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Square array read by antidiagonals: T(m,n) (m>=0, n>=0) are the coefficients in an expansion of the Weierstrass sigma-function.
5

%I #27 Sep 24 2021 22:46:50

%S 1,-1,-3,-9,-18,-54,69,513,4968,14904,321,33588,257580,502200,1506600,

%T 160839,2808945,20019960,162100440,796330440,2388991320,1416951,

%U -41843142,-376375410,-9465715080,-144916218720,-1289959784640,-3869879353920,-388946691,-6519779667

%N Square array read by antidiagonals: T(m,n) (m>=0, n>=0) are the coefficients in an expansion of the Weierstrass sigma-function.

%C On pages 635 to 637 of the Handbook, T(m, n) is denoted by a_{m, n}. Equation 18.5.6 is sigma(z) = Sum_{m, n>=0} a_{m, n} (1/2 g_2)^m (2 g_3)^n * z^(4m+6n+1) / (4m+6n+1)!. - _Michael Somos_, Sep 24 2021

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 637.

%H Nathaniel Johnston, <a href="/A188797/b188797.txt">Table of n, a(n) for n = 0..1325</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. See Table on p. 637.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WeierstrassSigmaFunction.html">Weierstrass Sigma Function</a>

%F T(0,0)=1, T(m,n)=0 if either m or n is negative; otherwise T(m,n)=3*(m+1)*T(m+1,n-1)+(16/3)*(n+1)*T(m-2,n+1)-(1/3)*(2*m+3*n-1)*(4*m+6*n-1)*T(m-1,n). [Abramowitz-Stegun, Eq. (18.5.8)].

%e Array begins:

%e 1 -3 -54 14904 ...

%e -1 -18 4968 502200 ...

%e -9 513 257580 162100440 ...

%e 69 33588 20019960 -9465715080 ...

%e 321 2808945 -376375410 -4582619446320 ...

%e 160839 -41843142 -210469286736 -1028311276281264 ...

%e ...

%Y First two rows and columns are in A188798, A188799, A188800, A188801.

%K sign,tabl,easy

%O 0,3

%A _N. J. A. Sloane_, Apr 10 2011

%E a(14)-a(29) from _Nathaniel Johnston_, Apr 11 2011