A118193 -id:A118193 - cp's OEIS frontend

A118190 Triangle T(n,k) = 5^(k*(n-k)) for n >= k >= 0, read by rows.

1, 1, 1, 1, 5, 1, 1, 25, 25, 1, 1, 125, 625, 125, 1, 1, 625, 15625, 15625, 625, 1, 1, 3125, 390625, 1953125, 390625, 3125, 1, 1, 15625, 9765625, 244140625, 244140625, 9765625, 15625, 1, 1, 78125, 244140625, 30517578125, 152587890625, 30517578125, 244140625, 78125, 1

Offset: 0

Paul D. Hanna, Apr 15 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. For example, the matrix inverse is defined by [T^-1](n,k) = A118193(n-k)*T(n,k); also, the matrix log is given by [log(T)](n,k) = A118194(n-k)*T(n,k).

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/(1-5^n*x).

			A(x,y) = 1/(1-x*y) + x/(1-5*x*y) + x^2/(1-25*x*y) + x^3/(1-125*x*y) + ...
Triangle begins:
  1;
  1,     1;
  1,     5,       1;
  1,    25,      25,         1;
  1,   125,     625,       125,         1;
  1,   625,   15625,     15625,       625,       1;
  1,  3125,  390625,   1953125,    390625,    3125,     1;
  1, 15625, 9765625, 244140625, 244140625, 9765625, 15625, 1; ...
The matrix inverse T^-1 starts:
         1;
        -1,       1;
         4,      -5,        1;
       -76,     100,      -25,     1;
      7124,   -9500,     2500,  -125,    1;
  -3326876, 4452500, -1187500, 62500, -625, 1; ...
where [T^-1](n,k) = A118193(n-k)*(5^k)^(n-k).

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

Cf. A118191 (row sums), A118192 (antidiagonal sums), A118193, A118194.

Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), this sequence (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15).

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

With[{m=3}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 29 2021 *)

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

flatten([[5^(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/(1-5^n*x*y).
G.f. satisfies: A(x,y) = 1/(1-x*y) + x*A(x,5*y).
T(n,k) = (1/n)*( 5^(n-k)*k*T(n-1,k-1) + 5^k*(n-k)*T(n-1,k) ), where T(i,j)=0 if j>i. - Tom Edgar, Feb 21 2014
T(n, k, m) = (m+2)^(k*(n-k)) with m = 3. - G. C. Greubel, Jun 29 2021

A118194 Column 0 of the matrix log of triangle A118190, after term in row n is multiplied by n: a(n) = n[log(A118190)](n,0), where A118190(n,k) = 5^(k(n-k)).

Original entry on oeis.org

0, 1, -3, 53, -4871, 2262505, -5269940619, 61424345593757, -3580474937256484367, 1043606492389898678125009, -1520932783784930699920673828115, 11082945991224258678496051788222656261, -403804307486446123171767495567989349951171863

Offset: 0

Paul D. Hanna, Apr 15 2006

sign

The entire matrix log of triangle A118190 is determined by column 0 (this sequence): [log(A118190)](n,k) = a(n-k)5^(k*(n-k))/(n-k) for n>k>=0.

			Column 0 of log(A118190) = [0, 1, -3/2, 53/3, -4871/4, ...].
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 + 7*x^7 + ...
  = x/(1-5*x) -3*x^2/(1-25*x) +53*x^3/(1-125*x) -4871*x^4/(1-625*x) + 2262505*x^5/(1-3125*x) - 5269940619*x^6/(1-15625*x) + ...

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A118190.

A118193[n_]:= A118193[n]= If[n<2, (-1)^n, -Sum[5^(j*(n-j))*A118193[j], {j, 0, n-1}]];
a[n_]:= a[n]= -Sum[5^(j*(n-j))*j*A118193[j], {j, 0, n}];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jun 29 2021 *)

{a(n)=local(T=matrix(n+1,n+1,r,c,if(r>=c,(5^(c-1))^(r-c))), L=sum(m=1,#T,-(T^0-T)^m/m));return(n*L[n+1,1])}

@CachedFunction
def A118193(n): return (-1)^n if (n<2) else -sum(5^(j*(n-j))*A118193(j) for j in (0..n-1))
def a(n): return (-1)*sum(5^(j*(n-j))*j*A118193(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/(1-5^n*x). By using the inverse transformation: a(n) = Sum_{k=0..n} k*A118193(n-k)*5^(k*(n-k)) for n>=0.

cp's OEIS Frontend

A118190 Triangle T(n,k) = 5^(k*(n-k)) for n >= k >= 0, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A118194 Column 0 of the matrix log of triangle A118190, after term in row n is multiplied by n: a(n) = n[log(A118190)](n,0), where A118190(n,k) = 5^(k(n-k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

cp's OEIS Frontend

A118190 Triangle T(n,k) = 5^(k*(n-k)) for n >= k >= 0, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A118194 Column 0 of the matrix log of triangle A118190, after term in row n is multiplied by n: a(n) = n*[log(A118190)](n,0), where A118190(n,k) = 5^(k*(n-k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A118194 Column 0 of the matrix log of triangle A118190, after term in row n is multiplied by n: a(n) = n[log(A118190)](n,0), where A118190(n,k) = 5^(k(n-k)).