OFFSET
0,2
COMMENTS
Pascal-like triangle: start with 1 at top; every subsequent entry is the sum of everything above you, plus 1.
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
G.f.: 1/(1 - 2*z - 2*w + 2*z*w).
T(n, k) = Sum_{j=0..n} (-1)^j*2^(n + k - j)*C(n, j)*C(n + k - j, n).
T(n, 0) = T(n, n) = A000079(n).
T(2*n, n) = A084773(n).
T(n, k) = 2^n*binomial(n, k)*hypergeom([-k, k - n], [-n], 1/2). - Peter Luschny, Nov 26 2021
From G. C. Greubel, May 21 2023: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..n} T(n, k) = A007070(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A077957(n).
T(n, 2) = 4*A049611(n-1). (End)
EXAMPLE
Triangle begins as:
n\k [0] [1] [2] [3] [4] [5] [6] ...
[0] 1;
[1] 2, 2;
[2] 4, 6, 4;
[3] 8, 16, 16, 8;
[4] 16, 40, 52, 40, 16;
[5] 32, 96, 152, 152, 96, 32;
[6] 64, 224, 416, 504, 416, 224, 64;
...
MAPLE
read transforms; SERIES2(1/(1-2*z-2*w+2*z*w), x, y, 12): SERIES2TOLIST(%, x, y, 12);
# Alternative
T := (n, k) -> 2^n*binomial(n, k)*hypergeom([-k, -n + k], [-n], 1/2):
for n from 0 to 10 do seq(simplify(T(n, k)), k = 0 .. n) end do; # Peter Luschny, Nov 26 2021
MATHEMATICA
Table[(-1)^k*2^n*JacobiP[k, -n-1, 0, 0], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Oct 04 2017; May 21 2023 *)
PROG
(Magma)
A059474:= func< n, k | (&+[(-1)^j*2^(n-j)*Binomial(n-k, j)*Binomial(n-j, n-k): j in [0..n-k]]) >;
[A059474(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 21 2023
(SageMath)
def A059474(n, k): return 2^n*binomial(n, k)*simplify(hypergeometric([-k, k-n], [-n], 1/2))
flatten([[A059474(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, May 21 2023
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, Feb 03, 2001; revised Jun 12 2005
STATUS
approved