A099239 Square array read by antidiagonals associated with sections of 1/(1-x-x^k).

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 8, 4, 1, 1, 16, 21, 13, 5, 1, 1, 32, 55, 41, 19, 6, 1, 1, 64, 144, 129, 69, 26, 7, 1, 1, 128, 377, 406, 250, 106, 34, 8, 1, 1, 256, 987, 1278, 907, 431, 153, 43, 9, 1, 1, 512, 2584, 4023, 3292, 1757, 686, 211, 53, 10, 1, 1, 1024, 6765, 12664, 11949, 7168, 3088, 1030, 281, 64, 11, 1

Offset

0,5

Comments

Rows include A099242, A099253. Columns include A034856. Main diagonal is A099240. Sums of antidiagonals are A099241.

Links

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Formula

T(n, k) = Sum_{j=0..n} binomial(k*n -(k-1)*(j-1), j), n, k>=0. (square array)

T(n, k) = Sum_{j=0..n} binomial(k + (n-1)*(j+1), n*(j+1) -1), n>0. (square array)

T(n, k) = Sum_{j=0..n-k} binomial(k*(n-k) - (k-1)*(j-1), j). (number triangle)

Rows of the square array are generated by 1/((1-x)^k-x).

Rows satisfy a(n) = a(n-1) - Sum_{k=1..n} (-1)^(k^binomial(n, k)) * a(n-k).

Example

Rows begin
  1, 1,  1,   1,   1, ...                               A000012;
  1, 2,  4,   8,  16, ...      1-section of 1/(1-x-x)   A000079;
  1, 3,  8,  21,  55, ....     bisection of 1/(1-x-x^2) A001906;
  1, 4, 13,  41, 129, ...     trisection of 1/(1-x-x^3) A052529; (essentially)
  1, 5, 19,  69, 250, ...  quadrisection of 1/(1-x-x^4) A055991;
  1, 6, 26, 106, 431, ...  quintisection of 1/(1-x-x^5) A079675; (essentially)

Mathematica

T[n_, k_]:= Sum[Binomial[k*(n-k) - (k-1)*(j-1), j], {j, 0, n-k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 09 2021 *)

Prog

(Sage)
def A099239(n, k): return sum( binomial(k*(n-k) -(k-1)*(j-1), j) for j in (0..n-k) )
flatten([[A099239(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2021
(Magma)
A099239:= func< n, k | (&+[Binomial(k*(n-k) -(k-1)*(j-1), j): j in [0..n-k]]) >;
[A099239(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 09 2021

Crossrefs

Cf. A034856, A099240, A099241, A099242, A099253.

Cf. A000079, A001906, A052529, A055991, A079675.

Sequence in context: A247286 A055587 A137743 * A167630 A322264 A009998

Adjacent sequences: A099236 A099237 A099238 * A099240 A099241 A099242

Keyword

easy,nonn,tabl

Author

Paul Barry, Oct 08 2004

Status

approved