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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

REFERENCES

A Hlavác, M Marvan, Nonlocal conservation laws of the constant astigmatism equation, arXiv preprint arXiv:1602.06861, 2016

LINKS

Vincenzo Librandi, Rows n = 0..100, flattened

E. Mendelson, Races with Ties, Math. Mag. 55 (1982), 170-175.

Index entries for triangles and arrays related to Pascal's triangle

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: A212306 A080242 A183927 * A103923 A186711 A061987

Adjacent sequences:  A035314 A035315 A035316 * A035318 A035319 A035320

KEYWORD

nonn,easy,tabl,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers

STATUS

approved

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Last modified March 24 07:54 EDT 2017. Contains 283985 sequences.