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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).

%I A128915 #18 Feb 16 2025 08:33:05
%S A128915 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,
%T A128915 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,
%U A128915 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
%N 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).
%C A128915 P_n(x) appears to have degree A035106(n).
%H A128915 Seiichi Manyama, <a href="/A128915/b128915.txt">Table of n, a(n) for n = 0..9523 (rows n=0..47 of triangle, flattened).</a>
%H A128915 A. V. Sills, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v10i1r13">Finite Rogers-Ramanujan type identities</a>, Electron. J. Combin. 10 (2003), Research Paper 13, 122 pp. See Identity 3-14, p. 25.
%H A128915 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%e A128915 Triangle begins:
%e A128915 1
%e A128915 1
%e A128915 1,0,1
%e A128915 1,0,1,1
%e A128915 1,0,1,1,1,0,1
%e A128915 1,0,1,1,1,1,1,1,1
%e A128915 1,0,1,1,1,1,2,1,2,1,1,0,1
%e A128915 1,0,1,1,1,1,2,2,2,2,2,1,2,1,1,1
%e A128915 1,0,1,1,1,1,2,2,3,2,3,2,3,2,3,2,2,1,1,0,1
%p A128915 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]);
%p A128915 lprint(seriestolist(series(P[n],q,M^2))); d:=[op(d),degree(P[n],q)]; od: d;
%Y A128915 Rows converge to A003114 (coefficients in expansion of the first Rogers-Ramanujan identities). Cf. A119469.
%Y A128915 Rows converge to A003106. Cf. A127836, A119469.
%K A128915 nonn,tabf
%O A128915 0,32
%A A128915 _N. J. A. Sloane_, Apr 24 2007