A156862 Triangle read by rows: T(n, k) = 2^k - binomial(n, k+1) + 2^(n-k) - binomial(n, n-k+1).

2, 2, 2, 3, 2, 3, 6, 2, 2, 6, 13, 3, 0, 3, 13, 28, 7, -3, -3, 7, 28, 59, 18, -6, -14, -6, 18, 59, 122, 44, -6, -32, -32, -6, 44, 122, 249, 101, 4, -58, -80, -58, 4, 101, 249, 504, 221, 39, -90, -162, -162, -90, 39, 221, 504, 1015, 468, 130, -119, -292, -356, -292, -119, 130, 468, 1015

Offset

0,1

Comments

Row sums are 2^(n+1): {2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, ...}.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

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

Example

Triangle begins as:
     2;
     2,   2;
     3,   2,   3;
     6,   2,   2,    6;
    13,   3,   0,    3,   13;
    28,   7,  -3,   -3,    7,   28;
    59,  18,  -6,  -14,   -6,   18,   59;
   122,  44,  -6,  -32,  -32,   -6,   44,  122;
   249, 101,   4,  -58,  -80,  -58,    4,  101, 249;
   504, 221,  39,  -90, -162, -162,  -90,   39, 221, 504;
  1015, 468, 130, -119, -292, -356, -292, -119, 130, 468, 1015;

Maple

f(n, m):= 2^m - binomial(n, m+1); seq(seq( f(n, k) + f(n, n-k), k=0..n), n=0..10); # G. C. Greubel, Dec 01 2019

Mathematica

f[n_, m_]:= 2^m - Binomial[n, m+1]; T[n_, k_]:= f[n, k] + f[n, n-k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(PARI) T(n, k) = my(f(n, m)=2^m - binomial(n, m+1)); f(n, k) + f(n, n-k); \\ G. C. Greubel, Dec 01 2019
(Magma)
f:= func< n, m | 2^m - Binomial(n, m+1) >;
[f(n, k)+f(n, n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 01 2019
(Sage)
def f(n, m): return 2^m - binomial(n, m+1)
def T(n, k): return f(n, k) + f(n, n-k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 01 2019
(GAP) Flat(List([0..10], n-> List([0..n], k-> 2^k - Binomial(n, k+1) + 2^(n-k) - Binomial(n, n-k+1) ))); # G. C. Greubel, Dec 01 2019

Crossrefs

Sequence in context: A050142 A092964 A183368 * A318583 A076709 A307774

Adjacent sequences: A156859 A156860 A156861 * A156863 A156864 A156865

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 17 2009

Status

approved