A038234 Triangle whose (n, k)-th entry is binomial(n, k)*4^(n - k)*4^k.

1, 4, 4, 16, 32, 16, 64, 192, 192, 64, 256, 1024, 1536, 1024, 256, 1024, 5120, 10240, 10240, 5120, 1024, 4096, 24576, 61440, 81920, 61440, 24576, 4096, 16384, 114688, 344064, 573440, 573440, 344064, 114688, 16384, 65536, 524288

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy

Comments

Also the absolute values of the coefficients of the Belyi Polynomial P_(i,i)(x). - R. J. Mathar, Oct 16 2008

Examples

			1 ;
4 4 ;
16 32 16 ;
64 192 192 64 ;
256 1024 1536 1024 256 ;
1024 5120 10240 10240 5120 1024 ;
4096 24576 61440 81920 61440 24576 4096 ;
16384 114688 344064 573440 573440 344064 114688 16384 ;
65536 524288 1835008 3670016 4587520 3670016 1835008 524288 65536 ;
262144 2359296 9437184 22020096 33030144 33030144 22020096 9437184 2359296 262144 ;
		

Links

• I. Bauer, F. Catanese, F. Grunewald, Chebycheff and Belyi Polynomials, Dessins de'Enfants, Beauville Surfaces and Group Theory, Med. J. Math. vol 3 no 2 (2006) 121-146. [From _R. J. Mathar_, Oct 16 2008]
• B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Programs

Maple

seq(print(seq(4^n*binomial(n, k), k=0..n)), n=0..9);  # Peter Luschny, Feb 07 2025

Formula

G.f.: 1/(1 - 4*x - 4*x*y). - Ilya Gutkovskiy, Apr 21 2017

T(n, k) = 2^(2*n)*JacobiP(n - k, k, -1/2 - n, 1). - Peter Luschny, Feb 07 2025