|
%I #60 Sep 08 2022 08:46:17
%S 0,2,11,39,114,300,741,1757,4052,9162,20415,44979,98214,212888,458633,
%T 982905,2097000,4456278,9436995,19922735,41942810,88080132,184549101,
%U 385875669,805306044,1677721250,3489660551,7247756907,15032385102,31138512432,64424508945
%N Accumulation of the upper left triangle used in binomial transform of nonnegative integers.
%C After 0, is this the second column of A108284? [_Bruno Berselli_, Sep 13 2016 - this comment may be removed if the property is confirmed.]
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (7,-19,25,-16,4).
%F O.g.f.: x*(2 - 3*x)/((1 - x)^3*(1 - 2*x)^2).
%F E.g.f.: x*exp(x)*(8*exp(x) - x - 4)/2.
%F a(n) = n*(2^(n+2) - n - 3)/2.
%F a(n) = 7*a(n-1) - 19*a(n-2) + 25*a(n-3) - 16*a(n-4) + 4*a(n-5) for n > 4.
%F a(n) = a(n-1) + A058877(n+1). - _R. J. Mathar_, Sep 14 2016
%F a(n) = Sum_{k=2..n+3} Sum_{i=2..n+3} k * C(n-i+3,k). - _Wesley Ivan Hurt_, Sep 20 2017
%e Starting from the triangle:
%e 0, 1, 2, 3, 4, 5, ...
%e 1, 3, 5, 7, 9, ...
%e 4, 8, 12, 16, ...
%e 12, 20, 28, ...
%e 32, 48, ...
%e 80, ...
%e ...
%e the first terms are:
%e a(0) = 0;
%e a(1) = a(0) + 1 + 1 = 2;
%e a(2) = a(1) + 4 + 3 + 2 = 11;
%e a(3) = a(2) + 12 + 8 + 5 + 3 = 39, etc.
%e First column is A001787: n*2^(n-1).
%p A276659:=n->n*(2^(n+2) - n - 3)/2: seq(A276659(n), n=0..50); # _Wesley Ivan Hurt_, Sep 16 2017
%t t[0, k_] := k; t[n_, k_] := t[n, k] = t[n - 1, k] + t[n - 1, k + 1]; a[n_] := Sum[t[m, k], {m, 0, n}, {k, 0, n - m}]; Table[a[n], {n, 0, 30}]
%t Table[(2^(n + 2) - n - 3) n / 2, {n, 0, 30}] (* _Vincenzo Librandi_, Sep 13 2016 *)
%o (Magma) [(2^(n+2)-n-3)*n/2: n in [0..40]]; // _Vincenzo Librandi_, Sep 13 2016
%o (PARI) x='x+O('x^99); concat(0, Vec(x*(2-3*x)/((1-x)^3*(1-2*x)^2))) \\ _Altug Alkan_, Sep 14 2017
%Y Cf. A001787, A058877, A062111, A152920.
%K nonn,easy
%O 0,2
%A _Jean-François Alcover_ and _Francois Alcover_, Sep 11 2016
%E Edited and extended by _Bruno Berselli_, Sep 13 2016
|
