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.

Showing 1-6 of 6 results.

A377670 Number of subwords of the form UDD in nondecreasing Dyck paths of length 2n.

Original entry on oeis.org

0, 0, 1, 4, 14, 45, 138, 411, 1200, 3454, 9836, 27779, 77938, 217493, 604222, 1672246, 4613030, 12689265, 34817418, 95320335, 260436588, 710278318, 1933906496, 5257545599, 14273273314, 38699274665, 104799960058, 283487736166, 766045036730, 2067997219629, 5577597593466, 15030365074659, 40470488092008
Offset: 0

Views

Author

Rigoberto Florez, Nov 03 2024

Keywords

Comments

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.
a(n) also represents the number of subwords of the form UUDDD in nondecreasing Dyck paths of length 2n.

Links

Crossrefs

Cf. A000032, A000045, A377679, A375995.

Programs

  • Mathematica
    Table[If[n<2, 0,(2*Fibonacci[2n-3] + n*LucasL[2n-3]+2 LucasL[2n-2]-5*2^(n-2))/5], {n,0,20}]

Formula

a(n) = (2*F(2*n-3) + n*L(2*n-3) + 2*L(2*n-2) - 5*2^(n-2))/5 for n>=2, where F(n) = A000045(n) and L(n) = A000032(n).
G.f.: x^2*(1-x)*(x^3-2*x^2+3*x-1)/((2*x-1)*(x^2-3*x+1)^2). - Alois P. Heinz, Nov 03 2024
E.g.f.: (4*exp(3*x/2)*(5*(10 - x)*cosh(sqrt(5)*x/2) - sqrt(5)*(18 - 5*x)*sinh(sqrt(5)*x/2)) - 25*(7 + exp(2*x) + 2*x))/100. - Stefano Spezia, Mar 04 2025

A377679 Number of subwords of the form DDD in nondecreasing Dyck paths of length 2n.

Original entry on oeis.org

0, 0, 0, 1, 6, 26, 97, 333, 1085, 3411, 10448, 31376, 92773, 270907, 783003, 2243815, 6383550, 18048494, 50755897, 142067625, 396014681, 1099863867, 3044737100, 8404071596, 23135752141, 63538808311, 174120317367, 476207551183
Offset: 0

Views

Author

Rigoberto Florez, Nov 03 2024

Keywords

Comments

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

Links

Crossrefs

Cf. A000032, A000045, A377670, A375995.

Programs

  • Mathematica
    Table[If[n<2,0,n Fibonacci[2 n-3]-LucasL[2 n-2]+2^(n-2)],{n,0,30}]

Formula

a(n) = n*F(2*n-3) - L(2*n-2) + 2^(n-2) for n>=2, where F(n) = A000045(n) and L(n) = A000032(n).
G.f.: x^3*(1 - 2*x + x^2 - x^3)/((1 - 2*x)*(1 - 3*x + x^2)^2).

A377857 Number of subwords of the form UUUD in nondecreasing Dyck paths of length 2n.

Original entry on oeis.org

0, 0, 0, 1, 5, 18, 60, 191, 589, 1775, 5257, 15360, 44394, 127171, 361595, 1021693, 2871245, 8031246, 22372344, 62096135, 171797257, 473928875, 1304007889, 3579517116, 9804791910, 26804181643, 73145473655, 199276078201, 542076556949, 1472491141770, 3994615719732
Offset: 0

Views

Author

Rigoberto Florez, Nov 09 2024

Keywords

Comments

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

Links

Crossrefs

Cf. A000032, A000045, A377679, A377670, A375995.

Programs

  • Mathematica
    Table[If[n<3,0,n Fibonacci[2n-5]-LucasL[2n-6]], {n,0,30}]

Formula

a(n) = n*F(2*n-5) - L(2*n-6) for n>=3, where F(n) = A000045(n) and L(n) = A000032(n).
G.f.: x^3*(1 - x)^2*(1 + x)/(1 - 3*x + x^2)^2.
a(n) = A317408(n-2)-A317408(n-3) = A030267(n-2)+A030267(n-3). - R. J. Mathar, Dec 16 2024

A377866 Number of subwords of the form DUUD or DDUUD in nondecreasing Dyck paths of length 2n.

