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A143805 Eigensequence of triangle A130534.

%I A143805 #30 Jul 18 2025 09:34:47
%S A143805 1,1,2,7,36,250,2229,24656,329883,5233837,96907908,2066551242,
%T A143805 50196458429,1375782397859,42203985613593,1438854199059479,
%U A143805 54180508061067099,2241000820010271224,101316373253530824771,4984697039955303538934,265819807417517749652933
%N A143805 Eigensequence of triangle A130534.
%C A143805 Triangle A130534 begins:
%C A143805    1;
%C A143805    1,  1;
%C A143805    2,  3,  1;
%C A143805    6, 11,  6,  1;
%C A143805   24, 50, 35, 10, 1;
%C A143805   ...
%C A143805 Shift the entire triangle down 1 place, with T(0,0) = 1. Let T = the new triangle:
%C A143805    1;
%C A143805    1;
%C A143805    1,  1;
%C A143805    2,  3,  1;
%C A143805    ...
%C A143805 This sequence is lim_{n -> infinity} T^n as a vector.
%H A143805 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Janjic/janjic42.html">Determinants and Recurrence Sequences</a>, Journal of Integer Sequences, 2012, Article 12.3.5. [_N. J. A. Sloane_, Sep 16 2012]
%F A143805 a(n) = Sum_{k=0..n-1} (-1)^(n-k-1) * Stirling1(n,k+1) * a(k) for n>0 with a(0)=1 (by definition). - _Paul D. Hanna_, Oct 01 2013
%F A143805 E.g.f.: Sum_{n>=0} a(n)*x^n/n! = 1 + Sum_{n>=1} a(n-1)*(-log(1-x))^n/n!. - _Paul D. Hanna_, May 20 2009
%F A143805 Conjecture: a(n) = R(n,0) where R(n,k) = R(n-1,n-1) + Sum_{j=0..k-1} (j+1)*R(n-1,j) for 0 <= k <= n with R(0,0) = 1. - _Mikhail Kurkov_, Jul 18 2025
%e A143805 From _Paul D. Hanna_, May 20 2009: (Start)
%e A143805 E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 36*x^4/4! + 250*x^5/5! + ...
%e A143805 A(x) = 1 - log(1-x) + log(1-x)^2/2! - 2*log(1-x)^3/3! + 7*log(1-x)^4/4! - 36*log(1-x)^5/5! +- ... (End)
%o A143805 (PARI) {a(n)=local(A=[1]);for(i=1,n,A=Vec(serlaplace(1+sum(k=1,#A,A[k]*(-log(1-x+x*O(x^n)))^k/k!))));A[n+1]} \\ _Paul D. Hanna_, May 20 2009
%o A143805 (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o A143805 {a(n)=if(n==0, 1, sum(k=0, n-1, (-1)^(n-k-1)*Stirling1(n, k+1)*a(k)))} \\ _Paul D. Hanna_, Oct 01 2013
%Y A143805 Cf. A143806.
%K A143805 nonn
%O A143805 0,3
%A A143805 _Gary W. Adamson_, Sep 01 2008
%E A143805 Extended by _Paul D. Hanna_, May 20 2009
%E A143805 Offset 0 by _Georg Fischer_, Apr 14 2024