A081495 - OEIS

%I #13 Sep 08 2022 08:45:09

%S 1,5,17,55,189,681,2519,9451,35765,136153,520695,1998745,7696467,

%T 29716025,115000947,445962899,1732525861,6741529113,26270128535,

%U 102501265057,400411345659,1565841089321,6129331763923,24014172955545,94163002754699,369507926510401

%N Start with Pascal's triangle; form a rhombus by sliding down n steps from top on both sides then sliding down inwards to complete the rhombus and then deleting the inner numbers; a(n) = sum of entries on perimeter of rhombus.

%H G. C. Greubel, <a href="/A081495/b081495.txt">Table of n, a(n) for n = 1..1000</a>

%F a(0)=1 for n>0 a(n)=binomial(2*n, n)-binomial(2*n-2, n-1)+2*n-3. - _Benoit Cloitre_, Sep 10 2003

%F G.f.: ((1-x)^3 - (1-2*x-x^3)*sqrt(1-4*x))/((1-x)^2*sqrt(1-4*x)). - _G. C. Greubel_, Aug 13 2019

%e The rhombus pertaining to n = 4 is obtained from the solid rhombus

%e .....1

%e ...1...1

%e .1...2...1

%e 1..3...3...1

%e ..4..6...4

%e ...10..10

%e .....20

%e giving

%e .....1

%e ...1...1

%e .1.......1

%e 1..........1

%e ..4......4

%e ...10..10

%e .....20

%e and the sum of all the numbers is 55, a(4) = 55.

%p seq(coeff(series(((1-x)^3 - (1-2*x-x^3)*sqrt(1-4*x))/((1-x)^2*sqrt(1-4*x) ), x, n+1), x, n), n = 1..25); # _G. C. Greubel_, Aug 13 2019

%t With[{C = CatalanNumber}, Table[If[n==1, 1, (n+1)*C[n] -n*C[n-1] +2*n-3], {n, 25}]] (* _G. C. Greubel_, Aug 13 2019 *)

%o (PARI) vector(25, n, b=binomial; if(n==1,1,b(2*n, n)-b(2*(n-1), n-1) +2*n -3)) \\ _G. C. Greubel_, Aug 13 2019

%o (Magma) C:=Catalan; [1] cat [(n+1)*C(n) -n*C(n-1) +2*n-3: n in [2..25]]; // _G. C. Greubel_, Aug 13 2019

%o (Sage) b=binomial; [1]+[b(2*n, n)-b(2*(n-1), n-1) +2*n -3 for n in (2..25)] # _G. C. Greubel_, Aug 13 2019

%o (GAP) B:=Binomial;; Concatenation([1], List([2..25], n-> B(2*n, n)-B(2*(n-1), n-1) +2*n -3)); # _G. C. Greubel_, Aug 13 2019

%Y Cf. A081494, A081496, A081497.

%K nonn

%O 1,2

%A _Amarnath Murthy_, Mar 25 2003

%E More terms from _Benoit Cloitre_, Sep 10 2003