cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A239928 Expansion of F(x^2, x) where F(x,y) is the g.f. of A239927.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 1, 1, 4, 3, 2, 5, 6, 4, 6, 10, 8, 9, 15, 15, 15, 22, 26, 26, 33, 43, 45, 52, 69, 76, 85, 109, 127, 141, 173, 209, 235, 278, 340, 390, 452, 550, 643, 742, 890, 1054, 1221, 1445, 1720, 2007, 2356, 2803, 3291, 3853, 4568, 5385, 6309, 7450, 8800, 10330, 12164, 14372, 16905, 19879
Offset: 0

Views

Author

Joerg Arndt, Mar 29 2014

Keywords

Comments

What does this sequence count?

Links

Crossrefs

Cf. A000108 (F(1, x)), A143951 (F(x, 1)), A005169 (F(x, x), with interlaced zeros), A227310 (F(x, x^2)).

Programs

  • PARI
    N=66; x='x+O('x^N);
F(x, y, d=0)=if (d>N, 1, 1 / (1-x*y * F(x, x^2*y, d+1) ) );
Vec( F(x^2, x) )

Formula

G.f.: 1/(1 - x^3/(1 - x^7/(1 - x^11/(1 - x^15/(1 - x^19/(1 - x^23/( ... ))))))).

A301832 G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x)/(1 - x^3*A(x)^3/(1 - x^5*A(x)^5/(1 - x^7*A(x)^7/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 5, 15, 49, 168, 595, 2160, 7998, 30095, 114751, 442402, 1721636, 6753869, 26680262, 106042264, 423750562, 1701476738, 6861334966, 27776206851, 112839216109, 459867381701, 1879624039171, 7703187691979, 31647457638073, 130314986803631, 537730217342715, 2223228743506792
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 27 2018

Keywords

Examples

			G.f. A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 49*x^5 + 168*x^6 + 595*x^7 + 2160*x^8 + 7998*x^9 + 30095*x^10 + ...

Links

Crossrefs

Cf. A143951, A301627.

Formula

a(n) = [x^n] (Sum_{k>=0} A143951(k)*x^k)^(n+1)/(n + 1).
a(n) ~ c * d^n / n^(3/2), where d = 4.36034166192381738574769007441081546251391... and c = 0.42401561796424536417811444539653002307... - Vaclav Kotesovec, Nov 04 2021

A366920 a(n) is the number times a Dyck path in an m X m box of any size has area n, counted to the lower right.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 1, 3, 3, 3, 2, 2, 5, 6, 7, 7, 5, 6, 8, 12, 15, 18, 16, 16, 15, 17, 24, 32, 40, 43, 45, 45, 42, 44, 53, 69, 87, 104, 115, 126, 125, 124, 124, 136, 160, 198, 240, 282, 321, 345, 360, 365, 367, 382, 417, 482, 574, 682, 791, 895, 976
Offset: 0

Views

Author

William J. Keith, Oct 28 2023

Keywords

Comments

A Dyck path in an m X m grid is a set of up steps U and right steps R from the lower left corner to the upper right corner, staying weakly above the diagonal.
For this statistic, count the boxes below and to the right of the path.
The first time an area appears in two different squares is at size 15, which appears in the 4 X 4 box below UUURURRR and in the 5 X 5 box below URURURURUR.

Examples

			The 0 X 0 box yields the trivial (empty) path of area 0.
The 1 X 1 box yields one Dyck path of area 1 (UR).
The 2 X 2 box yields one Dyck path each of area 3 (URUR) and 4 (UURR).
The 3 X 3 box yields one Dyck path of area 6 (URURUR), two of area 7 (UURRUR and URUURR), and one each of area 8 (UURURR) and 9 (UUURRR).

Crossrefs

Cf. A143951, A227543.

Formula

G.f.: 1 + q + q^3 + q^4 + q^6 + 2q^7 + ...
To construct the g.f., take A(x,q) as defined in A227543, and replace each instance of x^k with q^(k*(k+1)/2).

Extensions

a(45)-a(65) from Alois P. Heinz, Oct 29 2023
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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