A128915 Triangle read by rows: row n gives coefficients (lowest degree first) of P_n(x), where P_0(x) = P_1(x) = 1; P_n(x) = P_{n-1}(x) + x^n*P_{n-2}(x).

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 3, 3, 2, 2, 2, 1, 1

Offset

0,32

Comments

P_n(x) appears to have degree A035106(n).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..9523 (rows n=0..47 of triangle, flattened).

A. V. Sills, Finite Rogers-Ramanujan type identities, Electron. J. Combin. 10 (2003), Research Paper 13, 122 pp. See Identity 3-14, p. 25.

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Example

Triangle begins:
1
1
1,0,1
1,0,1,1
1,0,1,1,1,0,1
1,0,1,1,1,1,1,1,1
1,0,1,1,1,1,2,1,2,1,1,0,1
1,0,1,1,1,1,2,2,2,2,2,1,2,1,1,1
1,0,1,1,1,1,2,2,3,2,3,2,3,2,3,2,2,1,1,0,1

Maple

P[0]:=1; P[1]:=1; d:=[0, 0]; M:=14; for n from 2 to M do P[n]:=expand(P[n-1]+q^n*P[n-2]);
lprint(seriestolist(series(P[n], q, M^2))); d:=[op(d), degree(P[n], q)]; od: d;

Crossrefs

Rows converge to A003114 (coefficients in expansion of the first Rogers-Ramanujan identities). Cf. A119469.

Rows converge to A003106. Cf. A127836, A119469.

Sequence in context: A037888 A052308 A116510 * A063995 A280737 A322305

Adjacent sequences: A128912 A128913 A128914 * A128916 A128917 A128918

Keyword

nonn,tabf

Author

N. J. A. Sloane, Apr 24 2007

Status

approved