Atabey Kaygun - cp's OEIS frontend

A377936 Number of matchings in the complete planted binary tree with 2^n leaves.

2, 4, 24, 720, 712800, 666860040000, 597568733024952150000000, 474258018883889933710067708314342382812500000000

Offset: 0

Atabey Kaygun, Nov 11 2024

nonn

A planted binary tree has an initial root node with 1 child. The root is not considered to be a leaf. All internal nodes have degree 3. The total number of nodes is 2*n.

			The initial graphs for n=0..2 are:
      o        o                 o
      |        |                 |
      o        o                 o
             /   \             /   \
            o     o           o     o
                             / \   / \
                            o   o o   o

Andrew Howroyd, Table of n, a(n) for n = 0..11
Atabey Kaygun Hosoya Index of Balanced Binary Trees.
Eric Weisstein's World of Mathematics, Independent Edge Set.
Eric Weisstein's World of Mathematics, Matching.

Cf. A338293.

lista(n)={my(u=vector(n), v=vector(n)); u[1]=v[1]=1; for(n=1, #u-1, u[n+1]=v[n]^2; v[n+1]=u[n+1] + 2*v[n]*u[n]); v+u} \\ Andrew Howroyd, Nov 14 2024

a(n) = u(n) + v(n) where u(n) = v(n-1)^2 and v(n) = v(n-1)^2 + 2*v(n-1)*u(n-1) with u(1) = v(1) = 1. - Andrew Howroyd, Nov 14 2024

a(5) onwards from Andrew Howroyd, Nov 14 2024

A361254 Number of n-regular graphs on 2*n labeled nodes.

Original entry on oeis.org

1, 1, 3, 70, 19355, 66462606, 2977635137862, 1803595358964773088, 15138592322753242235338875, 1793196665025885172290508971592750, 3040059281615704147007085764679679740691838, 74597015246986083384362428357508730776063716190667288, 26737694395324301026230134763403079891362936970900741153038680278

Offset: 0

Atabey Kaygun, Mar 06 2023

nonn

These graphs share the same degree sequence as the complete bipartite graphs K(n,n).

Atabey Kaygun, Counting Graphs with a Prescribed Degree Sequence
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.

Cf. A001223, A059441, A339987, A360437.

\\ See Links in A295193 for GraphsByDegreeSeq.
a(n)={if(n==0, 1, vecsum(GraphsByDegreeSeq(2*n, n, (p, r)->valuation(p,x) >= n-r)[, 2])) } \\ Andrew Howroyd, Mar 06 2023

a(n) = A059441(2*n, n).

a(11)-a(12) from Andrew Howroyd, Mar 06 2023

A361267 Numbers k such that prime(k+2) - prime(k) = 6.

Original entry on oeis.org

3, 4, 5, 6, 7, 12, 13, 19, 25, 26, 27, 28, 43, 44, 48, 49, 59, 63, 64, 69, 88, 89, 112, 116, 142, 143, 147, 148, 151, 152, 181, 182, 206, 211, 212, 224, 225, 229, 234, 235, 236, 253, 261, 264, 276, 285, 286, 287, 301, 302, 313, 314, 322, 332, 336, 352, 384, 389

Offset: 1

Atabey Kaygun, Mar 06 2023

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Triplet
Wikipedia, Prime triplet

Cf. A000040, A000720, A007529, A022004, A022005.

