A002040

[N. J. A. Sloane]

nice work, Michael!  and thanks!

i corrected A002040 as follows:


%I A002040 M1159 N0442
%S A002040 1,2,4,8,21,52,131,316,765,1846,4494,10944,26654,64798,157502,
%T A002040 382868,931028,2264106,5505777,13387880,32553601,79156974,192479838,
%U A002040 468039888,1138098210,2767421826,6729311459,16363118556,39788886610
%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 = -1/F'(x), F(x) = Product (1-x^k), k=1..inf.
%Y A002040 Cf. A002039.
%E A002040 Formula corrected and sequence extended by Michael Somos (somos at grail.cba.csuohio.edu)

(i had already fixed A002039, but didn't see how to fix A002040)

incidentally F'/F gives A000203:


%I A002039 M2465 N0979
%S A002039 1,3,5,10,25,64,160,390,940,2270,5515,13440,32735,79610,193480,
%T A002039 470306,1143585,2781070,6762990,16445100,39987325,97232450,236432060,
%U A002039 574915770,1397981470,3399360474,8265943685,20099618590,48874630750
%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.
%F A002039 G.f.: Sum (-1)^n*a(n)*x^n = -F(x)/F'(x), F(x) = Product (1-x^k), k=1..inf.


%I A000203 M2329 N0921
%S A000203 1,3,4,7,6,12,8,15,13,18,12,28,14,24,24,31,18,39,20,42,32,36,24,60,31,42,
%T A000203 40,56,30,72,32,63,48,54,48,91,38,60,56,90,42,96,44,84,78,72,48,124,57,
%U A000203 93,72,98,54,120,72,120,80,90,60,168,62,96,104,127,84,144,68,126,96,144
%N A000203 sigma(n) = sum of divisors of n.
%D A000203 J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-1
67.
%D A000203 P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. S
oc., 19 (1919), 75-113.
%D A000203 G. Polya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ
 Press 1954, page 92.
%D A000203 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applie
d Math. Series 55, 1964 (and various reprintings), p. 840.
%K A000203 easy,core,nonn,nice
%O A000203 1,2
%A A000203 njas
%p A000203 with(numtheory): f:=n->sigma(n);
%F A000203 G.f.: -F'(x)/F(x), F(x) = Product (1-x^k), k=1..inf.

Neil Sloane