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A336622 a(n) = Sum_{k=0..n} Sum_{i=0..k} Sum_{j=0..i} (binomial(n,k) * binomial(k,i) * binomial(i,j))^n.

%I A336622 #15 Sep 01 2022 05:00:00
%S A336622 1,4,28,1192,591460,3441637504,219272057247376,185528149944660881488,
%T A336622 2405748000504972140803769860,349789137657321307953339196885516144,
%U A336622 652520795984468974632890750361094911319873648
%N A336622 a(n) = Sum_{k=0..n} Sum_{i=0..k} Sum_{j=0..i} (binomial(n,k) * binomial(k,i) * binomial(i,j))^n.
%H A336622 G. C. Greubel, <a href="/A336622/b336622.txt">Table of n, a(n) for n = 0..42</a>
%F A336622 a(n) = (n!)^n * [x^n] (Sum_{k>=0} x^k / (k!)^n)^4.
%t A336622 Table[Sum[Sum[Sum[(Binomial[n, k] Binomial[k, i] Binomial[i, j])^n, {j, 0, i}], {i, 0, k}], {k, 0, n}], {n, 0, 10}]
%t A336622 Table[(n!)^n SeriesCoefficient[Sum[x^k/(k!)^n, {k, 0, n}]^4, {x, 0, n}], {n, 0, 10}]
%o A336622 (Magma) B:=Binomial; [(&+[(&+[(&+[(B(n,j)*B(n-j,k-j)*B(k-j,k-i))^n: j in [0..i]]): i in [0..k]]): k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Aug 31 2022
%o A336622 (SageMath)
%o A336622 b=binomial
%o A336622 def A336622(n): return sum(sum(sum( (b(n,j)*b(n-j,k-j)*b(k-j,k-i))^n for j in (0..i)) for i in (0..k)) for k in (0..n))
%o A336622 [A336622(n) for n in (0..20)] # _G. C. Greubel_, Aug 31 2022
%Y A336622 Cf. A000302, A002895, A167010, A287699, A336270.
%K A336622 nonn
%O A336622 0,2
%A A336622 _Ilya Gutkovskiy_, Jul 29 2020