(defn next-prime [n]
  (if (= n 2)
      3
      (let [m (+ n 2)
            t (-> n Math/sqrt int (+ 2))]
          (if (some #(zero? (mod m %)) (range 2 t))
              (next-prime m)
              m))))
(def primes (lazy-seq (iterate next-prime 2)))
(defn triplet-primes-positions [n]
  (->> primes
       (take n)
       (partition 3 1)
       (map list (range))
       (filter (fn [[i xs]] (= 6 (- (last xs) (first xs)))))
       (map #(-> % first inc))))
(println (triplet-primes-positions 2000))

q:= n-> is(ithprime(n+2)-ithprime(n)=6):
select(q, [$1..400])[];  # Alois P. Heinz, Mar 06 2023

Select[Range[400], Prime[# + 2] - Prime[#] == 6 &] (* Michael De Vlieger, Mar 06 2023 *)
PrimePi/@(Select[Partition[Prime[Range[400]],3,1],#[[3]]-#[[1]]==6&][[;;,1]]) (* Harvey P. Dale, Sep 16 2023 *)

from itertools import count, islice
from sympy import nextprime, prime
def A361267_gen(startvalue=1): # generator of terms >= startvalue
    p = prime(m:=max(startvalue,1))
    q = nextprime(p)
    r = nextprime(q)
    for k in count(m):
        if r-p == 6:
            yield k
        p, q, r = q, r, nextprime(r)
A361267_list = list(islice(A361267_gen(),20)) # Chai Wah Wu, Mar 27 2023

a(n) = A000720(A007529(n)). - Alois P. Heinz, Mar 06 2023

A360437 The number of labeled graphs on n nodes whose degree sequences realize the first n even terms of A001223 (the prime gap sequence).

Original entry on oeis.org

0, 0, 0, 0, 1, 13, 126, 288, 8160, 110800, 8552205, 253822863, 21802799690, 1751958608840, 231900341949104, 8821242453833942, 1943505339202134370, 417503287597170065730, 18004075657522393347770, 7276527497090264178405240, 2096206309349359523650043220

Offset: 1

Atabey Kaygun, Feb 07 2023

nonn

Atabey Kaygun, Counting Graphs with a Prescribed Degree Sequence
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.

Cf. A001223, A339987.

A339987 The number of labeled graphs on 2n vertices that share the same degree sequence as any unrooted binary tree on 2n vertices.

Original entry on oeis.org

1, 4, 90, 8400, 1426950, 366153480, 134292027870, 67095690261600, 43893900947947050, 36441011093916429000, 37446160423265535041100, 46669357647008722700474400, 69367722399061403579194432500, 121238024532751529573125745790000, 246171692450596203263023527657431250

Offset: 1

Atabey Kaygun, Dec 25 2020

nonn

An unrooted binary tree is a tree in which all non-leaf vertices have degree 3. With 2n vertices there will be n+1 leaves and n-1 internal vertices.

Andrew Howroyd, Table of n, a(n) for n = 1..100 (terms 1..40 from Atabey Kaygun)
Atabey Kaygun, Counting Graphs with a Prescribed Degree Sequence.
Atabey Kaygun, Common LISP program that generates the sequence.
M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.

Cf. A001147, A001190, A002829.

\\ See Links in A295193 for GraphsByDegreeSeq.
a(n) = {if(n==1, 1, my(d=2*n-4, M=GraphsByDegreeSeq(n-1, 3, (p,r)-> subst(deriv(p),x,1) >= d-6*r), z=(2*n)!/(n-1)!); sum(i=1, matsize(M)[1], my(p=M[i,1], r=(subst(deriv(p), x, 1)-d)/2); M[i,2]*z / (2^polcoef(p,1) * 6^polcoef(p,0) * 2^r * r!)))} \\ Andrew Howroyd, Mar 01 2023

Conjectured recurrence: 32*(1 + n)*(2 + n)*(1 + 2*n)*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(11589 + 10844*n + 3300*n^2 + 328*n^3)*a(n) - 8*(2 + n)*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(148119 + 232328*n + 129460*n^2 + 30664*n^3 + 2624*n^4)*a(n+1) - 16*(3 + n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(341634 + 712135*n + 569267*n^2 + 219308*n^3 + 40852*n^4 + 2952*n^5)*a(n+2) + 8*(4 + n)*(7 + 2*n)*(9 + 2*n)*(527520 + 1057879*n + 818282*n^2 + 306380*n^3 + 55672*n^4 + 3936*n^5)*a(n+3) - 2*(5 + n)*(9 + 2*n)*(601452 + 1117119*n + 786236*n^2 + 264028*n^3 + 42472*n^4 + 2624*n^5)*a(n+4) + 3*(4 + n)*(6 + n)*(3717 + 5228*n + 2316*n^2 + 328*n^3)*a(n+5) = 0. - Manuel Kauers and Christoph Koutschan, Mar 01 2023

