This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A152260 #16 May 23 2023 17:29:10
%S A152260 1,1,2,1,10,9,1,32,113,64,1,86,786,1526,625,1,212,4182,18932,24337,
%T A152260 7776,1,498,19167,170332,477807,450066,117649,1,1136,80103,1266400,
%U A152260 6584615,12910704,9492289,2097152,1,2542,314928,8313394,72899230,254556594,375886768,225159022,43046721
%N A152260 Triangle T(n, k) = [x^k] p(n, x), where p(n, x) = (1/n)*(1-x)^(2*n) * Sum_{j >= 0} binomial(n+j-1, j) * j^n * x^(j-1).
%C A152260 A generalization of for n=m: q(x,n,m) = (1-x)^(n+m) * Sum_{j >= 0} binomial(m+j-1, j) * j^n * x^(j-1).
%D A152260 Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91
%H A152260 G. C. Greubel, <a href="/A152260/b152260.txt">Table of n, a(n) for n = 1..1275</a>
%F A152260 T(n, k) = [x^k] p(n, x), where p(n, x) = (1/n)*(1-x)^(2*n) * Sum_{j >= 0} binomial(n+j-1, j) * j^n * x^(j-1).
%F A152260 Sum_{k=1..n} T(n, k) = A006963(n).
%F A152260 T(n, m) = Sum_{k=1..m} binomial(n+k,k)*binomial(n-k,m-k)*k!*(-1)^(m-k) * Stirling2(n,k)*1/(n+k) (conjectured). - _Michael D. Weiner_, Jul 01 2020
%F A152260 From _G. C. Greubel_, May 23 2023: (Start)
%F A152260 T(n, k) = (1/n)*Sum_{j=0..k-1} (-1)^(k+j+1)*binomial(2*n, k-j-1) * binomial(n+j, j+1) * (j+1)^n.
%F A152260 T(n, n) = A000169(n).
%F A152260 T(n, n-1) = A176824(n). (End)
%e A152260 Polynomials, p(n, x), begin as:
%e A152260 p(1, x) = 1;
%e A152260 p(2, x) = 1 + 2*x;
%e A152260 p(3, x) = 1 + 10*x + 9*x^2;
%e A152260 p(4, x) = 1 + 32*x + 113*x^2 + 64*x^3;
%e A152260 p(5, x) = 1 + 86*x + 786*x^2 + 1526*x^3 + 625*x^4;
%e A152260 p(6, x) = 1 + 212*x + 4182*x^2 + 18932*x^3 + 24337*x^4 + 7776*x^5;
%e A152260 Triangle, T(n, k), begins as:
%e A152260 1;
%e A152260 1, 2;
%e A152260 1, 10, 9;
%e A152260 1, 32, 113, 64;
%e A152260 1, 86, 786, 1526, 625;
%e A152260 1, 212, 4182, 18932, 24337, 7776;
%e A152260 1, 498, 19167, 170332, 477807, 450066, 117649;
%e A152260 1, 1136, 80103, 1266400, 6584615, 12910704, 9492289, 2097152;
%t A152260 p[x_, n_]:= ((1-x)^(2*n)/n)*Sum[Binomial[k+n-1,k]*k^n*x^(k-1), {k, 0, Infinity}];
%t A152260 Table[CoefficientList[p[x, n], x], {n,12}]//Flatten
%t A152260 (* Second program *)
%t A152260 T[n_, k_]:= (1/n)*Sum[Binomial[2*n,k-j]*Binomial[n+j-1,j]*(-1)^(k+j) *j^n, {j,k}];
%t A152260 Table[T[n,k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, May 23 2023 *)
%o A152260 (Magma)
%o A152260 A152260:= func< n,k | (&+[(-1)^(k+j)*Binomial(2*n,k-j)*Binomial(n+j-1,j)*j^n: j in [1..k]])/n >;
%o A152260 [A152260(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, May 23 2023
%o A152260 (SageMath)
%o A152260 def A152260(n,k): return (1/n)*sum( (-1)^(k+j)*binomial(2*n,k-j)*binomial(n+j-1,j)*j^n for j in range(1,k+1) )
%o A152260 flatten([[A152260(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, May 23 2023
%Y A152260 Cf. A000169, A006963, A123125, A176824.
%K A152260 nonn,tabl
%O A152260 1,3
%A A152260 _Roger L. Bagula_, Dec 01 2008