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