A102054 Triangular matrix, read by rows, where T(n,k) = T(n-1,k) - [T^-1](n-1,k-1); also equals the matrix inverse of A060083 (Euler polynomials).

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 4, -2, 4, 1, 1, -13, 26, -10, 5, 1, 1, 142, -229, 116, -25, 6, 1, 1, -1931, 3181, -1567, 371, -49, 7, 1, 1, 36296, -59700, 29464, -6922, 952, -84, 8, 1, 1, -893273, 1469380, -725108, 170398, -23358, 2100, -132, 9, 1, 1, 27927346, -45938639, 22669816, -5327198, 730252, -65526, 4152

Offset

0,5

Comments

Column 1 forms A102055. Column 2 forms A102056.

Links

Table of n, a(n) for n=0..62.

Formula

T(n, k) = T(n-1, k) - A060083(n-1, k-1), for n>0, with T(0, 0)=1.

Example

T(5,3) = -10 = T(4,3) - A060083(4,2) = 4 - 14.
T(6,2) = -229 = T(5,2) - A060083(5,1) = 26 - 255.
Rows begin:
[1],
[1,1],
[1,2,1],
[1,1,3,1],
[1,4,-2,4,1],
[1,-13,26,-10,5,1],
[1,142,-229,116,-25,6,1],
[1,-1931,3181,-1567,371,-49,7,1],
[1,36296,-59700,29464,-6922,952,-84,8,1],...
The matrix inverse is equal to A060083:
[1],
[ -1,1],
[1,-2,1],
[ -3,5,-3,1],
[17,-28,14,-4,1],
[ -155,255,-126,30,-5,1],...

Prog

(PARI) {T(n, k)=local(M=matrix(n+1, n+1)); M[1, 1]=1; if(n>0, M[2, 1]=1; M[2, 2]=1); for(r=3, n+1, for(c=1, r, M[r, c]=if(c==1, M[r-1, 1], if(c==r, 1, M[r, c]=M[r-1, c]-((matrix(r-1, r-1, i, j, M[i, j]))^-1)[r-1, c-1])))); return(M[n+1, k+1])}

Crossrefs

Cf. A060083, A102055, A102056.

Sequence in context: A238710 A262905 A330794 * A111604 A101491 A276949

Adjacent sequences: A102051 A102052 A102053 * A102055 A102056 A102057

Keyword

sign,tabl

Author

Paul D. Hanna, Dec 28 2004

Status

approved