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A206306
Riordan array (1, x/(1-3*x+2*x^2)).
2
1, 0, 1, 0, 3, 1, 0, 7, 6, 1, 0, 15, 23, 9, 1, 0, 31, 72, 48, 12, 1, 0, 63, 201, 198, 82, 15, 1, 0, 127, 522, 699, 420, 125, 18, 1, 0, 255, 1291, 2223, 1795, 765, 177, 21, 1, 0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1
OFFSET
0,5
COMMENTS
The convolution triangle of the Mersenne numbers A000225. - Peter Luschny, Oct 09 2022
FORMULA
Triangle T(n,k), read by rows, given by (0, 3, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Diagonals sums are even-indexed Fibonacci numbers.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A204089(n), A204091(n) for x = 0, 1, 2 respectively.
G.f.: (1-3*x+2*x^)/(1-(3+y)*x+2*x^2).
From Philippe Deléham, Nov 17 2013; corrected Feb 13 2020: (Start)
T(n, n) = 1.
T(n+1, n) = 3n = A008585(n).
T(n+2, n) = A062725(n).
T(n,k) = 3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k), T(0,0)=T(1,1)=T(2,2)=1, T(1,0)=T(2,0)=0, T(2,1)=3, T(n,k)=0 if k<0 or if k>n. (End)
From G. C. Greubel, Dec 20 2022: (Start)
Sum_{k=0..n} (-1)^k*T(n,k) = [n=1] - A009545(n).
Sum_{k=0..n} (-2)^k*T(n,k) = [n=1] + A078020(n+1).
T(2*n, n+1) = A045741(n+2), n >= 0.
T(2*n+1, n+1) = A244038(n). (End)
EXAMPLE
Triangle begins:
1;
0, 1;
0, 3, 1;
0, 7, 6, 1;
0, 15, 23, 9, 1;
0, 31, 72, 48, 12, 1;
0, 63, 201, 198, 82, 15, 1;
0, 127, 522, 699, 420, 125, 18, 1;
0, 255, 1291, 2223, 1795, 765, 177, 21, 1;
0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1;
0, 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1;
MAPLE
# Uses function PMatrix from A357368.
PMatrix(10, n -> 2^n - 1); # Peter Luschny, Oct 09 2022
MATHEMATICA
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==n, 1, If[k==0, 0, 3*T[n- 1, k] +T[n-1, k-1] -2*T[n-2, k]]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 20 2022 *)
PROG
(Magma)
function T(n, k) // T = A206306
if k lt 0 or k gt n then return 0;
elif k eq n then return 1;
elif k eq 0 then return 0;
else return 3*T(n-1, k) +T(n-1, k-1) -2*T(n-2, k);
end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 20 2022
(SageMath)
def T(n, k): # T = A206306
if (k<0 or k>n): return 0
elif (k==n): return 1
elif (k==0): return 0
else: return 3*T(n-1, k) +T(n-1, k-1) -2*T(n-2, k)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 20 2022
CROSSREFS
Columns: A000007, A000225 (Mersenne numbers), A045618, A055582.
Sequence in context: A110504 A361475 A111246 * A178124 A354010 A143395
KEYWORD
easy,nonn,tabl
AUTHOR
Philippe Deléham, Feb 06 2012
STATUS
approved