A127836 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-1)*P_{n-2}(x).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 4, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 6, 6, 6, 5, 5

Offset: 0

Author

N. J. A. Sloane, Apr 07 2007

Keywords

• nonn
• tabf

Comments

P_n(x) has degree A002620(n).

Row sums are the Fibonacci numbers (A000045). - Emeric Deutsch, May 12 2007

T(n,k) is the number of Fibonacci words of length n-1 in which the sum of the positions of the 0's is equal to k. A Fibonacci binary word is a binary word having no 00 subword. Examples: T(5,4) = 2 because we have 1110 and 0101; T(7,6) = 3 because we have 111110, 101011 and 011101. - Emeric Deutsch, Jan 04 2009

Examples

			Triangle begins:
   1;
   1;
   1, 1;
   1, 1, 1;
   1, 1, 1, 1, 1;
   1, 1, 1, 1, 2, 1, 1;
   1, 1, 1, 1, 2, 2, 2, 1, 1, 1;
   1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1;
   1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 1, 1, 1;
   1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 4, 4, 3, 3, 2, 2, 1, 1;
   ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..9548 (rows n=0..48 of triangle, flattened).
• M. Barnabei, F. Bonetti, S. Elizalde, M. Silimbani, Descent sets on 321-avoiding involutions and hook decompositions of partitions, arXiv preprint arXiv:1401.3011 [math.CO], 2014.
• A. V. Sills, Finite Rogers-Ramanujan type identities, Electron. J. Combin. 10 (2003), Research Paper 13, 122 pp. See Identity 3-18, pp. 26-27.
• 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. A128915, 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-1)*P[n-2]);
lprint(seriestolist(series(P[n],q,M^2))); d:=[op(d),degree(P[n],q)]; od: d;

Mathematica

P[0] = P[1] = 1; P[n_] := P[n] = P[n-1] + x^(n-1) P[n-2];
Table[CoefficientList[P[n], x], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jul 23 2018 *)

Maxima

P(n, x) := if n = 0 or n = 1 then 1 else P(n - 1, x) + x^(n  - 1)*P(n - 2, x)$ create_list(ratcoef(expand(P(n, x)), x, k), n, 0, 10, k, 0, floor(n^2/4)); /* Franck Maminirina Ramaharo, Nov 30 2018 */