A006472 -id:A006472 - cp's OEIS frontend

A276482 a(n) = 5^nGamma(n+1/5)Gamma(n+1)/Gamma(1/5).

1, 1, 12, 396, 25344, 2661120, 415134720, 90084234240, 25944259461120, 9573431741153280, 4403778600930508800, 2470519795122015436800, 1660189302321994373529600, 1316530116741341538208972800, 1216473827868999581305090867200, 1295544626680484554089921773568000

Offset: 0

Ilya Gutkovskiy, Sep 05 2016

12-gonal (or dodecagonal) factorial numbers, also polygorial(n, 12).

More generally, the m-gonal factorial numbers (or polygorial(n, m)) is 2^(-n)*(m - 2)^n*Gamma(n+2/(m-2))*Gamma(n+1)/Gamma(2/(m-2)), m>2.

Robert Israel, Table of n, a(n) for n = 0..220
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.
Index entries for sequences related to factorial numbers

Cf. A000142, A008548, A051624.

Cf. similar sequences of m-gonal factorial numbers (or polygorial(n, m)): A006472 (m=3), A001044 (m=4), A084939 (m=5), A000680 (m=6), A084940 (m=7), A084941 (m=8), A084942 (m=9), A084943 (m=10), A084944 (m=11).

seq(mul(k*(5*k-4),k=1..n), n=0..20); # Robert Israel, Sep 18 2016

FullSimplify[Table[5^n Gamma[n + 1/5] (Gamma[n + 1]/Gamma[1/5]), {n, 0, 15}]]
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2),n]]; Array[polygorial[12, #] &, 16, 0] (* Robert G. Wilson v, Dec 13 2016 *)

a(n) = prod(k=1, n, k*(5*k - 4)); \\ Michel Marcus, Sep 06 2016

a(n) = Product_{k=1..n} k*(5*k - 4), a(0)=1.
a(n) = Product_{k=1..n} A051624(k), a(0)=1.
a(n) = A000142(n)*A008548(n).
a(n) ~ 2*Pi*5^n*n^(2*n+1/5)/(Gamma(1/5)*exp(2*n)).
Sum_{n>=0} 1/a(n) = BesselI(-4/5,2/sqrt(5))*Gamma(1/5)/5^(2/5) = Hypergeometric0F1(1/5, 1/5) = 2.085898421130914...

A306350 Number of paraphyletic coalescence sequences for 2n lineages, n each in 2 species.

Original entry on oeis.org

0, 4, 162, 23328, 9072000, 7873200000, 13367512620000, 40367907740160000, 201793403949096960000, 1578804075215377920000000, 18484433452834116768000000000, 312162837144268369009766400000000, 7374810540967959718955457331200000000

Offset: 1

Noah A Rosenberg, Feb 09 2019

nonn

Consider a binary tree evolving in time from a single node until the tree has 2n labeled leaves. Color the 2n leaves in 2 colors, red and blue, assigning n leaves to each color. Suppose coalescences of pairs of leaves happen at distinct times (i.e., no simultaneous mergers). A coalescence sequence is a sequence of coalescence events backward in time, tracing the reduction of the 2n leaves to the single ancestral node. A paraphyletic coalescence sequence is a sequence in which (1) all n red leaves have a common ancestor node that is not the ancestor of any blue leaves; or (2) all n blue leaves have a common ancestor node that is not the ancestor of any red leaves; but not both (1) and (2).

			For n=2, consider two red leaves R1 and R2 and two blue leaves B1 and B2. The a(2)=4 paraphyletic coalescence sequences, separated by semicolons, are (R1,R2), ((R1,R2),B1), (((R1,R2),B1),B2); (R1,R2), ((R1,R2),B2), (((R1,R2),B2),B1); (B1,B2), ((B1,B2),R1), (((B1,B2),R1),R2); and (B1,B2), ((B1,B2),R2), (((B1,B2),R2),R1).

N. A. Rosenberg, The shapes of neutral gene genealogies in two species: probabilities of monophyly, paraphyly, and polyphyly in a coalescent model, Evolution 57 (2003), 1465-1477.

The total number of coalescence sequences for n leaves, from among which the paraphyletic coalescence sequences are identified, follows A006472. Reciprocally monophyletic coalescence sequences, in which conditions (1) and (2) above both hold, follow A306266.

