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