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A336163 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * binomial(n,j)^3. 2
1, 1, 1, 1, 2, 1, 1, 3, 10, 1, 1, 4, 21, 56, 1, 1, 5, 34, 171, 346, 1, 1, 6, 49, 352, 1521, 2252, 1, 1, 7, 66, 605, 3946, 14283, 15184, 1, 1, 8, 85, 936, 8065, 46744, 138909, 104960, 1, 1, 9, 106, 1351, 14346, 113525, 573616, 1385163, 739162, 1, 1, 10, 129, 1856, 23281, 231876, 1656145, 7217536, 14072193, 5280932, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Column k is the diagonal of the rational function 1 / (1 + y + z + x*y + y*z + k*z*x + (k+1)*x*y*z).

Column k is the diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) - k*x*y*z).

LINKS

Table of n, a(n) for n=0..65.

EXAMPLE

Square array begins:

  1,    1,     1,     1,      1,      1, ...

  1,    2,     3,     4,      5,      6, ...

  1,   10,    21,    34,     49,     66, ...

  1,   56,   171,   352,    605,    936, ...

  1,  346,  1521,  3946,   8065,  14346, ...

  1, 2252, 14283, 46744, 113525, 231876, ...

MATHEMATICA

Unprotect[Power]; 0^0 = 1; T[n_, k_] := Sum[k^j * Binomial[n, j]^3, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 11 2020 *)

CROSSREFS

Columns k=0-6 give: A000012, A000172, A206178, A206180, A216483, A216636, A216698.

Main diagonal gives A241247.

Cf. A307883, A336179, A336187.

Sequence in context: A077385 A337219 A220898 * A066013 A212261 A014521

Adjacent sequences:  A336160 A336161 A336162 * A336164 A336165 A336166

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Jul 10 2020

STATUS

approved

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Last modified June 16 11:58 EDT 2021. Contains 345057 sequences. (Running on oeis4.)