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A188797
Square array read by antidiagonals: T(m,n) (m>=0, n>=0) are the coefficients in an expansion of the Weierstrass sigma-function.
5
1, -1, -3, -9, -18, -54, 69, 513, 4968, 14904, 321, 33588, 257580, 502200, 1506600, 160839, 2808945, 20019960, 162100440, 796330440, 2388991320, 1416951, -41843142, -376375410, -9465715080, -144916218720, -1289959784640, -3869879353920, -388946691, -6519779667
OFFSET
0,3
COMMENTS
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
REFERENCES
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.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. See Table on p. 637.
Eric Weisstein's World of Mathematics, Weierstrass Sigma Function
FORMULA
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)].
EXAMPLE
Array begins:
1 -3 -54 14904 ...
-1 -18 4968 502200 ...
-9 513 257580 162100440 ...
69 33588 20019960 -9465715080 ...
321 2808945 -376375410 -4582619446320 ...
160839 -41843142 -210469286736 -1028311276281264 ...
...
CROSSREFS
First two rows and columns are in A188798, A188799, A188800, A188801.
Sequence in context: A363534 A027101 A015656 * A056372 A181574 A101652
KEYWORD
sign,tabl,easy
AUTHOR
N. J. A. Sloane, Apr 10 2011
EXTENSIONS
a(14)-a(29) from Nathaniel Johnston, Apr 11 2011
STATUS
approved