A117403 -id:A117403 - cp's OEIS frontend

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

Paul D. Hanna, Mar 12 2006

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.

			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;

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.

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).

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

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

PARI
```
T(n,k)=if(n
				
```

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

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

A117402 Row sums of triangle A117401: a(n) = Sum_{k=0..n} 2^((n-k)*k) for n>=0.

Original entry on oeis.org

1, 2, 4, 10, 34, 162, 1090, 10370, 139522, 2654722, 71435266, 2718435330, 146299424770, 11134711111682, 1198484887715842, 182431106853797890, 39271952800672710658, 11955805018770498519042, 5147453397489773531365378

Offset: 0

Paul D. Hanna, Mar 12 2006

nonn

a(n) is the number of 2-colored labeled graphs (as in A047863) such that the black nodes are labeled with {1,2,...,k} where k, 0<=k<=n, is the number of black nodes and the white nodes are labeled with {k+1,k+2,...,n}. These graphs form the desired binomial poset (for the case q=2) in the "task left to the reader" in the Stanley reference below. - Geoffrey Critzer, May 31 2020

			A(x) = 1/(1-x) + x/(1-2x) + x^2/(1-4x) + x^3/(1-8x) + ...

R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, Cambridge, 2012, Example 3.18.3 e, page 323.

Robert Israel, Table of n, a(n) for n = 0..116

Cf. A117401 (triangle), A117403 (antidiagonal sums).

[(&+[2^(k*(n-k)): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jun 28 2021

N:= 25:
G:= series(add(x^n/(1-2^n*x),n=0..N),x,N+1):
seq(coeff(G,x,n),n=0..N)); # Robert Israel, Dec 11 2018

a[n_]:= Sum[2^((n-k)*k), {k,0,n}]; Array[a, 20, 0] (* Amiram Eldar, Dec 12 2018 *)

PARI
```
a(n)=sum(k=0,n,2^((n-k)*k))
```

[sum(2^(k*(n-k)) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Jun 28 2021

G.f.: A(x) = Sum_{n>=0} x^n/(1-2^n*x).
Let E(x) = Sum_{n>=0} x^n/2^C(n,2). Then E(x)^2 = Sum_{n>=0} a(n)*x^n/2^C(n,2). - Geoffrey Critzer, May 31 2020
a(n) ~ c * 2^(n^2/4), where c = EllipticTheta[3, 0, 1/2] = JacobiTheta3(0,1/2) = 2.128936827211877158669458548544951324612516539940878092889... if n is even and c = EllipticTheta[2, 0, 1/2] = JacobiTheta2(0,1/2) = 2.128931250513027558591613402575350180853805396958448940969... if n is odd. - Vaclav Kotesovec, Jun 28 2021

cp's OEIS Frontend

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

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