|
%I #33 Dec 27 2023 16:52:56
%S 1,3,6,15,26,41,62,92,129,169,224,287,366,443,548,656,793,919,1090,
%T 1255,1466,1653,1906,2140,2441,2701,3052,3367,3774,4119,4584,4992,
%U 5521,5963,6558,7071,7738,8289,9030,9660,10481,11153,12056,12815,13806,14611,15692,16592
%N Sum of numbers in n-th upward diagonal of triangle in A079826.
%H Bruno Berselli, <a href="/A079825/b079825.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,2,-2,-2,2,-1,1,1,-1).
%F G.f.: x*( 1+2*x+2*x^2+7*x^3+6*x^4+2*x^5+6*x^6+x^7+x^8 ) / ( (1+x)^3*(1-x)^4*(1+x^2)^2 ). - _R. J. Mathar_, Sep 05 2012
%F a(n) = ( (2*n+1)*(7*n^2+7*n+12) -3*n*(n-7)*(-1)^n +6*(2*n-3*(-1)^n+1)*i^(n*(n-1)) )/96, where i=sqrt(-1). - _Bruno Berselli_, Sep 05 2012
%F E.g.f.: (1/48)*( (6+24*x+30*x^2+7*x^3)*cosh(x) + (6+42*x+33*x^2+7*x^3)* sinh(x) + 6*(x-1)*cos(x) - 6*(x-2)*sin(x) ). - _G. C. Greubel_, Dec 10 2023
%p A079825 := proc(n)
%p local a, k;
%p a := 0 ;
%p for k from 1 to (n+1)/2 do
%p a := a+A056011(n-k+1,k) ;
%p end do:
%p a ;
%p end proc: # _R. J. Mathar_, Sep 05 2012
%t LinearRecurrence[{1, 1, -1, 2, -2, -2, 2, -1, 1, 1, -1}, {1, 3, 6, 15, 26, 41, 62, 92, 129, 169, 224}, 35] (* _Bruno Berselli_, Sep 05 2012 *)
%o (Magma) m:=35; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+2*x^2+7*x^3+6*x^4+2*x^5+6*x^6+x^7+x^8)/((x^2+1)^2*(1+x)^3*(x-1)^4))); // _Bruno Berselli_, Sep 05 2012
%o (Magma) [((2*n+1)*(7*n^2+7*n+12) -3*(-1)^n*n*(n-7) +6*(-1)^Binomial(n,2)*(2*n-3*(-1)^n+1))/96: n in [1..41]]; // _G. C. Greubel_, Dec 10 2023
%o (Maxima) makelist(expand(((2*n+1)*(7*n^2+7*n+12)-3*n*(n-7)*(-1)^n+6*(2*n-3*(-1)^n+1)*%i^(n*(n-1)))/96), n, 1, 35); /* _Bruno Berselli_, Sep 05 2012 */
%o (SageMath) [((2*n+1)*(7*n^2+7*n+12)-3*(-1)^n*n*(n-7)+6*(-1)^binomial( n,2)*(2*n-3*(-1)^n+1))/96 for n in range(1,41)] # _G. C. Greubel_, Dec 10 2023
%Y Cf. A056011, A079826.
%K nonn,easy
%O 1,2
%A _Amarnath Murthy_, Feb 11 2003
|
