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A081495
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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.
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4
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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