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.

1, 5, 17, 55, 189, 681, 2519, 9451, 35765, 136153, 520695, 1998745, 7696467, 29716025, 115000947, 445962899, 1732525861, 6741529113, 26270128535, 102501265057, 400411345659, 1565841089321, 6129331763923, 24014172955545, 94163002754699, 369507926510401

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

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

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

Example

The rhombus pertaining to n = 4 is obtained from the solid rhombus
.....1
...1...1
.1...2...1
1..3...3...1
..4..6...4
...10..10
.....20
giving
.....1
...1...1
.1.......1
1..........1
..4......4
...10..10
.....20
and the sum of all the numbers is 55, a(4) = 55.

Maple

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

Mathematica

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 *)

Prog

(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
(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
(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
(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

Crossrefs

Cf. A081494, A081496, A081497.

Sequence in context: A027091 A183712 A244235 * A191645 A146240 A112410

Adjacent sequences: A081492 A081493 A081494 * A081496 A081497 A081498

Keyword

nonn

Author

Amarnath Murthy, Mar 25 2003

Extensions

More terms from Benoit Cloitre, Sep 10 2003

Status

approved