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A309010 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{j=0..n} binomial(n,j)^k. 11
1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 6, 8, 5, 1, 2, 10, 20, 16, 6, 1, 2, 18, 56, 70, 32, 7, 1, 2, 34, 164, 346, 252, 64, 8, 1, 2, 66, 488, 1810, 2252, 924, 128, 9, 1, 2, 130, 1460, 9826, 21252, 15184, 3432, 256, 10, 1, 2, 258, 4376, 54850, 206252, 263844, 104960, 12870, 512, 11 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

T(n,k) is the constant term in the expansion of (Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0. - Seiichi Manyama, Oct 27 2019

Let B_k be the binomial poset containing all k-tuples of equinumerous subsets of {1,2,...} ordered by inclusion componentwise (described in Stanley reference below).  Then T(k,n) is the number of elements in any n-interval of B_k. - Geoffrey Critzer, Apr 16 2020

Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) - Product_{j=1..k} x_j) for k>0. - Seiichi Manyama, Jul 11 2020

REFERENCES

R. P. Stanley, Enumerative Combinatorics Vol I, Second Edition, Cambridge, 2011, Example 3.18.3 d, page 366.

LINKS

Seiichi Manyama, Antidiagonals n = 0..100, flattened

FORMULA

Sum_n>=0 T(n,k) x^n/(n!^k) = (Sum_n>=0 x^n/(n!^k))^2. - Geoffrey Critzer, Apr 17 2020

EXAMPLE

Square array begins:

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

   2,  2,   2,    2,     2,      2, ...

   3,  4,   6,   10,    18,     34, ...

   4,  8,  20,   56,   164,    488, ...

   5, 16,  70,  346,  1810,   9826, ...

   6, 32, 252, 2252, 21252, 206252, ...

MATHEMATICA

nn = 8; Table[ek[x_] := Sum[x^n/n!^k, {n, 0, nn}]; Range[0, nn]!^k CoefficientList[Series[ek[x]^2, {x, 0, nn}], x], {k, 0, nn}] // Transpose // Grid (* Geoffrey Critzer, Apr 17 2020 *)

CROSSREFS

Columns k=0..10 give A000027(n+1), A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.

Main diagonal gives A167010.

T(n,n+1) gives A328812.

Cf. A328747, A328748, A328807.

Sequence in context: A145111 A104795 A116925 * A308500 A210950 A214314

Adjacent sequences:  A309007 A309008 A309009 * A309011 A309012 A309013

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Jul 06 2019

STATUS

approved

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Last modified April 20 06:59 EDT 2021. Contains 343125 sequences. (Running on oeis4.)