Original entry on oeis.org

0, 0, 0, 1, 5, 18, 59, 185, 564, 1685, 4957, 14406, 41455, 118321, 335400, 945193, 2650229, 7398330, 20573219, 57013865, 157517532, 433993661, 1192779085, 3270835566, 8950887895, 24448816993, 66665369424, 181489721425, 493361278949
Offset: 0

Views

Author

Rigoberto Florez, Nov 10 2024

Keywords

Comments

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

Links

Crossrefs

Cf. A000032, A000045, A377679, A377670, A375995.

Programs

  • Mathematica
    Table[If[n<3,0,(2*n*LucasL[2*n-5]-6*Fibonacci[2*n-6]-Fibonacci[2*n-7])/5], {n,0,20}]

Formula

a(n) = (2*n*L(2*n-5) - 6*F(2*n-6) - F(2*n-7))/5 for n>=3, where F(n)=A000045(n) and L(n)=A000032(n).
G.f.: -x^3*(x^2+x-1)/ (x^2-3*x+1)^2.
E.g.f.: exp(3*x/2)*(5*(35 - 8x)*cosh(sqrt(5)*x/2) - sqrt(5)*(79 - 20*x)*sinh(sqrt(5)*x/2))/25 - 7 - x. - Stefano Spezia, Nov 10 2024

A377867 Number of subwords of the form DDDD in nondecreasing Dyck paths of length 2n.

Original entry on oeis.org

0, 0, 0, 0, 1, 7, 33, 131, 473, 1608, 5242, 16567, 51123, 154793, 461525, 1358646, 3957088, 11420995, 32707809, 93040751, 263113505, 740238852, 2073098086, 5782387855, 16070206191, 44516728277, 122956408493, 338707969266, 930787894348, 2552224341403, 6984100641117
Offset: 0

Views

Author

Rigoberto Florez, Nov 10 2024

Keywords

Comments

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

Links

Crossrefs

Cf. A000032, A000045, A377679, A377670, A375995.

Programs

  • Mathematica
    Table[If[n < 3, 0, (3*(n-2)*LucasL[2*n-4]-3*Fibonacci[2*n+1])/5+(n+9)*2^(n-4)], {n,0,20}]

Formula

a(n) = (3*(n-2)*L(2*n-4) - 3*F(2*n+1))/5 + (n+9)*2^(n-4) for n>=3, where F(n) = A000045(n) and L(n) = A000032(n).
G.f.: x^4*(1 - 3*x + 2*x^2 + x^4)/((1 - 2*x)^2*(1 - 3*x + x^2)^2).

A378383 Number of subwords of the form UDDD in nondecreasing Dyck paths of length 2*n.

Original entry on oeis.org

0, 0, 0, 1, 5, 19, 64, 202, 612, 1803, 5206, 14809, 41650, 116114, 321478, 885169, 2426462, 6627499, 18048088, 49026874, 132901176, 359625015, 971639014, 2621683741, 7065545950, 19022080034, 51163908874, 137499581917, 369235213742, 990822728623, 2657069356996
Offset: 0

Views

Author

Rigoberto Florez, Nov 24 2024

Keywords

Comments

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

Links

Crossrefs

Cf. A000032, A000045, A377679, A377670, A375995, A377857, A377866, A377867.

Programs

  • Mathematica
    Table[If[n < 3, 0, (1/5)((n-3)LucasL[2n-5]+LucasL[2n-3]+Fibonacci[2n+2]-5(n+5) 2^(n-4))], {n,0,26}]

Formula

a(n) =((n-3)*L(2n-5)+L(2n-3)+F(2n+2) -5*(n+5)*2^(n-4))/5 for n>=3, where F(n) = A000045(n) and L(n) = A000032(n).
G.f.: (-x^5+2 x^4-5 x^3+8 x^2-5 x+1)*x^3/(2 x^3-7 x^2+5 x-1)^2.
Showing 1-6 of 6 results.

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