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.

A193489 Number of rooted trees with n nodes and omega-valency 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 7, 13, 55, 127, 435, 1110, 3458, 9261, 27460, 75305, 217619, 603618, 1720712, 4799095, 13580179, 37979149, 107045222, 299800569, 843242264, 2363693561, 6641675073, 18628827667, 52325428340
Offset: 1

Views

Author

N. J. A. Sloane, Jul 27 2011

Keywords

Comments

See A003120 for definitions.

Links

Crossrefs

Cf. A003120, A193487, A193489, A193490, A193491.

A193487 Number of rooted trees with n nodes and omega-valency 2.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 9, 29, 62, 181, 422, 1166, 2885, 7788, 19893, 53319, 138931, 372027, 982379, 2635898, 7026483, 18914903, 50781803, 137201419, 370442111, 1004512375, 2724867782, 7414635185, 20193165834, 55125632502
Offset: 1

Views

Author

N. J. A. Sloane, Jul 27 2011

Keywords

Comments

See A003120 for definitions.

Links

Crossrefs

Cf. A003120, A193488, A193489, A193490, A193491.

A193490 Number of rooted trees with n nodes and omega-valency 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 8, 15, 70, 165, 608, 1601, 5259, 14564, 45040, 127521, 381691, 1090130, 3202382, 9171574, 26642413, 76312777, 220134939, 629965221, 1808975528, 5170502225, 14801940226, 42257089390
Offset: 1

Views

Author

N. J. A. Sloane, Jul 27 2011

Keywords

Comments

See A003120 for definitions.

Links

Crossrefs

Cf. A003120, A193487, A193488, A193489, A193491.

A193491 Irregular triangle read by rows: row n gives numbers of rooted trees with n nodes (n >= 1) and omega-valency k (k >= 1).

Original entry on oeis.org

1, 1, 2, 3, 1, 7, 1, 1, 13, 5, 1, 1, 31, 9, 6, 1, 1, 66, 29, 11, 7, 1, 1, 159, 62, 42, 13, 8, 1, 1, 365, 181, 92, 55, 15, 9, 1, 1, 900, 422, 294, 127, 70, 17, 10, 1, 1, 2162, 1166, 720, 435, 165, 86, 19, 11, 1, 1, 5417, 2885, 2119, 1110, 608, 208, 104, 21, 12, 1, 1
Offset: 1

Views

Author

N. J. A. Sloane, Jul 27 2011

Keywords

Comments

See A003120 for definitions.

Examples

			Triangle begins:
     1
     1
     2
     3,    1
     7,    1,    1
    13,    5,    1,    1
    31,    9,    6,    1,   1
    66,   29,   11,    7,   1,   1
   159,   62,   42,   13,   8,   1,   1
   365,  181,   92,   55,  15,   9,   1,  1
   900,  422,  294,  127,  70,  17,  10,  1,  1
  2162, 1166,  720,  435, 165,  86,  19, 11,  1, 1
  5417, 2885, 2119, 1110, 608, 208, 104, 21, 12, 1, 1
  ...

Links

Crossrefs

Columns give A003120, A193487, A193488, A193489, A193490. Row sums give A000081.

A193488 Number of rooted trees with n nodes and omega-valency 3.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 6, 11, 42, 92, 294, 720, 2119, 5460, 15477, 40989, 114165, 306980, 848268, 2301794, 6341450, 17308926, 47659837, 130628590, 359924353, 989642184, 2730224431, 7526539869, 20795818601, 57457010219
Offset: 1

Views

Author

N. J. A. Sloane, Jul 27 2011

Keywords

Comments

See A003120 for definitions.

Links

Crossrefs

Cf. A003120, A193487, A193489, A193490, A193491.

A193530 Expansion of (1 - 2*x - 2*x^2 + 3*x^3 + x^5)/((1-x)*(1-2*x-x^2)*(1-2*x^2-x^4)).

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 31, 66, 159, 363, 876, 2065, 4985, 11915, 28765, 69156, 166957, 402373, 971414, 2343519, 5657755, 13654969, 32966011, 79577190, 192116331, 463786191, 1119678912, 2703086893, 6525829037, 15754607063, 38034986041, 91824246216, 221683340569, 535190123593, 1292063254826
Offset: 0

Views

Author

F. Chapoton and N. J. A. Sloane, Jul 29 2011

Keywords

Comments

This sequence was initially confused with A003120, but they are different sequences. The g.f. used here as the definition was found by Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009.

Links

Crossrefs

Cf. A000129, A001333, A002203, A135153.

Programs

  • Magma
    m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-2*x-2*x^2 +3*x^3+x^5)/((1-x)*(1-2*x-x^2)*(1-2*x^2-x^4)) )); // Vincenzo Librandi, Aug 28 2016
  • Maple
    f:=n->if n mod 2 = 0 then (1/4)*(A001333(n-2)+A001333((n-2)/2)+A001333((n-4)/2)+1) else (1/4)*(A001333(n-2)+A001333((n-1)/2)+A001333((n-3)/2)+1); fi; # produces the sequence with a different offset
  • Mathematica
    LinearRecurrence[{3,1,-7,3,-1,1,1}, {1,1,2,3,7,13,31}, 40] (* Vincenzo Librandi, Aug 28 2016 *)
Table[(2 +LucasL[n, 2] +2*(1+(-1)^n)*Fibonacci[(n+2)/2, 2] + 2*(1-(-1)^n)*Fibonacci[(n+1)/2, 2])/8, {n, 0, 40}] (* G. C. Greubel, May 21 2021 *)
  • Sage
    @CachedFunction
def Pell(n): return n if (n<2) else 2*Pell(n-1) + Pell(n-2)
def A193530(n): return (1 + Pell(n+1) - Pell(n) + (1 + (-1)^n)*Pell((n+2)/2) + (1-(-1)^n)*Pell((n+1)/2) )/4
[A193530(n) for n in (0..40)] # G. C. Greubel, May 21 2021

Formula

a(n) = 1 + A005409(floor((n+3)/2)) + A107769(n).
From G. C. Greubel, May 21 2021: (Start)
a(n) = (1 + A001333(n) + A135153(n+2))/4.
a(n) = (2 + Q(n) + 2*(1+(-1)^n)*Pell((n+2)/2) + 2*(1-(-1)^n)*Pell((n+1)/2))/8.
a(2*n) = (2 + Q(2*n) + 4*Pell(n+1))/8.
a(2*n+1) = (2 + Q(2*n+1) + 4*Pell(n+1))/8, where Pell(n) = A000129(n), and Q(n) = A002203. (End)
Showing 1-6 of 6 results.

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

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