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Revision History for A123163

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A123163

Triangle T(n, k) = binomial((n-k)^2, k^2) read by rows.
(history; published version)

#23 by Alois P. Heinz at Wed Jul 19 08:56:20 EDT 2023

STATUS

proposed

approved

#22 by Robert C. Lyons at Wed Jul 19 08:29:42 EDT 2023

STATUS

editing

proposed

#21 by Robert C. Lyons at Wed Jul 19 08:29:39 EDT 2023

NAME

Triangle T(n, k) = binomial((n-k)^2, k^2) read by rows,.

STATUS

approved

editing

#20 by OEIS Server at Wed Jul 19 07:57:13 EDT 2023

LINKS

G. C. Greubel, <a href="/A123163/b123163_1.txt">Rows n = 0..50 of the triangle, flattened</a>

#19 by Michael De Vlieger at Wed Jul 19 07:57:13 EDT 2023

STATUS

reviewed

approved

Discussion

Wed Jul 19

07:57

OEIS Server: Installed first b-file as b123163.txt.

#18 by Michel Marcus at Wed Jul 19 00:14:44 EDT 2023

STATUS

proposed

reviewed

#17 by G. C. Greubel at Wed Jul 19 00:06:52 EDT 2023

STATUS

editing

proposed

#16 by G. C. Greubel at Wed Jul 19 00:06:44 EDT 2023

NAME

Triangle read by rows: T(n, k) = binomial[((n-mk)^2,m k^2].) read by rows,

LINKS

G. C. Greubel, <a href="/A123163/b123163_1.txt">Rows n = 0..50 of the triangle, flattened</a>

FORMULA

aT(n,m k) = (n^2 - 2*n*m k + mk^2)!/((mk^2)!(n^2 - 2*n*mk)!).

From G. C. Greubel, Jul 18 2023: (Start)

T(n, 0) = T(2*n, n) = 1.

T(n, n) = A000007(n).

Sum_{k=0..n} T(n, k) = A123165(n). (End)

EXAMPLE

n\m k | 0 1 2 3 4 5 6 7

----+---------------------------------

----+--------------------------------------------

0 | 1;

1 | 1, 0;

2 | 1, 1, 0;

3 | 1, 4, 0, 0;

4 | 1, 9, 1, 0, 0;

5 | 1, 16, 126, 0, 0, 0;

6 | 1, 25, 1820, 1, 0, 0, 0;

7 | 1, 36, 12650, 11440, 0, 0, 0, 0;

MATHEMATICA

tT[n_, m_k_] = (n^2 - 2*n*m k+ mk^2)!/((mk^2)!(n^2 - 2*n*mk)!); a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[a]

Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

PROG

(Magma) [Binomial((n-k)^2, k^2): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 18 2023

(SageMath) flatten([[binomial((n-k)^2, k^2) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 18 2023

CROSSREFS

Cf. A000007, A011973, A123165.

STATUS

approved

editing

#15 by Susanna Cuyler at Mon Jan 28 08:08:26 EST 2019

STATUS

proposed

approved

#14 by Seiichi Manyama at Mon Jan 28 07:04:46 EST 2019

STATUS

editing

proposed