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 A099173 #22 Feb 17 2023 10:09:15
%S A099173 0,0,1,0,1,2,0,1,2,3,0,1,2,4,4,0,1,2,5,8,5,0,1,2,6,12,16,6,0,1,2,7,16,
%T A099173 29,32,7,0,1,2,8,20,44,70,64,8,0,1,2,9,24,61,120,169,128,9,0,1,2,10,
%U A099173 28,80,182,328,408,256,10,0,1,2,11,32,101,256,547,896,985,512,11
%N A099173 Array, A(k,n), read by diagonals: g.f. of k-th row x/(1-2*x-(k-1)*x^2).
%H A099173 G. C. Greubel, <a href="/A099173/b099173.txt">Antidiagonals n = 0..50, flattened</a>
%H A099173 Ralf Stephan, <a href="https://arxiv.org/abs/math/0409509">Prove or disprove. 100 Conjectures from the OEIS</a>, #16, arXiv:math/0409509 [math.CO], 2004.
%F A099173 A(n, k) = Sum_{i=0..floor(k/2)} n^i * C(k, 2*i+1) (array).
%F A099173 Recurrence: A(n, k) = 2*A(n, k-1) + (n-1)*A(n, k-2), with A(n, 0) = 0, A(n, 1) = 1.
%F A099173 T(n, k) = A(n-k, k) (antidiagonal triangle).
%F A099173 T(2*n, n) = A357502(n).
%F A099173 A(n, k) = ((1+sqrt(n))^k - (1-sqrt(n))^k)/(2*sqrt(n)). - _Jean-François Alcover_, Jan 21 2019
%e A099173 Square array, A(n, k), begins as:
%e A099173 0, 1, 2, 3, 4, 5, 6, 7, 8, ... A001477;
%e A099173 0, 1, 2, 4, 8, 16, 32, 64, 128, ... A000079;
%e A099173 0, 1, 2, 5, 12, 29, 70, 169, 408, ... A000129;
%e A099173 0, 1, 2, 6, 16, 44, 120, 328, 896, ... A002605;
%e A099173 0, 1, 2, 7, 20, 61, 182, 547, 1640, ... A015518;
%e A099173 0, 1, 2, 8, 24, 80, 256, 832, 2688, ... A063727;
%e A099173 0, 1, 2, 9, 28, 101, 342, 1189, 4088, ... A002532;
%e A099173 0, 1, 2, 10, 32, 124, 440, 1624, 5888, ... A083099;
%e A099173 0, 1, 2, 11, 36, 149, 550, 2143, 8136, ... A015519;
%e A099173 0, 1, 2, 12, 40, 176, 672, 2752, 10880, ... A003683;
%e A099173 0, 1, 2, 13, 44, 205, 806, 3457, 14168, ... A002534;
%e A099173 0, 1, 2, 14, 48, 236, 952, 4264, 18048, ... A083102;
%e A099173 0, 1, 2, 15, 52, 269, 1110, 5179, 22568, ... A015520;
%e A099173 0, 1, 2, 16, 56, 304, 1280, 6208, 27776, ... A091914;
%e A099173 Antidiagonal triangle, T(n, k), begins as:
%e A099173 0;
%e A099173 0, 1;
%e A099173 0, 1, 2;
%e A099173 0, 1, 2, 3;
%e A099173 0, 1, 2, 4, 4;
%e A099173 0, 1, 2, 5, 8, 5;
%e A099173 0, 1, 2, 6, 12, 16, 6;
%e A099173 0, 1, 2, 7, 16, 29, 32, 7;
%e A099173 0, 1, 2, 8, 20, 44, 70, 64, 8;
%e A099173 0, 1, 2, 9, 24, 61, 120, 169, 128, 9;
%e A099173 0, 1, 2, 10, 28, 80, 182, 328, 408, 256, 10;
%t A099173 A[k_, n_]:= Which[k==0, n, n==0, 0, True, ((1+Sqrt[k])^n - (1-Sqrt[k])^n)/(2 Sqrt[k])]; Table[A[k-n, n]//Simplify, {k, 0, 12}, {n, 0, k}]//Flatten (* _Jean-François Alcover_, Jan 21 2019 *)
%o A099173 (PARI) A(k,n)=sum(i=0,n\2,k^i*binomial(n,2*i+1))
%o A099173 (Magma)
%o A099173 A099173:= func< n,k | (&+[n^j*Binomial(k,2*j+1): j in [0..Floor(k/2)]]) >;
%o A099173 [A099173(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 17 2023
%o A099173 (SageMath)
%o A099173 def A099173(n,k): return sum( n^j*binomial(k, 2*j+1) for j in range((k//2)+1) )
%o A099173 flatten([[A099173(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Feb 17 2023
%Y A099173 Rows m: A001477 (m=0), A000079 (m=1), A000129 (m=2), A002605 (m=3), A015518 (m=4), A063727 (m=5), A002532 (m=6), A083099 (m=7), A015519 (m=8), A003683 (m=9), A002534 (m=10), A083102 (m=11), A015520 (m=12), A091914 (m=13).
%Y A099173 Columns q: A000004 (q=0), A000012 (q=1), A009056 (q=2), A008586 (q=3).
%Y A099173 Main diagonal gives A357502.
%K A099173 nonn,tabl
%O A099173 0,6
%A A099173 _Ralf Stephan_, Oct 13 2004