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 A168422 #16 Mar 31 2023 14:41:04
%S A168422 1,1,1,7,4,1,71,39,9,1,1001,536,126,16,1,18089,9545,2270,310,25,1,
%T A168422 398959,208524,49995,7120,645,36,1,10391023,5394991,1301139,190435,
%U A168422 18445,1197,49,1,312129649,161260336,39066076,5828704,589750,41776,2044,64,1
%N A168422 Number triangle with row sums given by quadruple factorial numbers A001813.
%C A168422 Reversal of coefficient array for the polynomials P(n,x) = Sum_{k=0..n} (C(n+k,2k)*(2k)!/k!)*x^k*(1-x)^(n-k).
%C A168422 Note that P(n,x) = Sum_{k=0..n} A113025(n,k)*x^k*(1-x)^(n-k). Row sums are A001813.
%H A168422 William P. Orrick, <a href="/A168422/b168422.txt">Table of n, a(n) for n = 0..10010</a>
%F A168422 T(n,k) = (1/k!)*Sum_{j=k..n} (-1)^(j-k)*(2*n-j)!/((n-j)!*(j-k)!).
%e A168422 Triangle begins
%e A168422 1
%e A168422 1 1
%e A168422 7 4 1
%e A168422 71 39 9 1
%e A168422 1001 536 126 16 1
%e A168422 18089 9545 2270 310 25 1
%e A168422 398959 208524 49995 7120 645 36 1
%e A168422 10391023 5394991 1301139 190435 18445 1197 49 1
%e A168422 312129649 161260336 39066076 5828704 589750 41776 2044 64 1
%e A168422 Production matrix begins
%e A168422 1 1
%e A168422 6 3 1
%e A168422 40 20 5 1
%e A168422 336 168 42 7 1
%e A168422 3456 1728 432 72 9 1
%e A168422 42240 21120 5280 880 110 11 1
%e A168422 599040 299520 74880 12480 1560 156 13 1
%e A168422 9676800 4838400 1209600 201600 25200 2520 210 15 1
%e A168422 Complete this with a top row (1,0,0,0,...) and invert: we get
%e A168422 1
%e A168422 -1 1
%e A168422 -3 -3 1
%e A168422 -5 -5 -5 1
%e A168422 -7 -7 -7 -7 1
%e A168422 -9 -9 -9 -9 -9 1
%e A168422 -11 -11 -11 -11 -11 -11 1
%e A168422 -13 -13 -13 -13 -13 -13 -13 1
%e A168422 -15 -15 -15 -15 -15 -15 -15 -15 1
%e A168422 -17 -17 -17 -17 -17 -17 -17 -17 -17 1
%o A168422 (SageMath)
%o A168422 def T(n,k):
%o A168422 return(sum((-1)^(j-k) * binomial(2*n-j,n) * binomial(n,j)\
%o A168422 * binomial(j,k) * factorial(n-j)\
%o A168422 for j in range(k,n+1))) # _William P. Orrick_, Mar 24 2023
%o A168422 (PARI) T(n,k)={sum(j=k, n, (-1)^(j-k)*(2*n-j)!/((n-j)!*(j-k)!))/k!} \\ _Andrew Howroyd_, Mar 24 2023
%Y A168422 Column 1 is |A002119|.
%Y A168422 Sum_{k=0..n} T(n,k) * 2^k, is A001517(n).
%Y A168422 Cf. A079267.
%K A168422 easy,nonn,tabl
%O A168422 0,4
%A A168422 _Paul Barry_, Nov 25 2009
%E A168422 Corrected and extended by _William P. Orrick_, Mar 24 2023