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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), ... 14
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

LINKS

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

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

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.

Sequence in context: A167637 A109754 A220074 * A124394 A086460 A136431

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 June 24 13:17 EDT 2019. Contains 324325 sequences. (Running on oeis4.)