A003113 - OEIS

%I M0270 #36 Oct 08 2023 11:33:05

%S 2,1,2,2,3,3,5,5,7,8,10,11,15,16,20,23,28,31,38,42,51,57,67,75,89,99,

%T 115,129,149,166,192,213,244,272,309,344,391,433,489,543,611,676,760,

%U 839,939,1038,1157,1276,1422,1565,1738,1913,2119,2328,2576,2826,3120

%N Coefficients in expansion of permanent of infinite tridiagonal matrix shown below.

%C 1 1 0 0 0 0 0 ...

%C 1 1 x 0 0 0 0 0 ...

%C 0 x 1 x^2 0 0 0 ...

%C 0 0 x^2 1 x^3 0 0 ...

%C 0 0 0 x^3 1 x^4 0 0 0 ...

%C ...................

%D D. H. Lehmer, Course on History of Mathematics, Univ. Calif. Berkeley, 1973.

%D H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vaclav Kotesovec, <a href="/A003113/b003113.txt">Table of n, a(n) for n = 0..1000</a>

%H Herman P. Robinson, <a href="/A003105/a003105.pdf">Letter to N. J. A. Sloane, Jan 1974</a>.

%F G.f.: 1 + sum(i>=1, x^(i*(i-1))/prod(j=1..i, 1-x^j)) - _Jon Perry_, Jul 04 2004

%F a(n) = A003114(n)+A003106(n). So this is the sum of the two famous Rogers-Ramanujan series. - _Vladeta Jovovic_, Jul 17 2004

%F G.f.: sum(n>=0,(q^(n^2)*(1+q^n)) / prod(k=1..n,1-q^k)). [_Joerg Arndt_, Oct 08 2012]

%F a(n) ~ (9+4*sqrt(5))^(1/4) * exp(2*Pi*sqrt(n/15)) / (2*3^(1/4)*sqrt(5)*n^(3/4)). - _Vaclav Kotesovec_, Jan 02 2016

%t nmax = 60; CoefficientList[1 + Series[Sum[x^(j*(j-1))/Product[1 - x^i, {i, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 02 2016 *)

%Y Cf. A119469, A003114, A003106.

%Y The generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] are A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. The present sequence, which is G[1]+G[2], plays the role of G[0].

%K nonn

%O 0,1

%A _N. J. A. Sloane_, _Herman P. Robinson_

%E More terms from _Vladeta Jovovic_, Aug 30 2001