A003113 Coefficients in expansion of permanent of infinite tridiagonal matrix shown below. (Formerly M0270)

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

0,1

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.

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

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 *)

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].

Sequence in context: A161256 A161281 A226916 * A225502 A152227 A246583

Adjacent sequences: A003110 A003111 A003112 * A003114 A003115 A003116

Keyword

nonn

Author

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

Extensions

More terms from Vladeta Jovovic, Aug 30 2001

Status

approved