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A322766 Row 1 of array in A322765.

%I A322766 #15 Jun 01 2022 09:07:13
%S A322766 1,4,26,249,3274,56135,1207433,31638625,987249425,36030130677,
%T A322766 1515621707692,72603595393584,3920675798922189,236615520916677436,
%U A322766 15840357595697061964,1168697367186883073296,94486667847573203169757,8328527812527985862657297,796762955545266206229493979
%N A322766 Row 1 of array in A322765.
%D A322766 D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778.
%H A322766 Seiichi Manyama, <a href="/A322766/b322766.txt">Table of n, a(n) for n = 0..300</a>
%F A322766 a(n) = A346500(n,n+1) = A346500(n+1,n). - _Alois P. Heinz_, Jul 21 2021
%p A322766 b:= proc(n) option remember; `if`(n=0, 1,
%p A322766       add(b(n-j)*binomial(n-1, j-1), j=1..n))
%p A322766     end:
%p A322766 A:= proc(n, k) option remember; `if`(n<k, A(k, n),
%p A322766      `if`(k=0, b(n), (A(n+1, k-1)+add(A(n-k+j, j)
%p A322766       *binomial(k-1, j), j=0..k-1)+A(n, k-1))/2))
%p A322766     end:
%p A322766 a:= n-> A(n, n+1):
%p A322766 seq(a(n), n=0..22);  # _Alois P. Heinz_, Jul 21 2021
%t A322766 b[n_] := b[n] = If[n == 0, 1,
%t A322766      Sum[b[n - j]*Binomial[n-1, j-1], {j, 1, n}]];
%t A322766 A[n_, k_] := A[n, k] = If[n < k, A[k, n],
%t A322766      If[k == 0, b[n], (A[n+1, k - 1] + Sum[A[n - k + j, j]*
%t A322766      Binomial[k-1, j], {j, 0, k - 1}] + A[n, k - 1])/2]];
%t A322766 a[n_] := A[n, n + 1];  Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Jun 01 2022, after _Alois P. Heinz_ *)
%Y A322766 Cf. A322765, A346500.
%K A322766 nonn
%O A322766 0,2
%A A322766 _N. J. A. Sloane_, Dec 30 2018