A134346 Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).

1, 3, 3, 7, 14, 7, 15, 45, 45, 15, 31, 124, 186, 124, 31, 63, 315, 630, 630, 315, 63, 127, 762, 1905, 2540, 1905, 762, 127, 255, 1785, 5355, 8925, 8925, 5355, 1785, 255, 511, 4088, 14308, 28616, 35770, 28616, 14308, 4088, 511

Offset: 0

Author

Gary W. Adamson, Oct 21 2007

Keywords

• nonn
• tabl
• easy

Comments

Inverse binomial transform: A134347.

From Wolfdieter Lang, Jul 27 2022: (Start)

Also the triangle t with offset 1 and elements t(n, m) = T(n-1, m-1) read by rows, giving in row n >= 1 the sums of the entries of A356028 of like m.

Also triangle t with offset 1 read by rows, giving in row n >= 1 the sum of the numbers from 1, 2, ..., 2^n - 1 with binary weight m, for m = 1, 2, ..., n. [Observation by Kevin Ryde.] (End)

T(n,k) is the sum of the entries in the (k+2)-th column of the Christmas tree pattern (A367562) of order n+1. - Paolo Xausa, Dec 20 2023

Examples

			First few rows of the triangle:
n\k    0    1     2     3      4      5     6     7    8   9 ...
0:     1
1:     3    3
2:     7   14     7
3:    15   45    45    15
4:    31  124   186   124     31
5:    63  315   630   630    315     63
6:   127  762  1905  2540   1905    762   127
7:   255 1785  5355  8925   8925   5355  1785   255
8:   511 4088 14308 28616  35770  28616 14308  4088  511
9:  1023 9207 36828 85932 128898 128898 85932 36828 9207 1023
... reformatted by _Wolfdieter Lang_, Aug 21 2022
----------------------------------------------------------------------------------
T(3, 1) = 12 + 10 + 9 + 6 + 5 + 3 = 45. (From A356028 row n = 4, m = 2.)
Recurrences: T(4, 1) = 45 + 15 + 4*16 = 2*(45 + 15) +4 = 124. - _Wolfdieter Lang_, Jul 27 2022
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).

Crossrefs

Cf. A000225, A006516(n+1) (row sums), A124929, A134347, A356028, A356117.

Cf. A367508, A367562.

Programs

Maple

A134346 := proc(n,k)
    (2^(n+1)-1)*binomial(n,k) ;
end proc:
seq(seq( A134346(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Aug 15 2022
ser := series((1/2 - x)^(k - n - 1) - (1 - x)^(k - n - 1), x, 10):
seq(seq(coeff(ser, x, k), k = 0..n), n = 0..9); # Peter Luschny, Aug 22 2022

Mathematica

A134346[n_,k_]:=(2^(n+1)-1)Binomial[n,k];
Table[A134346[n,k],{n,0,10},{k,0,n}] (* Paolo Xausa, Dec 20 2023 *)

PARI

T(n,k) = my(b=binomial(n,k)); b<<(n+1) - b; \\ Kevin Ryde, Aug 15 2022

Formula

T(n, m) = A000225(n+1)*A007318(n, m).

From Wolfdieter Lang, Aug 21 2022: (Start)

T(n, k) = 0 for n < k, T(n, 0) = 2^(n+1) - 1, and

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

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

(Proof for T(n-1, m-1) = t(n, m), offset 1, by separating in the list of the binary code of the numbers 1, 2, ..., 2^n-1 of length n and weight m the sublists with first entry 1 and 0. The total number of elements of the list for n and m is binomial(n, m).) (End)

T(n, k) = [x^k] ((1/2 - x)^(k - n - 1) - (1 - x)^(k - n - 1)). - Peter Luschny, Aug 22 2022

Extensions

Name simplified by R. J. Mathar, Aug 15 2022