A176125 Triangle, read by rows, T(n, k) = 1 - floor(n*(n-1)/4) + floor(binomial(n-1,k-1) * binomial(n, k-1)/(2*k)).

1, 1, 1, 1, 6, 1, 1, 19, 19, 1, 1, 43, 78, 43, 1, 1, 85, 232, 232, 85, 1, 1, 151, 571, 865, 571, 151, 1, 1, 249, 1239, 2625, 2625, 1239, 249, 1, 1, 386, 2449, 6904, 9676, 6904, 2449, 386, 1, 1, 573, 4505, 16303, 30460, 30460, 16303, 4505, 573, 1, 1, 820, 7827, 35354, 84904, 113218, 84904, 35354, 7827, 820, 1

Offset

3,5

Comments

Row sums are: {1, 2, 8, 40, 166, 636, 2311, 8228, 29156, 103684, 371030, 1336688, ...}.

Links

G. C. Greubel, Rows n = 3..100 of triangle, flattened

Formula

T(n, k) = 1 - floor(n*(n-1)/4) + floor(binomial(n-1,k-1) * binomial(n, k-1)/(2*k)).

Example

Triangle begins as:
  1;
  1,   1;
  1,   6,    1;
  1,  19,   19,    1;
  1,  43,   78,   43,    1;
  1,  85,  232,  232,   85,    1;
  1, 151,  571,  865,  571,  151,    1;
  1, 249, 1239, 2625, 2625, 1239,  249,   1;
  1, 386, 2449, 6904, 9676, 6904, 2449, 386, 1;

Maple

b:=binomial; T(n, k):= 1 -floor(n*(n-1)/4) +floor(b(n-1, k-1)*b(n, k-1)/(2*k)); seq(seq(T(n, k), k=2..n-1), n=3..15); # G. C. Greubel, Nov 27 2019

Mathematica

T[n_, k_]:= 1 -Floor[n*(n-1)/4] +Floor[Binomial[n-1, k-1]*Binomial[n, k-1]/(2*k)];
Table[T[n, k], {n, 3, 15}, {k, 2, n-1}]//Flatten

Prog

(PARI) b=binomial; T(n, k)= 1 -(n*(n-1))\4 +(b(n-1, k-1)*b(n, k-1))\(2*k); \\ G. C. Greubel, Nov 27 2019
(Magma) B:=Binomial; [1 -Floor(n*(n-1)/4) +Floor(B(n-1, k-1)*B(n, k-1)/(2*k)): k in [2..n-1], n in [3..15]]; // G. C. Greubel, Nov 27 2019
(SageMath) b=binomial; [[1 -floor(n*(n-1)/4) +floor(b(n-1, k-1)*b(n, k-1)/(2*k)) for k in (2..n-1)] for n in (3..15)] # G. C. Greubel, Nov 27 2019
(GAP) B:=Binomial;; Flat(List([3..15], n-> List([2..n-1], k-> 1 -Int(n*(n-1)/4) +Int(B(n-1, k-1)*B(n, k-1)/(2*k)) ))); # G. C. Greubel, Nov 27 2019

Crossrefs

Sequence in context: A146959 A157632 A328888 * A168289 A141690 A318408

Adjacent sequences: A176122 A176123 A176124 * A176126 A176127 A176128

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 09 2010

Status

approved