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A322769 Main diagonal of array in A322765.

%I A322769 #15 Apr 29 2022 05:47:41
%S A322769 1,4,92,5133,537813,91914202,23456071495,8411911367949,
%T A322769 4055497274641836,2540939492105630071,2014322292658946180922,
%U A322769 1977121111959534634757742,2360026677940190304494287625,3374607252811005168634470847052,5706308288951111509370981721908854
%N A322769 Main diagonal of array in A322765.
%D A322769 D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778.
%H A322769 Seiichi Manyama, <a href="/A322769/b322769.txt">Table of n, a(n) for n = 0..200</a>
%F A322769 a(n) = A346500(2n,n). - _Alois P. Heinz_, Jul 20 2021
%p A322769 b:= proc(n) option remember; `if`(n=0, 1,
%p A322769       add(b(n-j)*binomial(n-1, j-1), j=1..n))
%p A322769     end:
%p A322769 A:= proc(n, k) option remember; `if`(n<k, A(k, n),
%p A322769      `if`(k=0, b(n), (A(n+1, k-1)+add(A(n-k+j, j)
%p A322769       *binomial(k-1, j), j=0..k-1)+A(n, k-1))/2))
%p A322769     end:
%p A322769 a:= n-> A(2*n, n):
%p A322769 seq(a(n), n=0..15);  # _Alois P. Heinz_, Jul 21 2021
%t A322769 P[m_, n_] := P[m, n] = If[n == 0, BellB[m], (1/2)(P[m+2, n-1] + P[m+1, n-1] + Sum[Binomial[n-1, k] P[m, k], {k, 0, n-1}])];
%t A322769 a[n_] := P[n, n];
%t A322769 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Apr 29 2022 *)
%Y A322769 Cf. A322765, A346500.
%K A322769 nonn
%O A322769 0,2
%A A322769 _N. J. A. Sloane_, Dec 30 2018