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%I #19 Jul 26 2022 13:48:44
%S 1,3,10,38,172,944,6208,47696,417952,4101824,44491648,528068096,
%T 6804155392,94559581184,1409615239168,22434345998336,379633330204672,
%U 6805952938041344,128854632579186688,2568966172926181376
%N Number of maximal initial consecutive columns ending at the same level, summed over all deco polyominoes of height n.
%C A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.
%H G. C. Greubel, <a href="/A140710/b140710.txt">Table of n, a(n) for n = 1..440</a>
%H E. Barcucci, A. Del Lungo and R. Pinzani, <a href="http://dx.doi.org/10.1016/0304-3975(95)00199-9">"Deco" polyominoes, permutations and random generation</a>, Theoretical Computer Science, 159, 1996, 29-42.
%F a(n) = 2^(n-1) * (1 + Sum_{j=1..n-1} j*j!/2^j ).
%F a(n) = (n-1)!*(n-1) + 2*a(n-1) with a(1) = 1.
%F a(n) = Sum_{k=1..n} k*A140709(n,k).
%F (1 + x + 2*x^2 + 4*x^3 + 8*x^4 + ...)*(1 + 2*x + 6*x^2 + 24*x^3 + 120*x^4 + ...) = (1 + 3*x + 10*x^2 + 38*x^3 + 172*x^4 + ...) which is (Sum_{n>=0} A011782(n)*x^n) * (Sum_{n>=0} A000142(n+1)*x^n) = Sum_{n>=0} a(n+1)*x^n. - _Gary W. Adamson_, Feb 24 2012
%F a(n) = Sum_{j=0..n} (j+1)!*A011782(n-j) = (n+1)! + Sum_{j=0..n-1} 2^(n-k-1)*(j+1)!. - _G. C. Greubel_, May 03 2021
%F D-finite with recurrence a(n) +(-n-3)*a(n-1) +3*n*a(n-2) +2*(-n+2)*a(n-3)=0. - _R. J. Mathar_, Jul 26 2022
%e a(3)=10 because the 6 deco polyominoes of height 3 have columns ending at levels 3, 22, 12, 111, 22, 122, respectively and 1+2+1+3+2+1=10.
%p a:=proc(n) options operator, arrow: 2^(n-1)*(1+sum(j^2*factorial(j-1)/2^j, j= 1..n-1)) end proc: seq(a(n),n=1..20);
%t Table[2^(n-1)*(1 + Sum[j*j!/2^j, {j,n-1}]), {n,30}] (* _G. C. Greubel_, May 02 2021 *)
%o (Magma) [2^(n-1)*(&+[j*Factorial(j)/2^j: j in [1..n-1]]): n in [1..30]]; // _G. C. Greubel_, May 02 2021
%o (Sage) [2^(n-1)*sum(j*factorial(j)/2^j for j in (1..n-1)) for n in (1..30)] # _G. C. Greubel_, May 02 2021
%Y Cf. A000142, A011782, A140709.
%Y Row sums of A227550/2.
%K nonn
%O 1,2
%A _Emeric Deutsch_, Jun 03 2008
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