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A207974 Triangle related to A152198. 2
1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 2, 2, 1, 1, 5, 2, 4, 1, 1, 1, 6, 3, 6, 3, 2, 1, 1, 7, 3, 9, 3, 5, 1, 1, 1, 8, 4, 12, 6, 8, 4, 2, 1, 1, 9, 4, 16, 6, 14, 4, 6, 1, 1, 1, 10, 5, 20, 10, 20, 10, 10, 5, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Row sums are A027383(n).
Diagonal sums are alternately A014739(n) and A001911(n+1).
The matrix inverse starts
1;
-1,1;
1,-2,1;
1,-1,-1,1;
-1,2,0,-2,1;
-1,1,2,-2,-1,1;
1,-2,-1,4,-1,-2,1;
1,-1,-3,3,3,-3,-1,1;
-1,2,2,-6,0,6,-2,-2,1;
-1,1,4,-4,-6,6,4,-4,-1,1;
1,-2,-3,8,2,-12,2,8,-3,-2,1;
apparently related to A158854. - R. J. Mathar, Apr 08 2013
From Gheorghe Coserea, Jun 11 2016: (Start)
T(n,k) is the number of terms of the sequence A057890 in the interval [2^n,2^(n+1)-1] having binary weight k+1.
T(n,k) = A007318(n,k) (mod 2) and the number of odd terms in row n of the triangle is 2^A000120(n).
(End)
LINKS
Gheorghe Coserea, Rows n = 0..200, flattened
FORMULA
T(n,k) = T(n-1,k-1) - (-1)^k*T(n-1,k), k>0 ; T(n,0) = 1.
T(2n,2k) = T(2n+1,2k) = binomial(n,k) = A007318(n,k).
T(2n+1,2k+1) = A110813(n,k).
T(2n+2,2k+1) = 2*A135278(n,k).
T(n,2k) + T(n,2k+1) = A152201(n,k).
T(n,2k) = A152198(n,k).
T(n+1,2k+1) = A152201(n,k).
T(n,k) = T(n-2,k-2) + T(n-2,k).
T(2n,n) = A128014(n+1).
T(n,k) = card {p, 2^n <= A057890(p) <= 2^(n+1)-1 and A000120(A057890(p)) = k+1}. - Gheorghe Coserea, Jun 09 2016
P_n(x) = Sum_{k=0..n} T(n,k)*x^k = ((2+x+(n mod 2)*x^2)*(1+x^2)^(n\2) - 2)/x. - Gheorghe Coserea, Mar 14 2017
EXAMPLE
Triangle begins :
n\k [0] [1] [2] [3] [4] [5] [6] [7] [8] [9]
[0] 1;
[1] 1, 1;
[2] 1, 2, 1;
[3] 1, 3, 1, 1;
[4] 1, 4, 2, 2, 1;
[5] 1, 5, 2, 4, 1, 1;
[6] 1, 6, 3, 6, 3, 2, 1;
[7] 1, 7, 3, 9, 3, 5, 1, 1;
[8] 1, 8, 4, 12, 6, 8, 4, 2, 1;
[9] 1, 9, 4, 16, 6, 14, 4, 6, 1, 1;
[10] ...
MAPLE
A207974 := proc(n, k)
if k = 0 then
1;
elif k < 0 or k > n then
0 ;
else
procname(n-1, k-1)-(-1)^k*procname(n-1, k) ;
end if;
end proc: # R. J. Mathar, Apr 08 2013
PROG
(PARI)
seq(N) = {
my(t = vector(N+1, n, vector(n, k, k==1 || k == n)));
for(n = 2, N+1, for (k = 2, n-1,
t[n][k] = t[n-1][k-1] + (-1)^(k%2)*t[n-1][k]));
return(t);
};
concat(seq(10)) \\ Gheorghe Coserea, Jun 09 2016
(PARI)
P(n) = ((2+x+(n%2)*x^2) * (1+x^2)^(n\2) - 2)/x;
concat(vector(11, n, Vecrev(P(n-1)))) \\ Gheorghe Coserea, Mar 14 2017
CROSSREFS
Cf. Diagonals : A000012, A000034, A052938, A097362
Related to thickness: A000120, A027383, A057890, A274036.
Sequence in context: A105535 A182980 A244051 * A239454 A108888 A124021
KEYWORD
easy,nonn,tabl
AUTHOR
Philippe Deléham, Feb 22 2012
STATUS
approved

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Last modified July 14 05:06 EDT 2024. Contains 374291 sequences. (Running on oeis4.)