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 New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: 825 B. Generating function 11. Relations A. Recurrences B. Checks n m =O C (-1)"-mrn! s;p=1 n 2n-m k=O C. Numerical analysis - s(rn) Amf(z)=m! c + f ( n ) ( ~ ) if convergent n=m 111. Asymptotics and Special Values * lim rn-" S;rn)=(m!)-l n-) m rnZn S L Y m - a for n=o(m+) 24.2. Partitions 24.2.1 Unrestricted Partitions I. Definitions A. p(n) is the number of decompositions of n into integer summands without regard to order. E.~.,5=1+4=2+3=1+1+3=1+2+2=1+1+ 1 +2= I + 1 + 1 + 1 + 1 so that p(5)=7. 14<1 C. Closed form in-2, - where ((z))=2-[2]-3 if z is not an integer =O if z is an integer 11. Relations A. Recurrence =- l n c gl(k)P(n--k) n k=l B. Check 111. Asymptotics 1 rmdii 24.2.2 Partitions Into Distinct Parts I. Definitions A. q(n> is the number of decompositions of n into distinct integer summands without regard to order. E.g., 5=1+4=2+3 so that q(5)=3. B. Generating function m 0 m n=O c q(n)z"= n=l n (l+z")= n=l II (l-ZZn-1)-1 1Z1<1 C. Closed form where &(z) is the Bessel function of order 0 and A2k-l(n) was defined in part IL!. of the previous subsection. *See page 11. The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States.