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G. C. Greubel, <a href="/A110522/b110522.txt">Table Rows n = 0..50 of n, a(n) for the first 50 rows, triangle, flattened</a>

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P. Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry2/barry231.html">A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays</a>, J. Int. Seq. 16 (2013) #13.5.4.

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From G. C. Greubel, dec Dec 28 2023: (Start)

(PARI) concat([1], forA110522(n=1, 20, for(, k) = if(n==0, n, print1( 1, sum(j=0, n, (-1)^(n-j)*(-3)^(j-k)*binomial(n, j)*binomial(k, j-k)), ", ")))) \\ _G. C. Greubel_, Aug 30 2017;

for(n=0, 12, for(k=0, n, print1(A110522(n, k), ", "))) \\ G. C. Greubel, Aug 30 2017; Dec 28 2023

[A110522(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, dec Dec 28 2023

flatten([[A110522(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, dec Dec 28 2023

Riordan array (1/(1+x), x*(1-2x2*x)/(1+x)^2).

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, -1

Inverse of A110519.

Inverse of A110519. Row sums are A110523. Diagonal sums are A110524. Product of inverse binomial transform matrix (1/(1+x), x/(1+x)) and (1, x*(1-3x3*x)) (A110517).

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).

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)

T[0, 0] := 1; 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 *)

Table[T[n, k], {n, 0, 20}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 30 2017 *)

(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

Cf. A110519 (inverse), A110523 (row sums), A110524 (diagonal sums).

Cf. A000326, A016813, A033999, A052924, A078005, A110517, A236267.

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