Table[3*(n!)^2*(2n - 2)!*(n - 1)/((n + 1)(2^(2 n - 3))), {n, 1, 30}]

a(n) = 3*(n!)^2*(2*n-2)!*(n-1)/((n+1)*2^(2*n-3)).

a(n) ~ 24*exp(-4*n)*n^(4*n-1/2)*Pi^(3/2). - Stefano Spezia, Apr 30 2024

A306390 Size of one subtree in the unlabeled binary rooted tree shape of size n whose leaf-labeled trees have the largest number of coalescence sequences.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32

Offset: 3

Noah A Rosenberg, Feb 12 2019

nonn

Consider the unlabeled, rooted, binary tree shapes (A001190). For each unlabeled shape, assign a labeling of the leaves: a labeled topology. This topology is a leaf-labeled, rooted, binary tree (chosen from among all such trees, A001147). From among the set of possible coalescence sequences (A006472), count all coalescence sequences, or labeled histories, that give rise to the specified labeled topology. For the unlabeled shape of size n whose labeled topologies have the largest number of coalescence sequences, the two subtrees immediately descended from the root have sizes a(n) and n-a(n) (Harding 1971, Eq. 5.7).

The decomposition of trees of n leaves into subtrees with sizes a(n) and n-a(n) also describes the set of unlabeled tree shapes whose labeled topologies have the largest number of root ancestral configurations (Disanto & Rosenberg 2017, Proposition 4).

			For n=5, the three unlabeled shapes are ((((.,.),.),.),.), (((.,.),(.,.)),.), and (((.,.),.),(.,.)). The formula gives a(5)=2, so that the shape with a subtree of size 2, (((.,.),.),(.,.)), is the one whose labeled topologies have the largest number of coalescence sequences. A labeled topology (((A,B),C),(D,E)) has 3 coalescence sequences, whereas ((((A,B),C),D),E) has 1 and (((A,B),(C,D)),E) has 2.

E. H. Dickey and N. A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307. (see Theorem 4)
F. Disanto and N. A. Rosenberg, Enumeration of ancestral configurations for matching gene trees and species trees, J. Comput. Biol. 24 (2017), 831-850.
E. F. Harding, The probabilities of rooted tree-shapes generated by random bifurcation, Adv. Appl. Prob. 3 (1971), 44-77.

Cf. A001147, A001190, A006472.

Table[2^(1 + Floor[Log2[(n - 1)/3]]), {n, 3, 100}]

a(n) = 2^(1 + log((n-1)/3)\log(2)); \\ Michel Marcus, Mar 08 2019

a(n) = 2^(1 + floor(log_2((n-1)/3))).

A381536 Number of labeled histories for rooted 4-furcating trees with 3n+1 leaves if simultaneous 4-furcations are not allowed.

Original entry on oeis.org

1, 1, 35, 7350, 5255250, 9564555000, 37072215180000, 271183254041700000, 3430468163627505000000, 70238835650273164875000000, 2210064963735845132791875000000, 102493972758213553878355995000000000, 6769214430816214165896021689775000000000, 618638506832293812621237422228537250000000000

Offset: 0

Noah A Rosenberg, Feb 26 2025

nonn

Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307 (see Table 1).

Cf. A006472, A339411, A381523.

a:= n-> (3*n+1)!/24^n*mul(3*i-2, i=1..n):
seq(a(n), n=0..13);  # Alois P. Heinz, Feb 26 2025

a(n) = ((3*n+1)!/24^n) * Product_{i=1..n} (3*i-2).

A120476 Triangle read by rows: a(n,m)=(2n-1)(n-m)*(n+m+1)/2, where n is the column and m the row index.

Original entry on oeis.org

1, 3, -2, 6, -5, -9, 10, -9, -21, -20, 15, -14, -36, -45, -35, 21, -20, -54, -75, -77, -54, 28, -27, -75, -110, -126, -117, -77, 36, -35, -99, -150, -182, -189, -165, -104, 45, -44, -126, -195, -245, -270, -264, -221, -135, 55, -54, -156, -245, -315, -360, -374, -351, -285, -170

Offset: 0

Roger L. Bagula, Jul 19 2006

Triangular array based on recurrence in Laplace function in J. W. S. Rayleigh.

			1,
3, -2,
6, -5, -9,
10, -9, -21,-20,
15, -14,-36,-45, -35

J. W. S. Rayleigh, The Theory of Sound, volume 2, page 237,Dover, New York,1945

Cf. A006472.

a = Table[Table[(m + 1)*(2*n - 1)*(n - m)*(n + m + 1)/(2*(m + 1)), {n, 0, m - 1}], {m, 1, 10}] Flatten[a]

a(n,m) = (2n-1)*[A000217(n)-A000217(m)] = (1-2n)*A049777(n,m) . - R. J. Mathar, Dec 05 2007

Row sums: sum_{n=0..m-1} a(n,m) = -m(m+1)(3m^2-5m-4)/12. [From R. J. Mathar, Jan 15 2009]

Edited by N. J. A. Sloane, Oct 01 2006

A233003 (n!)^2 mod Pt(n), where Pt(n) is product of first n positive triangular numbers (A000217).

