A101980 Matrix logarithm of A008459 (squared entries of Pascal's triangle), read by rows.

0, 1, 0, -1, 4, 0, 4, -9, 9, 0, -33, 64, -36, 16, 0, 456, -825, 400, -100, 25, 0, -9460, 16416, -7425, 1600, -225, 36, 0, 274800, -463540, 201096, -40425, 4900, -441, 49, 0, -10643745, 17587200, -7416640, 1430016, -161700, 12544, -784, 64, 0, 530052880, -862143345, 356140800, -66749760, 7239456

Offset

0,5

Comments

Column 0 (A101981) is essentially a signed offset version of A002190 and is related to Bessel functions. Row sums form A101982.

Links

Table of n, a(n) for n=0..49.

Formula

T(n, k) = A101981(n-k)*C(n, k)^2.

Example

Rows begin:
[0],
[1,0],
[ -1,4,0],
[4,-9,9,0],
[ -33,64,-36,16,0],
[456,-825,400,-100,25,0],
[ -9460,16416,-7425,1600,-225,36,0],
[274800,-463540,201096,-40425,4900,-441,49,0],
[ -10643745,17587200,-7416640,1430016,-161700,12544,-784,64,0],...
and equal the term-by-term product of column 0:
A101981 = {0,1,-1,4,-33,456,-9460,274800,-10643745,...}
with the rows of the squared Pascal's triangle (A008459):
[0],
[1*1^2, 0*1^2],
[ -1*1^2, 1*2^2, 0*1^2],
[4*1^2, -1*3^2, 1*3^2, 0*1^2],
[ -33*1^2, 4*4^2, -1*6^2, 1*4^2, 0*1^2],
[456*1^2, -33*5^2, 4*10^2, -1*10^2, 1*5^2, 0*1^2],...

Prog

(PARI) {T(n, k)=if(n<k||k<0, 0, sum(m=1, n, (-1)^(m-1)* (matrix(n+1, n+1, i, j, if(i>j, binomial(i-1, j-1)^2))^m/m)[n+1, k+1]))}

Crossrefs

Cf. A008459, A002190, A101981, A101982.

Sequence in context: A088374 A156732 A200341 * A209134 A281297 A058536

Adjacent sequences: A101977 A101978 A101979 * A101981 A101982 A101983

Keyword

sign,tabl

Author

Paul D. Hanna, Dec 23 2004

Status

approved