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A059259 Triangle read by rows giving coefficient T(i,j) of x^i y^j in 1/(1-x-x*y-y^2) = 1/((1+y)(1-x-y)) for (i,j) = (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), ... 17
1, 1, 0, 1, 1, 1, 1, 2, 2, 0, 1, 3, 4, 2, 1, 1, 4, 7, 6, 3, 0, 1, 5, 11, 13, 9, 3, 1, 1, 6, 16, 24, 22, 12, 4, 0, 1, 7, 22, 40, 46, 34, 16, 4, 1, 1, 8, 29, 62, 86, 80, 50, 20, 5, 0, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 1, 10, 46, 128, 239, 314, 296, 200 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

This sequence provides the general solution to the recurrence a(n) = a(n-1) + k*(k+1)*a(n-2), a(0)=a(1)=1. The solution is (1, 1, k^2 + k + 1, 2*k^2 + 2*k + 1, ...) whose coefficients can be read from the rows of the triangle. The row sums of the triangle are given by the case k=1. These are the Jacobsthal numbers, A001045. Viewed as a square array, its first row is (1,0,1,0,1,...) with e.g.f. cosh(x), g.f. 1/(1-x^2) and subsequent rows are successive partial sums given by 1/((1-x)^n * (1-x^2)). - Paul Barry, Mar 17 2003

Conjecture: every second column of this triangle is identical to a column in the square array A071921. For example, column 4 of A059259 (1, 3, 9, 22, 46, ...) appears to be the same as column 3 of A071921; column 6 of A059259 (1, 4, 16, 50, 130, 296, ...) appears to be the same as column 4 of A071921; and in general column 2k of A059259 appears to be the same as column k+1 of A071921. Furthermore, since A225010 is a transposition of A071921 (ignoring the latter's top row and two leftmost columns), there appears to be a correspondence between column 2k of A059259 and row k of A225010. - Mathew Englander, May 17 2014

T(n,k) is the number of n-tilings of a (one-dimensional) board that use k (1,1)-fence tiles and n-k squares. A (1,1)-fence is a tile composed of two pieces of width 1 separated by a gap of width 1. - Michael A. Allen, Jun 25 2020

See the Edwards-Allen 2020 paper, page 14, for proof of Englander's conjecture. - Michael De Vlieger, Dec 10 2020

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Kenneth Edwards and Michael A. Allen, New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile, arXiv:2009.04649 [math.CO], 2020.

FORMULA

G.f.: 1/(1 - x - x*y - y^2).

As a square array read by antidiagonals, this is T(n, k) = Sum_{i=0..n} (-1)^(n-i)*C(i+k, k). - Paul Barry, Jul 01 2003

T(2*n,n) = A026641(n). - Philippe Deléham, Mar 08 2007

T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = T(2,2)=1, T(1,1)=0, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 24 2013

T(n,0) = 1, T(n,n) = (1+(-1)^n)/2, and T(n,k) = T(n-1,k) + T(n-1,k-1) for 0 < k < n. - Mathew Englander, May 24 2014

From Michael A. Allen, Jun 25 2020: (Start)

T(n,k) + T(n-1,k-1) = binomial(n,k) if n >= k > 0.

T(2*n-1,2*n-2) = T(2*n,2*n-1) = n, T(2*n,2*n-2) = n^2, T(2*n+1,2*n-1) = n*(n+1) for n > 0.

T(n,2) = binomial(n-2,2) + n - 1 for n > 1 and T(n,3) = binomial(n-3,3) + 2*binomial(n-2,2) for n > 2.

T(2*n-k,k) = A123521(n,k). (End)

EXAMPLE

Triangle begins:

  1;

  1,  0;

  1,  1,  1;

  1,  2,  2,   0;

  1,  3,  4,   2,   1;

  1,  4,  7,   6,   3,   0;

  1,  5, 11,  13,   9,   3,   1;

  1,  6, 16,  24,  22,  12,   4,   0;

  1,  7, 22,  40,  46,  34,  16,   4,  1;

  1,  8, 29,  62,  86,  80,  50,  20,  5,  0;

  1,  9, 37,  91, 148, 166, 130,  70, 25,  5, 1;

  1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6, 0;

...

MAPLE

read transforms; 1/(1-x-x*y-y^2); SERIES2(%, x, y, 12); SERIES2TOLIST(%, x, y, 12);

MATHEMATICA

T[n_, 0]:= 1; T[n_, n_]:= (1+(-1)^n)/2; T[n_, k_]:= T[n, k] = T[n-1, k] + T[n-1, k-1]; Table[T[n, k], {n, 0, 10} , {k, 0, n}]//Flatten (* G. C. Greubel, Jan 03 2017 *)

PROG

(Sage)

def A059259_row(n):

    @cached_function

    def prec(n, k):

        if k==n: return (-1)^n

        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)^(n-k+1)*prec(n+1, k) for k in (1..n)]

for n in (1..12): print(A059259_row(n)) # Peter Luschny, Mar 16 2016

(PARI) {T(n, k) = if(k==0, 1, if(k==n, (1+(-1)^n)/2, T(n-1, k) +T(n-1, k-1)) )};

for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Apr 29 2019

CROSSREFS

See A059260 for an explicit formula.

Diagonals of this triangle are given by A006498.

Similar to the triangles A035317, A080242, A108561, A112555.

Cf. A123521, A157897.

Sequence in context: A343489 A109754 A220074 * A124394 A086460 A321884

Adjacent sequences:  A059256 A059257 A059258 * A059260 A059261 A059262

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Jan 23 2001

STATUS

approved

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Last modified May 9 05:06 EDT 2021. Contains 343688 sequences. (Running on oeis4.)