Original entry on oeis.org

0, 1, 0, 36, 900, 8100, 0, 25401600, 514382400, 12859560000, 6224027040000, 56016243360000, 9466745127840000, 1855482045056640000, 0, 6679735362203904000000, 13513104637738497792000000, 156365925093831188736000000, 225792395835492236534784000000, 22579239583549223653478400000000

Offset: 1

Alex Ratushnyak, Dec 03 2013

nonn

Indices of zeros appear to be 2^k-1.

			a(4) = 1*4*9*16 mod 1*3*6*10 = 576 mod 90 = 36.

Cf. A000142, A000217, A000290, A001044(n!^2).
Cf. A006472 (triangular factorial, essentially equal to Pt(n)).
Cf. A006788 (floor(n!^2/Pt)).

s=t=1
for n in range(1,33):
  s*=n*n
  t*=n*(n+1)//2
  print(s%t, end=', ')

A233004 Pt(n) mod n!, where Pt(n) is product of first n positive triangular numbers (A000217).

Original entry on oeis.org

0, 1, 0, 12, 60, 540, 0, 20160, 181440, 907200, 19958400, 359251200, 1556755200, 32691859200, 0, 10461394944000, 177843714048000, 1600593426432000, 60822550204416000, 608225502044160000, 38318206628782080000, 702500454861004800000, 12926008369442488320000

Offset: 1

Alex Ratushnyak, Dec 03 2013

nonn

Pt(n) = n!*(n+1)! / 2^n.

Pt(n) mod n! = 0 if and only if 2^n divides (n+1)!, that is, n+1 is a power of 2. Thus indices of zeros are of the form 2^k-1.

Cf. A000142, A000217.
Cf. A006472 (triangular factorial, essentially equal to Pt(n)).
Cf. A067667 (Pt(n)/n! for n's of the form 2^k-1).
Cf. A069902, A007917.

f=t=1
for n in range(1,33):
  t*=n*(n+1)//2
  f*=n
  print(t%f, end=', ')

A259345 Number of labeled single-linkage dendrograms.

Original entry on oeis.org

1, 1, 1, 30, 750

Offset: 1

N. J. A. Sloane, Jun 27 2015, following a suggestion by Colin Mallows from 1981

O. Frank and K. Svensson, On probability distributions of single-linkage dendrograms, Journal of Statistical Computation and Simulation, 12 (1981), 121-131. (Annotated scanned copy)
C. L. Mallows, Note to N. J. A. Sloane circa 1979.

A306391 Number of polyphyletic coalescence sequences for 2n lineages, n each in 2 species.

Original entry on oeis.org

0, 12, 2484, 1557792, 2560572000, 9326330280000, 66250877823900000, 834917902101803520000, 17373747843395915811840000, 564479089176417832085760000000, 27382950623629177584815808000000000, 1912097851374544604017590025267200000000, 186429568131038636125345650494922854400000000

Offset: 1

Noah A Rosenberg, Feb 12 2019

nonn

Consider a binary tree evolving in time from a single node until the tree has 2n labeled leaves. Color the 2n leaves in 2 colors, red and blue, assigning n leaves to each color. Suppose coalescences of pairs of leaves happen at distinct times (i.e., no simultaneous mergers). A coalescence sequence is a sequence of coalescence events backward in time, tracing the reduction of the 2n leaves to the single ancestral node. Consider two possible features of the resulting tree: (1) all n red leaves have a common ancestor node that is not the ancestor of any blue leaves; (2) all n blue leaves have a common ancestor node that is not the ancestor of any red leaves. A reciprocally monophyletic sequence satisfies both (1) and (2). A paraphyletic coalescence sequence satisfies (1) or (2) but not both. A polyphyletic coalescence sequence does not satisfy either (1) or (2).

			For n=2, consider two red leaves R1 and R2 and two blue leaves B1 and B2. The a(2)=12 polyphyletic coalescence sequences, separated by semicolons, are (B1,R1), ((B1,R1),B2), (((B1,R1),B2),R2); (B1,R1), ((B1,R1),R2), (((B1,R1),R2),B2); (B1,R2), ((B1,R2),B2), (((B1,R2),B2),R1); (B1,R2), ((B1,R2),R1), (((B1,R2),R1),B2); (B2,R1), ((B2,R1),B1), (((B2,R1),B1),R2); (B2,R1), ((B2,R1),R2), (((B2,R1),R2),B1); (B2,R2), ((B2,R2),B1), (((B2,R2),B1),R1); (B2,R2), ((B2,R2),R1), (((B2,R2),R1),B1); (B1,R1), (B2,R2), ((B1,R1),(B2,R2)); (B1,R2), (B2,R1), ((B1,R2),(B2,R1)); (B2,R1), (B1,R2), ((B2,R1),(B1,R2)); (B2,R2), (B1,R1), ((B2,R2),(B1,R1)).

