A274477 - OEIS

%I #7 Jul 01 2016 17:35:36

%S 0,1,2,2,-64,424,100224,-14252064,-2465516544,5349979645056,

%T -2284492223508480,-32535188427388377600,248972687504267095941120,

%U 2418389754391936927997061120,-246866186803082697567984052961280,4557699858167315245689789135670272000,3413580835595898531780379863867877923225600,-1141255428747144951607112250069973499037619814400,-531525888535995992527627827436464215788606797801062400

%N Column 0 of the matrix logarithm of triangular matrix A134049.

%C This sequence forms the coefficients in column 0 of the matrix logarithm L of triangular matrix A134049, where L[n,k] = L[n-k,0] * 2^((n-k+1)*k).

%C Triangular matrix T = A134049 obeys T(n,k) = [T^(2^k)](n-k,0) * 2^((n-k)*k) for n>=k>=0.

%H Paul D. Hanna, <a href="/A274477/b274477.txt">Table of n, a(n) for n = 0..51</a>

%e E.g.f.: A(x) = x + 2*x^2/2! + 2*x^3/3! - 64*x^4/4! + 424*x^5/5! + 100224*x^6/6! - 14252064*x^7/7! - 2465516544*x^8/8! + 5349979645056*x^9/9! - 2284492223508480*x^10/10! - 32535188427388377600*x^11/11! + 248972687504267095941120*x^12/12! +...

%e AS COEFFICIENTS IN MATRIX LOG.

%e Let L denote the matrix logarithm of triangular matrix A134049, such that exp(L) = A134049, then L begins:

%e 0;

%e 1, 0;

%e 2/2!, 2^2, 0;

%e 2/3!, 2*2^3/2!, 2^4, 0;

%e -64/4!, 2*2^4/3!, 2*2^6/2!, 2^6, 0;

%e 424/5!, -64*2^5/4!, 2*2^8/3!, 2*2^9/2!, 2^8, 0;

%e 100224/6!, 424*2^6/5!, -64*2^10/4!, 2*2^12/3!, 2*2^12/2!, 2^10, 0;

%e -14252064/7!, 100224*2^7/6!, 424*2^12/5!, -64*2^15/4!, 2*2^16/3!, 2*2^15/2!, 2^12, 0;

%e -2465516544/8!, -14252064*2^8/7!, 100224*2^14/6!, 424*2^18/5!, -64*2^20/4!, 2*2^20/3!, 2*2^18/2!, 2^14, 0;

%e 5349979645056/9!, -2465516544*2^9/8!, -14252064*2^16/7!, 100224*2^21/6!, 424*2^24/5!, -64*2^25/4!, 2*2^24/3!, 2*2^21/2!, 2^16, 0; ...

%e in which L[n,k] = L[n-k,0] * 2^((n-k+1)*k) for n>=0, k=0..n.

%e Triangular matrix A134049 begins:

%e 1;

%e 1, 1;

%e 3, 4, 1;

%e 23, 40, 16, 1;

%e 512, 1072, 576, 64, 1;

%e 34939, 84736, 56064, 8704, 256, 1;

%e 7637688, 20930240, 16261120, 3190784, 135168, 1024, 1; ...

%o (PARI) /* Print as column 0 of triangle A134049 */

%o {LOGT(n, k)=local(M=Mat(1), L, R);

%o for(i=1, n,

%o L=sum(j=1, #M, -(M^0 - M)^j/j);

%o M=sum(j=0, #L, (L/2^(#L-1))^j/j!);

%o R=matrix(#M+1, #M+1, r, c, if(r>=c, if(r<=#M, M[r, c], 2^((c-1)*(#M+1-c)))));

%o M=R^(2^(#M-1)) );

%o L=sum(j=1, #M, -(M^0 - M)^j/j);

%o L[n+1, k+1]}

%o for(n=0, 20, print1(LOGT(n, 0)*n!, ", "));

%Y Cf. A134049.

%K sign

%O 0,3

%A _Paul D. Hanna_, Jul 01 2016