A118188 -id:A118188 - cp's OEIS frontend

A118185 Triangle T(n,k) = 4^(k*(n-k)) for n>=k>=0, read by rows.

1, 1, 1, 1, 4, 1, 1, 16, 16, 1, 1, 64, 256, 64, 1, 1, 256, 4096, 4096, 256, 1, 1, 1024, 65536, 262144, 65536, 1024, 1, 1, 4096, 1048576, 16777216, 16777216, 1048576, 4096, 1, 1, 16384, 16777216, 1073741824, 4294967296, 1073741824, 16777216, 16384, 1

Offset: 0

Paul D. Hanna, Apr 15 2006

For any column vector C, the matrix product of T*C transforms the g.f. of C: Sum_{n>=0} c(n)*x^n into the g.f.: Sum_{n>=0} c(n)*x^n/(1-4^n*x).

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. For example, the matrix inverse is defined by [T^-1](n,k) = A118188(n-k)*T(n,k); also, the matrix log is given by [log(T)](n,k) = A118189(n-k)*T(n,k).

			A(x,y) = 1/(1-xy) + x/(1-4xy) + x^2/(1-16xy) + x^3/(1-64xy) + ...
Triangle begins:
  1;
  1,    1;
  1,    4,       1;
  1,   16,      16,        1;
  1,   64,     256,       64,        1;
  1,  256,    4096,     4096,      256,       1;
  1, 1024,   65536,   262144,    65536,    1024,    1;
  1, 4096, 1048576, 16777216, 16777216, 1048576, 4096, 1; ...
The matrix inverse T^-1 starts:
        1;
       -1,      1;
        3,     -4,       1;
      -33,     48,     -16,     1;
     1407,  -2112,     768,   -64,    1;
  -237057, 360192, -135168, 12288, -256, 1; ...
where [T^-1](n,k) = A118188(n-k)*4^(k*(n-k)).

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A118186 (row sums), A118187 (antidiagonal sums), A118188, A118189.
Cf. A117401 (m=0), A118180 (m=1), this sequence (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15).
T(2n,n) gives A060757.

[4^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 29 2021

Table[4^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 29 2021 *)

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

flatten([[4^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 29 2021

G.f.: A(x,y) = Sum_{n>=0} x^n/(1-4^n*x*y).
G.f. satisfies: A(x,y) = 1/(1-x*y) + x*A(x,4*y).
T(n,k) = (1/n)*( 4^(n-k)*k*T(n-1,k-1) + 4^k*(n-k)*T(n-1,k) ), where T(i,j)=0 if j>i. - Tom Edgar, Feb 20 2014
T(n, k, m) = (m+2)^(k*(n-k)) with m = 2. - G. C. Greubel, Jun 29 2021

A118189 Column 0 of the matrix log of triangle A118185, after term in row n is multiplied by n: a(n) = n[log(A118185)](n,0), where A118185(n,k) = 4^(k(n-k)).

Original entry on oeis.org

0, 1, -2, 19, -764, 125701, -83499002, 222705979399, -2379643407695864, 101770765968904486921, -17414214749792087566712822, 11920352399707142353576549941259, -32640155138015817553201240150152052724, 357505372216293786145503061380504161718632461

Offset: 0

Paul D. Hanna, Apr 15 2006

sign

The entire matrix log of triangle A118185 is determined by column 0 (this sequence): [log(A118185)](n,k) = a(n-k)*4^(k*(n-k))/(n-k) for n > k >= 0.

			Column 0 of log(A118185) = [0, 1, -2/2, 19/3, -764/4, 125701/5, ...].
The g.f. is illustrated by:
x/(1-x)^2 = x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + ...
  = x/(1-4*x) - 2*x^2/(1-16*x) + 19*x^3/(1-64*x) - 764*x^4/(1-256*x) + 125701*x^5/(1-1024*x) - 83499002*x^6/(1-4096*x) + 222705979399*x^7/(1-16384*x) + ...
From _Paul D. Hanna_, Oct 14 2009: (Start)
Illustrate the logarithmic g.f. by:
L(x) = x/2^1 - 2*x^2/(2*2^4) + 19*x^3/(3*2^9) - 764*x^4/(4*2^16) +- ...
where exp(L(x)) = 1 + x/2^1 + x^2/2^4 + x^3/2^9 + x^4/2^16 + ... (End)

G. C. Greubel, Table of n, a(n) for n = 0..55

Cf. A118185 (triangle), A118188.

A118188[n_]:= A118188[n]= If[n<2, (-1)^n, -Sum[4^(j*(n-j))*A118188[j], {j,0,n-1}]];
a[n_]:= a[n]= -Sum[4^(j*(n-j))*j*A118188[j], {j, 0, n}];
Table[a[n], {n, 0, 15}] (* G. C. Greubel, Jun 29 2021 *)

{a(n)=local(T=matrix(n+1,n+1,r,c,if(r>=c,(4^(c-1))^(r-c))), L=sum(m=1,#T,-(T^0-T)^m/m));return(n*L[n+1,1])}

{a(n)=n*2^(n^2)*polcoeff(log(sum(m=0,n,x^m/2^(m^2))+x*O(x^n)),n)} \\ Paul D. Hanna, Oct 14 2009

@CachedFunction
def A118188(n): return (-1)^n if (n<2) else -sum(4^(j*(n-j))*A118188(j) for j in (0..n-1))
def a(n): return (-1)*sum(4^(j*(n-j))*j*A118188(j) for j in (0..n))
[a(n) for n in (0..30)] # G. C. Greubel, Jun 29 2021

G.f.: x/(1-x)^2 = Sum_{n>=0} a(n)*x^n/(1-4^n*x).
By using the inverse transformation: a(n) = Sum_{k=0..n} k*A118188(n-k)*4^(k*(n-k)) for n>=0.
a(2^n) is divisible by 2^n.
L.g.f.: Sum_{n>=1} a(n)*x^n/[n*2^(n^2)] = log( Sum_{n>=0} x^n/2^(n^2) ). - Paul D. Hanna, Oct 14 2009

cp's OEIS Frontend

A118185 Triangle T(n,k) = 4^(k*(n-k)) for n>=k>=0, read by rows.

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A118189 Column 0 of the matrix log of triangle A118185, after term in row n is multiplied by n: a(n) = n[log(A118185)](n,0), where A118185(n,k) = 4^(k(n-k)).

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cp's OEIS Frontend

A118185 Triangle T(n,k) = 4^(k*(n-k)) for n>=k>=0, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A118189 Column 0 of the matrix log of triangle A118185, after term in row n is multiplied by n: a(n) = n*[log(A118185)](n,0), where A118185(n,k) = 4^(k*(n-k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Sage

Formula

A118189 Column 0 of the matrix log of triangle A118185, after term in row n is multiplied by n: a(n) = n[log(A118185)](n,0), where A118185(n,k) = 4^(k(n-k)).