A002040

[Michael Somos]

> I am attempting to extend A002040.

I have not been able to get the AMM article since the library has put many
math journals from before 1971 in storage. Here is the sequence entries:

============================================================================
%I A002039 M2465 N0979
%S A002039 1,3,5,10,25,64,160,390,940,2270,5515,13440,32735,79610,193480,470306,
           1143585,
%T A002039 2781070,6762990,16445100,39987325,97232450,236432060,574915770,
           1397981470
%N A002039 Related to partitions (g.f. is inverse to A000203).
%D A002039 J. M. Gandhi, On numbers related to partitions of a number, Amer.
           Math. Monthly, 76 (1969), 1033-1036.
%O A002039 0,2
%A A002039 njas, sp.
%K A002039 nonn,nice,easy
%Y A002039 Cf. A002040.
============================================================================
%I A002040 M1159 N0442
%S A002040 1,2,4,8,21,52,131,316,765,1846,4494
%N A002040 Related to partitions.
%D A002040 J. M. Gandhi, On numbers related to partitions of a number, Amer.
           Math. Monthly, 76 (1969), 1033-1036.
%O A002040 0,2
%A A002040 njas
%K A002040 nonn,easy,nice,more
%F A002040 G.f.: Sum (-1)^n*a(n)*x^n = -F(x)/F'(x), F(x) = Product (1-x^k), k=1
           ..inf.
%Y A002040 Cf. A002039.
============================================================================

However, it appears that the formulas need a bit of correction based on the
following (lightly edited to fold long lines) computation that works:

============================================================================
                   GP/PARI CALCULATOR Version 2.0.14 (alpha)
                i686 running linux (ix86 kernel) 32-bit version
                  (readline enabled, extended help available)

                           Copyright (C) 1989-1999 by
          C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier.

Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical) support.

   realprecision = 28 significant digits
   seriesprecision = 16 significant terms
   format = g0.28

parisize = 4000000, primelimit = 500000
gp> F=prod(k=1,32,1-x^k,1+O(x^33))
%1 = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + x^26 + O(x^33)
gp> Vec(subst(-F/F',x,-x))
%2 = [1, 3, 5, 10, 25, 64, 160, 390, 940, 2270, 5515, 13440, 32735, 79610,
      193480, 470306, 1143585, 2781070, 6762990, 16445100, 39987325, 97232450,
      236432060, 574915770, 1397981470, 3399360474, 8265943685, 20099618590,
      48874630750, 118844554220, 288984832976, 702701288580]
gp> Vec(subst(-1/F',x,-x))
%3 = [1, 2, 4, 8, 21, 52, 131, 316, 765, 1846, 4494, 10944, 26654, 64798,
      157502, 382868, 931028, 2264106, 5505777, 13387880, 32553601, 79156974,
      192479838, 468039888, 1138098210, 2767421826, 6729311459, 16363118556,
      39788886610, 96751470494, 235262832929, 572069790768]
gp> \q
Good bye!
> exit
============================================================================
Shalom, Michael

-- 
Michael Somos <somos at grail.cba.csuohio.edu>     Cleveland State University
http://grail.cba.csuohio.edu/~somos/            Cleveland, Ohio, USA 44115