A103244 Unreduced numerators of the elements T(n,k)/(n-k)!, read by rows, of the triangular matrix P^-1, which is the inverse of the matrix defined by P(n,k) = (-k^2-k)^(n-k)/(n-k)! for n >= k >= 1.

1, 2, 1, 20, 6, 1, 512, 108, 12, 1, 25392, 4104, 336, 20, 1, 2093472, 273456, 17568, 800, 30, 1, 260555392, 28515456, 1500288, 54800, 1620, 42, 1, 45819233280, 4311418752, 191549952, 5808000, 140400, 2940, 56, 1, 10849051434240, 894918533760, 34352605440, 887256000, 18033840, 313992, 4928, 72, 1

Offset

1,2

Comments

Define triangular matrix P by P(n,k) = (-k^2-k)^(n-k)/(n-k)!, then M = P*D*P^-1 = A103238 satisfies: M^2 + M = SHIFTUP(M) where D is the diagonal matrix consisting of {1,2,3,...}. The operation SHIFTUP(M) shifts each column of M up 1 row.

First column is A103353.

Links

Table of n, a(n) for n=1..45.

Formula

For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(-m^2-m)^(n-m)*T(m, k).

For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(-k^2-k)^(j-k)*T(n, j).

Example

This triangle begins:
1;
2, 1;
20, 6, 1;
512, 108, 12, 1;
25392, 4104, 336, 20, 1;
2093472, 273456, 17568, 800, 30, 1;
260555392, 28515456, 1500288, 54800, 1620, 42, 1;
45819233280, 4311418752, 191549952, 5808000, 140400, 2940, 56, 1;
10849051434240, 894918533760, 34352605440, 887256000, 18033840, 313992, 4928, 72, 1; ...
Rows of unreduced fractions T(n,k)/(n-k)! begin:
[1/0!],
[2/1!, 1/0!],
[20/2!, 6/1!, 1/0!],
[512/3!, 108/2!, 12/1!, 1/0!],
[25392/4!, 4104/3!, 336/2!, 20/1!, 1/0!],
[2093472/5!, 273456/4!, 17568/3!, 800/2!, 30/1!, 1/0!],...
forming the inverse of matrix P where P(n,k) = (-1)^(n-k)*(k^2+k)^(n-k)/(n-k)!:
[1/0!],
[ -2/1!, 1/0!],
[4/2!, -6/1!, 1/0!],
[ -8/3!, 36/2!, -12/1!, 1/0!],
[16/4!, -216/3!, 144/2!, -20/1!, 1/0!], ...

Mathematica

nmax = 9;
P = Table[If[n >= k, (-k^2-k)^(n-k)/(n-k)!, 0], {n, 1, nmax}, {k, 1, nmax}] // Inverse;
T[n_, k_] := If[n < k || k < 1, 0, P[[n, k]]*(n - k)!];
Table[T[n, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 09 2018, from PARI *)

Prog

(PARI) {T(n, k)=local(P); if(n>=k&k>=1, P=matrix(n, n, r, c, if(r>=c, (-c^2-c)^(r-c)/(r-c)!))); return(if(n<k||k<1, 0, (P^-1)[n, k]*(n-k)!))}
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A261642, A103238, A103353.

Sequence in context: A012907 A317996 A066753 * A328921 A185169 A353914

Adjacent sequences: A103241 A103242 A103243 * A103245 A103246 A103247

Keyword

nonn,tabl,frac

Author

Paul D. Hanna, Feb 02 2005

Status

approved