A117401 Triangle T(n,k) = 2^(k*(n-k)), read by rows.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 16, 8, 1, 1, 16, 64, 64, 16, 1, 1, 32, 256, 512, 256, 32, 1, 1, 64, 1024, 4096, 4096, 1024, 64, 1, 1, 128, 4096, 32768, 65536, 32768, 4096, 128, 1, 1, 256, 16384, 262144, 1048576, 1048576, 262144, 16384, 256, 1

Offset: 0

Author

Paul D. Hanna, Mar 12 2006

Keywords

• nonn
• tabl

Comments

Matrix power T^m satisfies: [T^m](n,k) = [T^m](n-k,0)*T(n,k) for all m and so the triangle has an invariant character.

Examples

			A(x,y) = 1/(1-xy) + x/(1-2xy) + x^2/(1-4xy) + x^3/(1-8xy) + ...
Triangle begins:
  1;
  1,   1;
  1,   2,     1;
  1,   4,     4,      1;
  1,   8,    16,      8,       1;
  1,  16,    64,     64,      16,       1;
  1,  32,   256,    512,     256,      32,      1;
  1,  64,  1024,   4096,    4096,    1024,     64,     1;
  1, 128,  4096,  32768,   65536,   32768,   4096,   128,   1;
  1, 256, 16384, 262144, 1048576, 1048576, 262144, 16384, 256, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Crossrefs

Cf. A117402 (row sums), A117403 (antidiagonal sums), A002416 (central terms).

Cf. this sequence (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15).

Programs

Magma

A117401:= func< n, k, m | (m+2)^(k*(n-k)) >;
[A117401(n, k, 0): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 28 2021

Mathematica

Table[2^((n-k)k),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Jan 09 2017 *)

PARI

T(n,k)=if(n
				

Sage

def A117401(n, k, m): return (m+2)^(k*(n-k))
flatten([[A117401(n, k, 0) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 28 2021

Formula

G.f.: A(x,y) = Sum_{n>=0} x^n/(1 - 2^n*x*y).

G.f. satisfies: A(x,y) = 1/(1 - x*y) + x*A(x,2*y).

Equals ConvOffsStoT transform of the 2^n series: (1, 2, 4, 8, ...); e.g., ConvOffs transform of (1, 2, 4, 8) = (1, 8, 16, 8, 1). - Gary W. Adamson, Apr 21 2008

T(n,k) = (1/n)*( 2^(n-k)*k*T(n-1,k-1) + 2^k*(n-k)*T(n-1,k) ), where T(i,j)=0 if j>i. - Tom Edgar, Feb 20 2014

Let E(x) = Sum_{n>=0} x^n/2^C(n,2). Then E(x)*E(y*x) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/2^C(n,2). - Geoffrey Critzer, May 31 2020

T(n, k, m) = (m+2)^(k*(n-k)) with m = 0. - G. C. Greubel, Jun 28 2021