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 A372260 #17 Mar 29 2025 19:43:09
%S A372260 1,0,2,2,8,8,8,60,96,48,60,544,1248,1152,384,544,6040,17920,24000,
%T A372260 15360,3840,6040,79008,287520,503040,472320,230400,46080,79008,
%U A372260 1190672,5131392,11067840,13655040,9838080,3870720,645120,1190672,20314880,101153024,259187712,395566080,375889920,219340800,72253440,10321920
%N A372260 Triangle read by rows: T(n, k) = (T(n-1, k-1) + T(n-1, k)) * 2 * n with initial values T(n, 0) = Sum_{i=0..n} (-1)^(n-i) * binomial(n, i) * A001147(i) and T(i, j) = 0 if j > i.
%F A372260 T(n, 0) = (2*n - 2) * (T(n-1, 0) + T(n-2, 0)) for n > 1 with initial values T(n, 0) = 1 - n for n < 2 (see A053871).
%F A372260 T(n, k) = (Sum_{i=0..k} binomial(k, i) * T(n-i, 0)) * 2^(2*k) * binomial(n, k) / binomial(2*k, k).
%F A372260 E.g.f. of column k: (exp(-t) / sqrt(1 - 2*t)) * (2*t / (1 - 2*t))^k.
%F A372260 E.g.f.: exp((2*x / (1 - 2*t) - 1) * t) / sqrt(1 - 2*t).
%e A372260 Triangle T(n, k) starts:
%e A372260 n\k : 0 1 2 3 4 5 6 7
%e A372260 ====================================================================
%e A372260 0 : 1
%e A372260 1 : 0 2
%e A372260 2 : 2 8 8
%e A372260 3 : 8 60 96 48
%e A372260 4 : 60 544 1248 1152 384
%e A372260 5 : 544 6040 17920 24000 15360 3840
%e A372260 6 : 6040 79008 287520 503040 472320 230400 46080
%e A372260 7 : 79008 1190672 5131392 11067840 13655040 9838080 3870720 645120
%e A372260 etc.
%p A372260 T := proc(n, k) option remember; `if`(k > n, 0, `if`(k = n, 2^n * n!, `if`(k = 0, `if`(n < 2, 1 - n, (2*n - 2) * (T(n-1, k) + T(n-2, k))), (T(n-1, k-1) + T(n-1, k)) * 2*n))) end:
%p A372260 for n from 0 to 7 do seq(T(n, k), k = 0..n) od; # _Peter Luschny_, Apr 25 2024
%t A372260 T[n_,k_]:=n!SeriesCoefficient[(Exp[-t]/Sqrt[1 - 2*t])*(2*t/(1-2*t))^k,{t,0,n}]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* _Stefano Spezia_, Apr 25 2024 *)
%o A372260 (PARI) { T(n, k) = if(k>n, 0, if(k==n, 2^n * n!, if(k==0, if(n<2, 1-n,
%o A372260 (2*n-2) * (T(n-1, k) + T(n-2, k))), (T(n-1, k-1) + T(n-1, k)) * 2*n))) }
%o A372260 (PARI) memo = Map(); memoize(f, A[..]) =
%o A372260 { my(res);
%o A372260 if(!mapisdefined(memo, [f, A], &res), res = call(f, A);
%o A372260 mapput(memo, [f, A], res)); res; }
%o A372260 T(n, k) =
%o A372260 { if(k>n, 0, if(k==n, 2^n * n!, if(k==0, if(n<2, 1 - n,
%o A372260 (2 * n - 2) * (memoize(T, n-1, k) + memoize(T, n-2, k))),
%o A372260 (memoize(T, n-1, k-1) + memoize(T, n-1, k)) * 2 * n))); }
%Y A372260 Cf. A053871 (column 0), 2*A179540 (column 1), A000165 (main diagonal).
%Y A372260 Cf. A001147.
%K A372260 nonn,easy,tabl
%O A372260 0,3
%A A372260 _Werner Schulte_, Apr 24 2024