A122542 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
1, 0, 1, 0, 2, 1, 0, 2, 4, 1, 0, 2, 8, 6, 1, 0, 2, 12, 18, 8, 1, 0, 2, 16, 38, 32, 10, 1, 0, 2, 20, 66, 88, 50, 12, 1, 0, 2, 24, 102, 192, 170, 72, 14, 1, 0, 2, 28, 146, 360, 450, 292, 98, 16, 1, 0, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1
Offset: 0
Examples
Triangle begins: 1; 0, 1; 0, 2, 1; 0, 2, 4, 1; 0, 2, 8, 6, 1; 0, 2, 12, 18, 8, 1; 0, 2, 16, 38, 32, 10, 1; 0, 2, 20, 66, 88, 50, 12, 1; 0, 2, 24, 102, 192, 170, 72, 14, 1; 0, 2, 28, 146, 360, 450, 292, 98, 16, 1; 0, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1;
Links
- Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
- Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
- Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 1.
Crossrefs
Programs
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Haskell
a122542 n k = a122542_tabl !! n !! k a122542_row n = a122542_tabl !! n a122542_tabl = map fst $ iterate (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) $ zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [0, 1]) -- Reinhard Zumkeller, Jul 20 2013, Apr 17 2013 -
Magma
function T(n, k) // T = A122542 if k eq 0 then return 0^n; elif k eq n then return 1; else return T(n-1,k) + T(n-1,k-1) + T(n-2,k-1); end if; end function; [T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 27 2024
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Mathematica
CoefficientList[#, y]& /@ CoefficientList[(1-x)/(1 - (1+y)x - y x^2) + O[x]^11, x] // Flatten (* Jean-François Alcover, Sep 09 2018 *) (* Second program *) T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, 0, T[n-1,k-1] +T[n-1,k] +T[n-2,k- 1] ]]; (* T = A122542 *) Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 27 2024 *)
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Sage
def A122542_row(n): @cached_function def prec(n, k): if k==n: return 1 if k==0: return 0 return prec(n-1,k-1)+2*sum(prec(n-i,k-1) for i in (2..n-k+1)) return [prec(n, k) for k in (0..n)] for n in (0..10): print(A122542_row(n)) # Peter Luschny, Mar 16 2016
Formula
Sum_{k=0..n} x^k*T(n,k) = A000007(n), A001333(n), A104934(n), A122558(n), A122690(n), A091928(n) for x = 0, 1, 2, 3, 4, 5. - Philippe Deléham, Jan 25 2012
Sum_{k=0..n} 3^(n-k)*T(n,k) = A086901(n).
Sum_{k=0..n} 2^(n-k)*T(n,k) = A007483(n-1), n >= 1. - Philippe Deléham, Oct 08 2006
T(2*n, n) = A123164(n).
T(n, k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1), n > 1. - Philippe Deléham, Jan 25 2012
G.f.: (1-x)/(1-(1+y)*x-y*x^2). - Philippe Deléham, Mar 02 2012
From G. C. Greubel, Oct 27 2024: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = A057077(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A001590(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A078016(n). (End)
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