A117401 - OEIS

%I #36 Feb 07 2022 00:29:30

%S 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,

%T 1,64,1024,4096,4096,1024,64,1,1,128,4096,32768,65536,32768,4096,128,

%U 1,1,256,16384,262144,1048576,1048576,262144,16384,256,1

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

%C 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.

%H G. C. Greubel, <a href="/A117401/b117401.txt">Rows n = 0..50 of the triangle, flattened</a>

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

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

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

%F 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

%F 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

%F 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

%F T(n, k, m) = (m+2)^(k*(n-k)) with m = 0. - _G. C. Greubel_, Jun 28 2021

%e A(x,y) = 1/(1-xy) + x/(1-2xy) + x^2/(1-4xy) + x^3/(1-8xy) + ...

%e Triangle begins:

%e   1;

%e   1,   1;

%e   1,   2,     1;

%e   1,   4,     4,      1;

%e   1,   8,    16,      8,       1;

%e   1,  16,    64,     64,      16,       1;

%e   1,  32,   256,    512,     256,      32,      1;

%e   1,  64,  1024,   4096,    4096,    1024,     64,     1;

%e   1, 128,  4096,  32768,   65536,   32768,   4096,   128,   1;

%e   1, 256, 16384, 262144, 1048576, 1048576, 262144, 16384, 256, 1;

%t Table[2^((n-k)k),{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Jan 09 2017 *)

%o (PARI) T(n,k)=if(n<k || k<0,0,2^((n-k)*k))

%o (Magma)

%o A117401:= func< n, k, m | (m+2)^(k*(n-k)) >;

%o [A117401(n, k, 0): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 28 2021

%o (Sage)

%o def A117401(n, k, m): return (m+2)^(k*(n-k))

%o flatten([[A117401(n, k, 0) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 28 2021

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

%Y 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).

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Mar 12 2006