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)).

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

Offset

0,3

Comments

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.

Links

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

Formula

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.

Example

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) + ...

Mathematica

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 *)

Prog

(PARI) {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])}
(Sage)
@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

Crossrefs

Cf. A118190.

Sequence in context: A300606 A301348 A296680 * A325725 A173357 A144537

Adjacent sequences: A118191 A118192 A118193 * A118195 A118196 A118197

Keyword

sign

Author

Paul D. Hanna, Apr 15 2006

Status

approved