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%I #20 Sep 08 2022 08:45:38
%S 1,16,201,3736,95545,2738856,82152385,2526574240,78991868961,
%T 2498812448320,79748142780361,2562704059745688,82808387862053113,
%U 2687915950092986536,87578455481326007745,2862650767821013111936
%N Row sums in A144630.
%H Klaus Brockhaus, <a href="/A144632/b144632.txt">Table of n, a(n) for n = 1..100</a>
%F Recurrence: (n-2)*(n-1)^2*(2*n-7)*(2*n-5)*(6*n^4 - 60*n^3 + 220*n^2 - 350*n + 203)*a(n) = 2*(2*n-7)*(216*n^8 - 3636*n^7 + 26190*n^6 - 105112*n^5 + 255968*n^4 - 384806*n^3 + 345331*n^2 - 166378*n + 31919)*a(n-1) - 2*(n-2)*(2*n-7)*(2*n-1)*(3*n^2-12*n+11)*(70*n^4 - 560*n^3 + 1524*n^2 - 1616*n + 539)*a(n-2) + 2*(2*n-1)*(216*n^8 - 3276*n^7 + 21150*n^6 - 75896*n^5 + 165408*n^4 - 223706*n^3 + 182571*n^2 - 81550*n + 15031)*a(n-3) - (n-2)*(2*n-3)*(2*n-1)*(6*n^4 - 36*n^3 + 76*n^2 - 66*n + 19)*(n-3)^2*a(n-4). - _Vaclav Kotesovec_, Aug 07 2013
%F a(n) ~ 2^(1/4)*(17+12*sqrt(2))^n/(64*Pi^(3/2)*sqrt(n)). - _Vaclav Kotesovec_, Aug 07 2013
%p invH := proc(n,i,j) (-1)^(i+j)*(i+j-1)*binomial(n+i-1,n-j)*binomial(n+j-1,n-i)* (binomial(i+j-2,i-1))^2 ; end: A144630 := proc(n,k) local T,i,j ; T := 0 ; for i from n-k+1 to n do for j from n-k+1 to n do T := T+invH(n,i,j) ; od; od; RETURN(T) ; end: A144632 := proc(n) local k; add(A144630(n,k),k=1..n) ; end: for n from 1 to 30 do printf("%a,",A144632(n)) : od: # _R. J. Mathar_, Jan 21 2009
%t a = DifferenceRoot[Function[{y, n}, {-13632 n^5 - 136320 n^4 - 540336 n^3 - 1060896 n^2 + (-1728 n^5 - 16484 n^4 - 60648 n^3 - 106194 n^2 - 86888 n - 25970) y[n+1] + (3360 n^5 + 33600 n^4 + 154334 n^3 + 388404 n^2 + 503246 n + 257468) y[n+2] + (-1728 n^5 - 18076 n^4 - 73384 n^3 - 145038 n^2 - 140376 n - 53518) y[n+3] + (n+1)^2 (n+2)(2n + 5)(24 n + 61) y[n] + (n + 2)(n + 3)^2 (2n + 3)(24n + 35) y[n+4] - 1031616 n - 397440 == 0, y[1] == 1, y[2] == 16, y[3] == 201, y[4] == 3736}]];
%t Array[a, 30] (* _Jean-François Alcover_, Mar 31 2020 *)
%o (Magma) [ &+[ &+[I[i][j]: i, j in [k..n] ]: k in [n..1 by -1] ] where I:=H^-1 where H:=Matrix(Rationals(), n, n, [ < i, j, 1/(i+j-1) >: i, j in [1..n] ] ): n in [1..16] ]; // _Klaus Brockhaus_, Jan 21 2009
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Jan 20 2009
%E Extended by _Klaus Brockhaus_ and _R. J. Mathar_, Jan 21 2009
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