Conjecture: a(n) ~ 2^(5*n - 1/2) * n^(2*n - 3/2) / (sqrt(Pi) * 3^(n-1) * exp(2*n + 21/16)), based on the recurrence by Manuel Kauers and Christoph Koutschan. - Vaclav Kotesovec, Mar 07 2023

A339847 The number of labeled 6-regular graphs on n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 105, 30016, 11180820, 5188453830, 2977635137862, 2099132870973600, 1803595358964773088, 1872726690127181663775, 2329676580698022197516875, 3443086402825299720403673760, 5997229769947050271535917422040, 12218901113752712984458458475480428

Offset: 0

Atabey Kaygun, Dec 21 2020

nonn

Marni Mishna, Table of n, a(n) for n = 0..195 (terms 0..36 from Andrew Howroyd, terms 37..40 from Atabey Kaygun)
Frédéric Chyzak and Marni Mishna Differential equations satisfied by generating functions of 5-, 6-, and 7-regular labelled graphs: a reduction-based approach, arXiv:2406.04753 [math.CO], 2024.
Atabey Kaygun, Counting Graphs with a Prescribed Degree Sequence.
Atabey Kaygun, Common LISP program that generates the sequence.
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.
Marni Mishna, Maple code to generate terms.

Column k=6 of A059441.

Cf. A165627 (unlabeled case), A295193.

\\ Needs GraphsByDegreeSeq from links in A295193.
a(n)={my(M=GraphsByDegreeSeq(n, 6, (p,r)->6-valuation(p,x) <= r)); if(n>=7, vecsum(M[,2]), n==0)} \\ Andrew Howroyd, Dec 26 2020

Terms a(14) and beyond from Andrew Howroyd, Dec 26 2020

A338978 Number of labeled 5-regular graphs on 2n nodes.

Original entry on oeis.org

1, 0, 0, 1, 3507, 66462606, 2977635137862, 283097260184159421, 52469332407700365320163, 17647883828569858659972268092, 10148613081040117624319536901932188, 9494356410654311931931879706070629989407, 13859154719468565627065764000731047706917194485

Offset: 0

Atabey Kaygun, Dec 18 2020

nonn

Marni Mishna, Table of n, a(n) for n = 0..110 (first 51 terms from Brendan Mackay)
Frédéric Chyzak and Marni Mishna, Differential equations satisfied by generating functions of 5-, 6-, and 7-regular labelled graphs: a reduction based approach, arXiv:2406.04753 [math.CO], 2024.
Élie de Panafieu, Asymptotic expansion of regular and connected regular graphs, arXiv:2408.12459 [math.CO], 2024. See p. 9.
Atabey Kaygun, Counting Graphs with a Prescribed Degree Sequence.
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.

With interspersed zeros, column k=5 of A059441.

Cf. A001205, A002829, A005815, A165626 (unlabeled case).

cp's OEIS Frontend

User: Atabey Kaygun

A377936 Number of matchings in the complete planted binary tree with 2^n leaves.

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A361254 Number of n-regular graphs on 2*n labeled nodes.

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A361267 Numbers k such that prime(k+2) - prime(k) = 6.

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A360437 The number of labeled graphs on n nodes whose degree sequences realize the first n even terms of A001223 (the prime gap sequence).

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A339987 The number of labeled graphs on 2n vertices that share the same degree sequence as any unrooted binary tree on 2n vertices.

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PARI

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A339847 The number of labeled 6-regular graphs on n nodes.

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A338978 Number of labeled 5-regular graphs on 2n nodes.

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