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A106513 A Pell-Pascal matrix. 4
1, 2, 1, 5, 3, 1, 12, 8, 4, 1, 29, 20, 12, 5, 1, 70, 49, 32, 17, 6, 1, 169, 119, 81, 49, 23, 7, 1, 408, 288, 200, 130, 72, 30, 8, 1, 985, 696, 488, 330, 202, 102, 38, 9, 1, 2378, 1681, 1184, 818, 532, 304, 140, 47, 10, 1, 5741, 4059, 2865, 2002, 1350, 836, 444, 187, 57 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row sums are A106514. Antidiagonal sums are A106515.

This triangle gives the iterated partial sums of the Pell sequence A000129(n+1), n>=0. - Wolfdieter Lang, Oct 05 2014

LINKS

Table of n, a(n) for n=0..63.

FORMULA

Riordan array (1/(1-2x-x^2), x/(1-x)).

Number triangle T(n,0) = A000129(n+1), T(n,k) = T(n-1,k-1) + T(n-1,k); T(n,k) = Sum_{j=0..floor((n+1)/2)} binomial(n+1, 2j+k+1)2^j.

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1) - T(n-3,k) - T(n-3,k-1), T(0,0)=1, T(1,0)=2, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 14 2014

From Wolfdieter Lang, Oct 05 2014: (Start)

O.g.f. for row polynomials R(n,x) = Sum_{k=0..n} T(n,k)*x^k: (1 - z)/((1 - 2*z - z^2)*(1 - (1+x)*z)).

O.g.f. column m: (1/(1 - 2*z - z^2))*(x/(1 - x)))^m, m >= 0. (Riordan property).

The alternating row sums are shown in A001333.

A-sequence: [1, 1] (see the three term recurrence given above). Z-sequence has o.g.f. (2 + 3*x)/(1 + x), [2, 1, repeat(-1,1)] (unsigned A054977). See the W. Lang link under A006232 for Riordan A- and Z-sequences.

The inverse Riordan triangle is shown in A248156. (End)

EXAMPLE

The triangle T(n,k) begins:

n\k    0    1    2    3    4   5   6   7  8  9 10 ...

0:     1

1:     2    1

2:     5    3    1

3:    12    8    4    1

4:    29   20   12    5    1

5:    70   49   32   17    6   1

6:   169  119   81   49   23   7   1

7:   408  288  200  130   72  30   8   1

8:   985  696  488  330  202 102  38   9  1

9:  2378 1681 1184  818  532 304 140  47 10  1

10: 5741 4059 2865 2002 1350 836 444 187 57 11  1

... Reformatted and extended. - Wolfdieter Lang, Oct 05 2014

-----------------------------------------------------

Recurrence from the Z-sequence (see the formula above) for T(0,n) in terms of the entries of row n-1. For example, 29 = T(4,0) = 2*12 + 1*8 + (-1)*4 + 1*1 = 29. - Wolfdieter Lang, Oct 05 2014

CROSSREFS

Cf. A000129, A001333, A106514, A106515, A248156.

Sequence in context: A105728 A120095 A130197 * A054446 A164981 A047858

Adjacent sequences:  A106510 A106511 A106512 * A106514 A106515 A106516

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, May 05 2005

STATUS

approved

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Last modified February 21 04:43 EST 2018. Contains 299389 sequences. (Running on oeis4.)