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A035317 Pascal-like triangle associated with A000670. 24
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
From Johannes W. Meijer, Jul 20 2011: (Start)
The triangle sums, see A180662 for their definitions, link this "Races with Ties" triangle with several sequences, see the crossrefs. Observe that the Kn4 sums lead to the golden rectangle numbers A001654 and that the Fi1 and Fi2 sums lead to the Jacobsthal sequence A001045.
The series expansion of G(x, y) = 1/((y*x-1)*(y*x+1)*((y+1)*x-1)) as function of x leads to this sequence, see the second Maple program. (End)
T(2*n+1,n) = A014301(n+1); T(2*n+1,n+1) = A026641(n+1). - Reinhard Zumkeller, Jul 19 2012
LINKS
Vincenzo Librandi, Rows n = 0..100, flattened
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.
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)
T(n, n-k) = A128176(n, k); T(n+k, n-k) = A158909(n, k); T(2*n-k, k) = A092879(n, k). (End)
EXAMPLE
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;
...
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
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 *)
PROG
(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
(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
CROSSREFS
Row sums are A000975, diagonal sums are A080239.
Central terms are A014300.
Similar to the triangles A059259, A080242, A108561, A112555.
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.
Sequence in context: A339708 A080242 A183927 * A368296 A103923 A349432
KEYWORD
nonn,easy,tabl,nice
AUTHOR
EXTENSIONS
More terms from James A. Sellers
STATUS
approved

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Last modified March 19 06:25 EDT 2024. Contains 370953 sequences. (Running on oeis4.)