OFFSET
0,5
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.
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.
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
EXAMPLE
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;
MATHEMATICA
Table[2^((n-k)k), {n, 0, 10}, {k, 0, n}]//Flatten (* Harvey P. Dale, Jan 09 2017 *)
PROG
(PARI) T(n, k)=if(n<k || k<0, 0, 2^((n-k)*k))
(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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 12 2006
STATUS
approved