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A110522
Riordan array (1/(1+x), x*(1-2*x)/(1+x)^2).
3
1, -1, 1, 1, -5, 1, -1, 12, -9, 1, 1, -22, 39, -13, 1, -1, 35, -115, 82, -17, 1, 1, -51, 270, -344, 141, -21, 1, -1, 70, -546, 1106, -773, 216, -25, 1, 1, -92, 994, -2954, 3199, -1466, 307, -29, 1, -1, 117, -1674, 6888, -10791, 7461, -2487, 414, -33, 1, 1, -145, 2655, -14484, 31179, -30645, 15060, -3900, 537, -37, 1
OFFSET
0,5
COMMENTS
Inverse of A110519.
Product of inverse binomial transform matrix (1/(1+x), x/(1+x)) and (1, x*(1-3*x)) (A110517).
FORMULA
Number triangle T(n, k) = Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-3)^(j-k)*C(k, j-k).
T(n, k) = Sum_{j=0..n} Sum_{i=0..k} C(k, i)*C(n+k-i-j-1, n-k-i-j)*(-1)^(n-k)*2^i.
Sum_{k=0..n} T(n, k) = A110523(n) (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = A110524(n) (diagonal sums).
T(n,k) = T(n-1,k-1) - 2*T(n-1,k) - T(n-2,k) - 2*T(n-2,k-1), T(0,0) = 1, T(1,0) = -1, T(1,1) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 12 2014
From G. C. Greubel, Dec 28 2023: (Start)
T(n, 0) = A033999(n).
T(n, 1) = (-1)^(n-1)*A000326(n), n >= 1.
T(n, n) = 1.
T(n, n-1) = -A016813(n-1), n >= 1.
T(n, n-2) = A236267(n-2), n >= 2.
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A052924(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^n*A078005(n). (End)
EXAMPLE
Rows begin
1;
-1, 1;
1, -5, 1;
-1, 12, -9, 1;
1, -22, 39, -13, 1;
-1, 35, -115, 82, -17, 1;
MATHEMATICA
T[n_, k_]:= Sum[(-1)^(n-j)*(-3)^(j-k)*Binomial[k, j- k]*Binomial[n, j], {j, 0, n}];
Table[T[n, k], {n, 0, 20}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 30 2017 *)
PROG
(PARI) A110522(n, k) = if(n==0, 1, sum(j=0, n, (-1)^(n-j)*(-3)^(j-k)*binomial(n, j)*binomial(k, j-k)));
for(n=0, 12, for(k=0, n, print1(A110522(n, k), ", "))) \\ G. C. Greubel, Aug 30 2017; Dec 28 2023
(Magma)
A110522:= func< n, k | (-1)^(n+k)*(&+[ 3^(j-k)*Binomial(k, j-k)*Binomial(n, j) : j in [0..n]] ) >;
[A110522(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 28 2023
(SageMath)
def A110522(n, k): return (-1)^(n+k)*sum(3^(j-k)*binomial(k, j-k)*binomial(n, j) for j in range(n+1))
flatten([[A110522(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 28 2023
CROSSREFS
Cf. A110519 (inverse), A110523 (row sums), A110524 (diagonal sums).
Sequence in context: A146954 A174949 A174861 * A146987 A297915 A298508
KEYWORD
easy,sign,tabl
AUTHOR
Paul Barry, Jul 24 2005
STATUS
approved