A035317 Pascal-like triangle associated with A000670.
1, 1, 1, 1, 2, 2, 1, 3, 4, 2, 1, 4, 7, 6, 3, 1, 5, 11, 13, 9, 3, 1, 6, 16, 24, 22, 12, 4, 1, 7, 22, 40, 46, 34, 16, 4, 1, 8, 29, 62, 86, 80, 50, 20, 5, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6, 1, 11, 56, 174, 367, 553, 610, 496, 295, 125
Offset: 0
Examples
Triangle begins: 1; 1, 1; 1, 2, 2; 1, 3, 4, 2; 1, 4, 7, 6, 3; 1, 5, 11, 13, 9, 3; 1, 6, 16, 24, 22, 12, 4; 1, 7, 22, 40, 46, 34, 16, 4; 1, 8, 29, 62, 86, 80, 50, 20, 5; 1, 9, 37, 91, 148, 166, 130, 70, 25, 5; 1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6; ...
Links
- Vincenzo Librandi, Rows n = 0..100, flattened
- Joseph Briggs, Alex Parker, Coy Schwieder, and Chris Wells, Frogs, hats and common subsequences, arXiv preprint arXiv:2404.07285 [math.CO], 2024. See p. 28.
- A. Hlavác, M. Marvan, Nonlocal conservation laws of the constant astigmatism equation, arXiv preprint arXiv:1602.06861 [nlin.SI], 2016.
- E. Mendelson, Races with Ties, Math. Mag. 55 (1982), 170-175.
- Index entries for triangles and arrays related to Pascal's triangle
Crossrefs
Central terms are A014300.
Cf. A059260.
Triangle sums (see the comments): A000975 (Row1), A059841 (Row2), A080239 (Kn11), A052952 (Kn21), A129696 (Kn22), A001906 (Kn3), A001654 (Kn4), A001045 (Fi1, Fi2), A023435 (Ca2), Gi2 (A193146), A190525 (Ze2), A193147 (Ze3), A181532 (Ze4). - Johannes W. Meijer, Jul 20 2011
Cf. A181971.
Programs
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Haskell
a035317 n k = a035317_tabl !! n !! k a035317_row n = a035317_tabl !! n a035317_tabl = map snd $ iterate f (0, [1]) where f (i, row) = (1 - i, zipWith (+) ([0] ++ row) (row ++ [i])) -- Reinhard Zumkeller, Jul 09 2012
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Maple
A035317 := proc(n,k): add((-1)^(i+k) * binomial(i+n-k+1, i), i=0..k) end: seq(seq(A035317(n,k), k=0..n), n=0..10); # Johannes W. Meijer, Jul 20 2011 A035317 := proc(n,k): coeff(coeftayl(1/((y*x-1)*(y*x+1)*((y+1)*x-1)), x=0, n), y, k) end: seq(seq(A035317(n,k), k=0..n), n=0..10); # Johannes W. Meijer, Jul 20 2011
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Mathematica
t[n_, k_] := (-1)^k*(((-1)^k*(n+2)!*Hypergeometric2F1[1, n+3, k+2, -1])/((k+1)!*(n-k+1)!) + 2^(k-n-2)); Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* Jean-François Alcover, Dec 14 2011, after Johannes W. Meijer *) -
PARI
{T(n,k)=if(n==k,(n+2)\2,if(k==0,1,if(n>k,T(n-1,k-1)+T(n-1,k))))} for(n=0,12,for(k=0,n,print1(T(n,k),","));print("")) \\ Paul D. Hanna, Jul 18 2012 -
Sage
def A035317_row(n): @cached_function def prec(n, k): if k==n: return 1 if k==0: return 0 return -prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1)) return [(-1)^k*prec(n+2, k) for k in (1..n)] for n in (1..11): print(A035317_row(n)) # Peter Luschny, Mar 16 2016
Formula
T(n,k) = Sum_{j=0..floor(n/2)} binomial(n-2j, k-2j). - Paul Barry, Feb 11 2003
From Johannes W. Meijer, Jul 20 2011: (Start)
T(n, k) = Sum_{i=0..k}((-1)^(i+k) * binomial(i+n-k+1,i)). (Mendelson)
T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = floor(n/2) + 1. (Mendelson)
Sum_{k = 0..n}((-1)^k * (n-k+1)^n * T(n, k)) = A000670(n). (Mendelson)
Extensions
More terms from James Sellers
Comments