A119469 -id:A119469 - cp's OEIS frontend

A003113 Coefficients in expansion of permanent of infinite tridiagonal matrix shown below.

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

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

nonn

1 0 0 0 0 0 ...
1 x 0 0 0 0 0 ...
x 1 x^2 0 0 0 ...
0 x^2 1 x^3 0 0 ...
0 0 x^3 1 x^4 0 0 0 ...
...................

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).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.

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].

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 *)

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

More terms from Vladeta Jovovic, Aug 30 2001

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).

Original entry on oeis.org

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

N. J. A. Sloane, Apr 07 2007

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

			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;
   ...

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

Rows converge to A003114 (coefficients in expansion of the first Rogers-Ramanujan identities). Cf. A128915, A119469.

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;

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 *)

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 */

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).

Original entry on oeis.org

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

N. J. A. Sloane, Apr 24 2007

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

			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

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

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

Rows converge to A003106. Cf. A127836, A119469.

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;

A152881 Positions of those 1's that are followed by a 0, summed over all Fibonacci binary words of length n. A Fibonacci binary word is a binary word having no 00 subword.

Original entry on oeis.org

0, 1, 5, 15, 40, 95, 213, 455, 940, 1890, 3720, 7194, 13710, 25805, 48055, 88665, 162272, 294865, 532395, 955795, 1707110, 3034836, 5372400, 9473700, 16646700, 29155225, 50908793, 88644915, 153952120, 266726195, 461066385, 795320159

Offset: 1

Emeric Deutsch, Jan 04 2009

a(n) = Sum(k*A119469(n+1,k),k>=0).

For n>1, a(n-1) is the n-th antidiagonal sum of A213777. [Clark Kimberling, Jun 21 2012]

			a(4)=15 because the Fibonacci binary words of length 4 are 1110, 1111, 1101, 1010, 1011, 0110, 0111, 0101 and the positions of those 1's that are followed by a 0 are 3, 2, 1, 3, 1, 3 and 2; their sum is 15.

Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Cf. A119469.

G := z^2*(1+2*z)/(1-z-z^2)^3: Gser := series(G, z = 0, 38): seq(coeff(Gser, z, n), n = 1 .. 34);

G.f.: z^2*(1+2z)/(1-z-z^2)^3.

a(n) = A001628(n-1) + 2*A001628(n-2), n>1, a(0)=0, a(1)=1. [Vladimir Kruchinin, Apr 26 2011]

cp's OEIS Frontend

A003113 Coefficients in expansion of permanent of infinite tridiagonal matrix shown below.

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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).

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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).

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A152881 Positions of those 1's that are followed by a 0, summed over all Fibonacci binary words of length n. A Fibonacci binary word is a binary word having no 00 subword.

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