A003113 Coefficients in expansion of permanent of infinite tridiagonal matrix shown below.

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, 115, 129, 149, 166, 192, 213, 244, 272, 309, 344, 391, 433, 489, 543, 611, 676, 760, 839, 939, 1038, 1157, 1276, 1422, 1565, 1738, 1913, 2119, 2328, 2576, 2826, 3120

Offset: 0

Author

N. J. A. Sloane, Herman P. Robinson

Keywords

• nonn

Comments

1 1 0 0 0 0 0 ...

1 1 x 0 0 0 0 0 ...

0 x 1 x^2 0 0 0 ...

0 0 x^2 1 x^3 0 0 ...

0 0 0 x^3 1 x^4 0 0 0 ...

...................

References

• D. H. Lehmer, Course on History of Mathematics, Univ. Calif. Berkeley, 1973.
• H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.

Crossrefs

Cf. A119469, A003114, A003106.

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].

Programs

Mathematica

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 *)

Formula

G.f.: 1 + sum(i>=1, x^(i*(i-1))/prod(j=1..i, 1-x^j)) - Jon Perry, Jul 04 2004

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

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

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

Extensions

More terms from Vladeta Jovovic, Aug 30 2001