cp's OEIS Frontend

A081495 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.

%I A081495 #13 Sep 08 2022 08:45:09
%S A081495 1,5,17,55,189,681,2519,9451,35765,136153,520695,1998745,7696467,
%T A081495 29716025,115000947,445962899,1732525861,6741529113,26270128535,
%U A081495 102501265057,400411345659,1565841089321,6129331763923,24014172955545,94163002754699,369507926510401
%N A081495 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 A081495 G. C. Greubel, <a href="/A081495/b081495.txt">Table of n, a(n) for n = 1..1000</a>
%F A081495 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 A081495 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 A081495 The rhombus pertaining to n = 4 is obtained from the solid rhombus
%e A081495 .....1
%e A081495 ...1...1
%e A081495 .1...2...1
%e A081495 1..3...3...1
%e A081495 ..4..6...4
%e A081495 ...10..10
%e A081495 .....20
%e A081495 giving
%e A081495 .....1
%e A081495 ...1...1
%e A081495 .1.......1
%e A081495 1..........1
%e A081495 ..4......4
%e A081495 ...10..10
%e A081495 .....20
%e A081495 and the sum of all the numbers is 55, a(4) = 55.
%p A081495 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 A081495 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 A081495 (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 A081495 (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 A081495 (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 A081495 (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 A081495 Cf. A081494, A081496, A081497.
%K A081495 nonn
%O A081495 1,2
%A A081495 _Amarnath Murthy_, Mar 25 2003
%E A081495 More terms from _Benoit Cloitre_, Sep 10 2003