N. A. Rosenberg, The shapes of neutral gene genealogies in two species: probabilities of monophyly, paraphyly, and polyphyly in a coalescent model, Evolution 57 (2003), 1465-1477.

The total number of coalescence sequences for n leaves, from among which the polyphyletic coalescence sequences are identified, follows A006472. The sum of a(n), A306266(n), and A306350(n) is A006472(2n).

Table[(1 - (2 n! n!/(2 n)!) (7 n - 5)/((n + 1) (2 n - 1))) (2 n)! (2 n - 1)!/2^(2 n - 1), {n, 1, 30}]

a(n) = (1 - (2*n!*n!/(2*n)!)*(7*n-5)/((n+1)*(2*n-1)))*(2*n)!*(2*n-1)!/2^(2*n-1); \\ Michel Marcus, Feb 12 2019

a(n) = (1 - (2 n! n!/(2n)!)(7n-5)/((n+1)(2n-1))) (2n)! (2n-1)!/2^(2n-1).

a(n) ~ exp(-4*n)*n^(4*n-1)*(4^n + 3*4^(1+n)*n - 84*sqrt(n*Pi))*Pi/3. - Stefano Spezia, Apr 30 2024

a(12)-a(13) from Stefano Spezia, Apr 30 2024

A377461 Number of ranked labeled trees compatible with the 2-leaf perfect phylogeny of sample size n that possesses the largest number of compatible ranked labeled trees.

Original entry on oeis.org

1, 1, 2, 9, 54, 540, 6480, 113400, 2268000, 61236000, 1837080000, 70727580000, 2970558360000, 154469034720000, 8650265944320000, 583892951241600000, 42040292489395200000, 3573424861598592000000, 321608237543873280000000, 33608060823334757760000000

Offset: 2

Noah A Rosenberg, Jan 03 2025

nonn

The 2-leaf perfect phylogeny of sample size n that possesses the largest number of compatible ranked labeled trees is (floor(n/2), ceiling(n/2)); a(n) is the number of ranked labeled trees for this perfect phylogeny.

J. A. Palacios, A. Bhaskar, F. Disanto and N. A. Rosenberg, Enumeration of binary trees compatible with a perfect phylogeny, J. Math. Biol. 84 (2022), 54.

Cf. A001405, A006472, A306266.

a[n_] := ((n - 2)!/((Floor[n/2] - 1)! (n - 1 - Floor[n/2])!)) Product[Binomial[i, 2], {i, 2, Floor[n/2]}] Product[Binomial[i, 2], {i, 2, Ceiling[n/2]}]
a[n_] := ((n - 2)!/((Floor[n/2] - 1)! (n - 1 - Floor[n/2])!)) Floor[n/2]! (Floor[n/2] - 1)! Ceiling[n/2]! (Ceiling[n/2] - 1)! /(2^(Floor[n/2] - 1) 2^(Ceiling[n/2] - 1))

a(n) = ((n-2)! / ((floor(n/2)-1)! (n-1-floor(n/2))!)) * (floor(n/2))! (floor(n/2)-1)! (ceiling(n/2))! (ceiling(n/2)-1)! / (2^(floor(n/2)-1) 2^(ceiling(n/2)-1)).
a(n) = A001405(n-2)*A006472(floor(n/2))*A006472(ceiling(n/2)).
a(2n) = A306266(n).

cp's OEIS Frontend

A276482 a(n) = 5^n*Gamma(n+1/5)*Gamma(n+1)/Gamma(1/5).

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A306350 Number of paraphyletic coalescence sequences for 2n lineages, n each in 2 species.

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A306390 Size of one subtree in the unlabeled binary rooted tree shape of size n whose leaf-labeled trees have the largest number of coalescence sequences.

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A381536 Number of labeled histories for rooted 4-furcating trees with 3n+1 leaves if simultaneous 4-furcations are not allowed.

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A120476 Triangle read by rows: a(n,m)=(2*n-1)*(n-m)*(n+m+1)/2, where n is the column and m the row index.

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A233003 (n!)^2 mod Pt(n), where Pt(n) is product of first n positive triangular numbers (A000217).

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A233004 Pt(n) mod n!, where Pt(n) is product of first n positive triangular numbers (A000217).

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A259345 Number of labeled single-linkage dendrograms.

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A306391 Number of polyphyletic coalescence sequences for 2n lineages, n each in 2 species.

A276482 a(n) = 5^nGamma(n+1/5)Gamma(n+1)/Gamma(1/5).

A120476 Triangle read by rows: a(n,m)=(2n-1)(n-m)*(n+m+1)/2, where n is the column and m the row index.