(MAGMAMagma)

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From Philippe Deléham, Mar 15 2014: (Start)

As a number triangle, this is the Riordan array (1/(1-x), x*(1+4*x)/(1-x)). It has row sums A063727(n). - _Philippe Deléham_, Mar 15 2014

Sum_{k=0..n} T(n, k) = A063727(n). (End)

T(n, k, m) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*m^j, for m = 4. (End)

Sum_{k=0..n} T(n, k, 4) = A063727(n). (End)

Table[Hypergeometric2F1[-k, k-n, 1, 5], {n, 0, 1012}, {k, 0, n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

Cf. A016861, A063727, A081587, A081588.

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1, 1, 1, 1, 6, 1, 1, 11, 11, 1, 1, 16, 46, 16, 1, 1, 21, 106, 106, 21, 1, 1, 26, 191, 396, 191, 26, 1, 1, 31, 301, 1011, 1011, 301, 31, 1, 1, 36, 436, 2076, 3606, 2076, 436, 36, 1, 1, 41, 596, 3716, 9726, 9726, 3716, 596, 41, 1, 1, 46, 781, 6056, 21746, 33876, 21746, 6056, 781, 46, 1

Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) +4T 4*T(n-1, k-1) + T(n-1, k). Rows are the expansions of (1+4x)^k/(1-x)^(k+1).

Rows are the expansions of (1+4*x)^k/(1-x)^(k+1).

From G. C. Greubel, May 26 2021: (Start)

T(n, k, m) = Hypergeometric2F1([-k, k-n], [1], m+1), for m = 4.

T(n, k, m) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*m^j, for m = 4.

Sum_{k=0..n} T(n, k, 4) = A063727(n). (End)

Rows start

Square array begins as:

1 , 1 , 1 , 1 , 1 , ... A000012;

1 , 6 , 11 , 16 , 21 , ... A016861;

1 , 11 , 46 , 106 , 191 , ... A081587;

1 , 16 , 106 , 396 , 1011 , ... A081588;

1 , 21 , 191 , 1011 , 3606 , ...

1;

1, 1;

1, 6, 1;

1, 11, 11, 1;

1, 16, 46, 16, 1;

1, 21, 106, 106, 21, 1;

1, 26, 191, 396, 191, 26, 1;

1, 31, 301, 1011, 1011, 301, 31, 1;

1, 36, 436, 2076, 3606, 2076, 436, 36, 1;

1, 41, 596, 3716, 9726, 9726, 3716, 596, 41, 1;

1, 46, 781, 6056, 21746, 33876, 21746, 6056, 781, 46, 1; - Philippe Deléham, Mar 15 2014

Table[ Hypergeometric2F1[-k, k-n, 1, 5], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 24 2013 *)

(MAGMA)

A081579:= func< n, k, q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;

[A081579(n, k, 4): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021

(Sage) flatten([[hypergeometric([-k, k-n], [1], 5).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

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Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8).