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

Author

N. J. A. Sloane, Apr 24 2007

Keywords

• nonn
• tabf

Comments

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

Examples

			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
		

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

Crossrefs

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

Rows converge to A003106. Cf. A127836, A119469.

Programs

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;