
%e A155755      121,    168,    142,    168,    121;
%e A155755      721,   1064,    735,    735,   1064,    721;
%e A155755     5041,   8055,   5399,   3330,   5399,   8055,   5041;
%e A155755    40321,  69299,  49371,  22449,  22449,  49371,  69299,  40321;
%e A155755   362881, 663740, 509830, 223300, 109298, 223300, 509830, 663740, 362881;
%t A155755 (* First program *)
%t A155755 q[x_, n_]:= Product[x +n-i+1, {i,0,n-1}];
%t A155755 p[x_, n_]:= q[x, n] + x^n*q[1/x, n];
%t A155755 Table[CoefficientList[p[x, n], x], {n,0,12}]//Flatten (* modified by _G. C. Greubel_, Jun 06 2021 *)
%t A155755 (* Second program *)
%t A155755 A143491[n_, k_]:= (n-2)!*Sum[(n-k-j+1)*Abs[StirlingS1[j+k-2, k-2]]/(j+k-2)!, {j,0,n-k}];
%t A155755 A155755[n_, k_]:= A143491[n+2, k+2] + A143491[n+2, n-k+2];
%t A155755 Table[A155755[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 06 2021 *)
%o A155755 (Sage)
%o A155755 def A143491(n,k): return factorial(n-2)*sum( (n-k-j+1)*stirling_number1(j+k-2, k-2)/factorial(j+k-2) for j in (0..n-k) )
%o A155755 def A155755(n,k): return A143491(n+2, k+2) + A143491(n+2, n-k+2)
%o A155755 flatten([[A155755(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 06 2021
%Y A155755 Cf. A143491.
%K A155755 nonn,tabl
%O A155755 0,1
%A A155755 _Roger L. Bagula_, Jan 26 2009