A350771 Triangle read by rows: T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k), 0 <= k <= n-1.

0, 1, 1, 3, 4, 3, 7, 12, 12, 7, 15, 32, 36, 32, 15, 31, 80, 100, 100, 80, 31, 63, 192, 270, 280, 270, 192, 63, 127, 448, 714, 770, 770, 714, 448, 127, 255, 1024, 1848, 2128, 2100, 2128, 1848, 1024, 255, 511, 2304, 4680, 5880, 5796, 5796, 5880, 4680, 2304, 511, 1023, 5120, 11610, 16080, 16380, 15624, 16380, 16080, 11610, 5120, 1023

Offset: 1

Author

Ambrosio Valencia-Romero, Jan 14 2022

Keywords

• nonn
• tabl

Comments

The elements in T(n,k) result from the product of each element of A350770(n,k) and binomial(n-1,k).

Examples

			Triangle begins:
     0;
     1,    1;
     3,    4,     3;
     7,   12,    12,     7;
    15,   32,    36,    32,    15;
    31,   80,   100,   100,    80,    31;
    63,  192,   270,   280,   270,   192,    63;
   127,  448,   714,   770,   770,   714,   448,   127;
   255, 1024,  1848,  2128,  2100,  2128,  1848,  1024,   255;
   511, 2304,  4680,  5880,  5796,  5796,  5880,  4680,  2304,  511;
  1023, 5120, 11610, 16080, 16380, 15624, 16380, 16080, 11610, 5120, 1023;
...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
• Ambrosio Valencia-Romero and P. T. Grogan, The strategy dynamics of collective systems: Underlying hindrances beyond two-actor coordination, PLOS ONE 19(4): e0301394 (S1 Appendix).

Crossrefs

Column k=0 gives A000225(n-1).

Row sums give A056182(n-1) = 2*A001047(n-1).

Programs

Maple

T := n -> local k; seq((2^(n - k - 1) + 2^k - 2)*binomial(n - 1, k), k = 0 .. n - 1);
seq(T(n), n = 1 .. 11);

PARI

T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k) \\ Andrew Howroyd, Jan 05 2024

Formula

T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k).