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A098436 Triangle of 3rd central factorial numbers T(n,k).

%I A098436 #18 Jan 13 2025 11:33:57
%S A098436 1,1,1,1,9,1,1,73,36,1,1,585,1045,100,1,1,4681,28800,7445,225,1,1,
%T A098436 37449,782281,505280,35570,441,1,1,299593,21159036,33120201,4951530,
%U A098436 130826,784,1,1,2396745,571593565,2140851900,652061451,33209946,399738,1296,1
%N A098436 Triangle of 3rd central factorial numbers T(n,k).
%H A098436 M. Domaratzki, <a href="https://research.cs.queensu.ca/TechReports/Reports/2001-449.pdf">A Generalization of the Genocchi Numbers with Applications to Enumeration of Finite Automata</a>, Technical Report No. 2001-449, September 2001, 11 pages.
%H A098436 John Riordan, <a href="/A002720/a002720_2.pdf">Letter, Apr 28 1976.</a>
%F A098436 Recurrence: T(n, k) = (k+1)^3*T(n-1, k) + T(n-1, k-1), T(0, 0)=1.
%e A098436   1;
%e A098436   1,   1;
%e A098436   1,   9,    1;
%e A098436   1,  73,   36,   1;
%e A098436   1, 585, 1045, 100, 1;
%e A098436   ...
%p A098436 A098436 := proc(n,k)
%p A098436     option remember;
%p A098436     if k=0 or k = n then
%p A098436         1;
%p A098436     else
%p A098436         (k+1)^3*procname(n-1,k)+procname(n-1,k-1) ;
%p A098436     end if;
%p A098436 end proc:
%p A098436 seq(seq( A098436(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Jan 13 2025
%t A098436 T[n_, n_] = 1;
%t A098436 T[n_ /; n>=0, k_] /; 0<=k<=n := T[n, k] = (k+1)^3 T[n-1, k]+T[n-1, k-1];
%t A098436 T[_, _] = 0;
%t A098436 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 08 2022 *)
%Y A098436 First column is A023001, first diagonal is A000537.
%Y A098436 Row sums are in A098437.
%Y A098436 Replace in recurrence (k+1)^3 with k: A008277; with k^2: A008957 (note offsets).
%K A098436 tabl,nonn
%O A098436 0,5
%A A098436 _Ralf Stephan_, Sep 08 2004