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%I #38 Apr 21 2024 22:11:20
%S 1,1,1,3,1,1,5,2,1,7,6,1,1,9,4,1,11,11,2,1,13,15,3,1,15,25,10,1,1,17,
%T 8,1,19,21,4,1,21,28,6,1,23,44,19,2,1,25,39,9,1,27,58,27,3,1,29,68,34,
%U 4,1,31,90,65,15,1,1,33,16,1,35,41,8,1,37,54,12,1
%N Irregular table T(n, k), n >= 0, k > 0, read by rows of extended (due to binary expansion of n) Stirling numbers of the second kind.
%C Let U(n,m) = A329369(2*(A054429(n + 2^(A000523(n) + 1)) + 2^(A000523(n) + 2)*(2^m - 1))).
%C Experiments with WolframAlpha lead us to conjecture (which we subsequently check on a large number of values) that
%C U(n,m) = Sum_{k=1..wt(n) + 2} k!*k^(m+1)*R(n, k)*(-1)^(wt(n) - k + 2) for n > 0, m >= 0 where wt(n) = A000120(n) and where R(n, k) are unknown coefficients.
%C Then T(n,k) = R(A059893(n), k).
%C Row n length is A000120(n) + 2.
%F T(n, 1) = 1 for n > 0 with T(0, 1) = T(0, 2) = 1.
%F T(2n+1, k) = k*T(n, k) + T(n, k-1) for n >= 0, k > 1.
%F T(2n, k) = k*T(n, k) + T(n, k-1) - (T(2n, k-1) + T(n, k-1))/(k-1) for n > 0, k > 1.
%F T(2^n - 1, k) = Stirling2(n+2, k) for n >= 0, k > 0.
%F T(n, 2) = 2n+1 for n >= 0.
%F T(n, A000120(n) + 2) = A341392(n) for n >= 0.
%F Sum_{k=1..wt(n) + 2} k!*T(n, k)*(-1)^(wt(n) - k + 2) = A329369(n) for n >= 0 where wt(n) = A000120(n).
%F Sum_{k=1..wt(f(n)) + 2} k!*k^(A290255(A054429(n)) + 1)*T(A059893(f(n)), k)*(-1)^(wt(f(n)) - k + 2) = A329369(2n) for n > 0, A053645(n+1) > 0 where wt(n) = A000120(n) and where f(n) = A035327(n).
%e Irregular table begins:
%e 1, 1;
%e 1, 3, 1;
%e 1, 5, 2;
%e 1, 7, 6, 1;
%e 1, 9, 4;
%e 1, 11, 11, 2;
%e 1, 13, 15, 3;
%e 1, 15, 25, 10, 1;
%e 1, 17, 8;
%e 1, 19, 21, 4;
%e 1, 21, 28, 6;
%e 1, 23, 44, 19, 2;
%e 1, 25, 39, 9;
%e 1, 27, 58, 27, 3;
%e 1, 29, 68, 34, 4;
%e 1, 31, 90, 65, 15, 1;
%o (PARI) T(n, k)=if(n==0 || k==1, (n==0 && k<3) + (k==1 && n>0), k*T(n\2, k) + T(n\2, k-1) - if(n%2==0, (T(n, k-1) + T(n\2,k-1))/(k-1)))
%Y Cf. A000120, A000523, A008277, A035327, A053645, A054429, A059893, A290255, A329369, A341392, A357990.
%Y Similar tables: A358631.
%K nonn,base,tabf
%O 1,4
%A _Mikhail Kurkov_, Nov 23 2022 [verification needed]
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