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A213569 Principal diagonal of the convolution array A213568.

%I A213569 #44 Sep 08 2022 08:46:02
%S A213569 1,7,25,71,181,435,1009,2287,5101,11243,24553,53223,114661,245731,
%T A213569 524257,1114079,2359261,4980699,10485721,22020055,46137301,96468947,
%U A213569 201326545,419430351,872415181,1811939275,3758096329,7784628167
%N A213569 Principal diagonal of the convolution array A213568.
%C A213569 Create a triangle having first column T(n,1) = 2*n-1 for n = 1,2,3... The remaining terms are set to T(r,c) = T(r,c-1) + T(r-1,c-1). The sum of the terms in row n is a(n). The first five rows of the triangle are 1; 3,4; 5,8,12; 7,12,20,32; 9,16,28,48,80. - _J. M. Bergot_, Jan 17 2013
%C A213569 Starting at n=1, a(n) = (n+1)*2^n - 2*n - 1. A001787(n) = n*2^n. - _J. M. Bergot_, Jan 27 2013
%H A213569 Clark Kimberling, <a href="/A213569/b213569.txt">Table of n, a(n) for n = 1..1000</a>
%H A213569 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-13,12,-4).
%F A213569 a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4).
%F A213569 G.f.: x*(1 + x - 4*x^2)/( (1-2*x)^2*(1-x)^2 ).
%F A213569 a(n) = A001787(n+1)- 2*n - 1. - _J. M. Bergot_, Jan 22 2013
%F A213569 a(n) = Sum_{k=1..n} Sum_{i=0..n} (n-i) * C(k,i). - _Wesley Ivan Hurt_, Sep 19 2017
%p A213569 f:= gfun:-rectoproc({a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4),
%p A213569   a(1)=1,a(2)=7,a(3)=25,a(4)=71},a(n),remember):
%p A213569 map(f, [$1..30]); # _Robert Israel_, Sep 19 2017
%t A213569 (* First program *)
%t A213569 b[n_]:= 2^(n-1); c[n_]:= n;
%t A213569 t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213569 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213569 Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
%t A213569 r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213568 *)
%t A213569 d = Table[t[n, n], {n, 1, 40}] (* A213569 *)
%t A213569 s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
%t A213569 s1 = Table[s[n], {n, 1, 50}] (* A047520 *)
%t A213569 (* Additional programs *)
%t A213569 LinearRecurrence[{6,-13,12,-4},{1,7,25,71},30] (* _Harvey P. Dale_, Jan 06 2015 *)
%t A213569 Table[2^n*(n+1) -(2*n+1), {n,30}] (* _G. C. Greubel_, Jul 25 2019 *)
%o A213569 (PARI) my(x='x+O('x^30)); Vec(x*(1+x-4*x^2)/((1-2*x)^2*(1-x)^2)) \\ _Altug Alkan_, Sep 19 2017
%o A213569 (PARI) vector(30, n, 2^n*(n+1) -(2*n+1)) \\ _G. C. Greubel_, Jul 25 2019
%o A213569 (Magma) [2^n*(n+1) -(2*n+1): n in [1..30]]; // _G. C. Greubel_, Jul 25 2019
%o A213569 (Sage) [2^n*(n+1) -(2*n+1) for n in (1..30)] # _G. C. Greubel_, Jul 25 2019
%o A213569 (GAP) List([1..30], n-> 2^n*(n+1) -(2*n+1)); # _G. C. Greubel_, Jul 25 2019
%Y A213569 Cf. A001787, A213568, A213500.
%K A213569 nonn,easy
%O A213569 1,2
%A A213569 _Clark Kimberling_, Jun 18 2012