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Sum_{k=0..n}( (-1)^k*T(n, k) = (-1)^n*A108666(n+1) (alternating row sums).

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J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].

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G. C. Greubel, <a href="/A331431/b331431.txt">Rows n = 0..50 of the triangle, flattened</a>

T(n, 0) = (-1)^n*A000027(n+1).

T(n, 1) = A331433(n-1) = (-1)^(n+1)*A007531(n+2).

T(n, 2) = A331434(n-2) = (-1)^n*A054559(n+3).

T(n, n-2) = A002738(n-2).

T(n, n-1) = (-1)*A002736(n).

T(n, n) = A002457(n).

T(2*n, n) = (-1)^n*(3*n+1)!/(n!)^3 = (-1)^n*A331322(n).

Sum_{k=0..n} T(n, k) = A000290(n+1) (row sums).

Sum_{k=0..n}((-1)^k*T(n, k) = (-1)^n*A108666(n+1) (alternating row sums).

Sum_{k=0..n} T(n-k, k) = (-1)^n*A109188(n+1) (diagonal sums).

2^n*Sum_{k=0..n} T(n, k)/2^k = (-1)^floor(n/2)*A100071(n+1) (positive half sums).

(-2)^n*Sum_{k=0..n} T(n, k)/(-2)^k = A331323(n) (negative half sums).

T(2*n+1, n) = (-2)*(-27)^n*Pochhammer(4/3, n)*Pochhammer(5/3, n)/(n!*(n+1)!). - G. C. Greubel, Mar 22 2022

Table[(-1)^(n+k)*(n+k+1)*Binomial[2*k, k]*Binomial[n+k, n-k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 22 2022 *)

(Magma) [(-1)^(n+k)*(k+1)*(2*k+1)*Binomial(n+k+1, n-k)*Catalan(k): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 22 2022

(Sage) flatten([[(-1)^(n+k)*(2*k+1)*binomial(2*k, k)*binomial(n+k+1, n-k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Mar 22 2022

Row sums: Sum_{k=0..n} T(n, k) = A000290(n+1).

Alternating row sums: Sum_{k=0..n}((-1)^k*T(n, k) = (-1)^n*A108666(n+1).

Diagonal sums: Sum_{k=0..n} T(n-k, k) = (-1)^n*A109188(n+1).

Positive half sums: 2^n*Sum_{k=0..n} T(n, k)/2^k = (-1)^floor(n/2)*A100071(n+1).

Negative half sums: (-2)^n*Sum_{k=0..n} T(n, k)/(-2)^k = A331323(n).

Central values: T(2*n, n) = (-1)^n*(3*n+1)!/(n!)^3 = (-1)^n*A331322(n).

Cf. A000290 (row sums), A002457, , A100071, A108666 (alternating row sums), A109188 (diagonal sums), A331322, A331323, A331430, A331432.

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