OFFSET
0,2
COMMENTS
This triangle gives the iterated partial sums of the Pell sequence A000129(n+1), n>=0. - Wolfdieter Lang, Oct 05 2014
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
Riordan array (1/(1-2*x-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, 2*j+k+1)*2^j.
Sum_{k=0..n} T(n, k) = A106514(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A106515(n).
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
MATHEMATICA
T[n_, k_]= Sum[Binomial[n+1, 2*j+k+1]*2^j, {j, 0, Floor[(n+1)/2]}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 05 2021 *)
PROG
(Magma) [ (&+[Binomial(n+1, 2*j+k+1)*2^j: j in [0..Floor((n+1)/2)]]) : k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 05 2021
(Sage)
@CachedFunction
def T(n, k):
if (k<0 or k>n): return 0
elif (k==0): return lucas_number1(n+1, 2, -1)
else: return T(n-1, k-1) + T(n-1, k)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 05 2021
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, May 05 2005
STATUS
approved