A006472 -id:A006472 - cp's OEIS frontend

A306266 Number of reciprocally monophyletic coalescence sequences for 2n lineages, n each in 2 species.

1, 2, 54, 6480, 2268000, 1837080000, 2970558360000, 8650265944320000, 42040292489395200000, 321608237543873280000000, 3696886690566823353600000000, 61486619437507406017075200000000, 1433990938521547723130227814400000000

Offset: 1

Noah A Rosenberg, Feb 01 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 reciprocally monophyletic coalescence sequence is a sequence in which all n red leaves have a common ancestor node that is not the ancestor of any blue leaves, and all n blue leaves have a common ancestor node that is not the ancestor of any red leaves.

			For n=2, consider two red leaves R1 and R2 and two blue leaves B1 and B2. In a reciprocally monophyletic coalescence sequence, the pair of red leaves must coalesce, and the pair of blue leaves must coalesce. These events can occur in either of two orders (red first or blue first), so that a(2)=2.

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 reciprocally monophyletic coalescence sequences are identified, follows A006472.

a:= n-> n!^2*(2*n-2)!/2^(2*n-2):
seq(a(n), n=1..15);  - Alois P. Heinz, Feb 07 2019

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

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

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

A328299 Number of n-step walks on cubic lattice starting at (0,0,0), ending at (floor(n/3), floor((n+1)/3), floor((n+2)/3)), remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).

Original entry on oeis.org

1, 1, 3, 12, 41, 179, 909, 3968, 19680, 106368, 516905, 2717631, 15139485, 77813569, 422589823, 2395441908, 12734635078, 70577595746, 404540380566, 2199035619696, 12356298623126, 71368686011040, 394076753535029, 2236273925952447, 12988459939106601

Offset: 0

Alois P. Heinz, Oct 11 2019

			a(2) = 3: [(0,0,0),(1,0,0),(0,1,1)], [(0,0,0),(0,1,0),(0,1,1)], [(0,0,0),(0,0,1),(0,1,1)].

Alois P. Heinz, Table of n, a(n) for n = 0..1039
Wikipedia, Lattice path
Wikipedia, Self-avoiding walk

Cf. A006472, A022916, A328297, A328345.

b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
      add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
      sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
    end:
a:= n-> b([floor((n+i)/3)$i=0..2]):
seq(a(n), n=0..24);

b[l_] := b[l] = If[Last[l] == 0, 1, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, {-1, 0, 1}}, {j, {-1, 0, 1}}, {k, {-1, 0, 1}}]];
a[n_] := b[Table[Floor[(n+i)/3], {i, 0, 2}]];
a /@ Range[0, 24] (* Jean-François Alcover, May 12 2020, after Maple *)

A339411 Product of partial sums of odd squares.

Original entry on oeis.org

1, 1, 10, 350, 29400, 4851000, 1387386000, 631260630000, 429257228400000, 415950254319600000, 553213838245068000000, 979741707532015428000000, 2253405927323635484400000000, 6591212337421633791870000000000, 24084289880938649875492980000000000, 108258883014819231190340945100000000000

Offset: 0

Werner Schulte, Dec 03 2020

a(n) is also the number of labeled histories across all trifurcating labeled topologies on trees with non-simultaneous trifurcations, where the number of leaves is 2n+1. - Noah A Rosenberg, Feb 24 2025

			a(4) = (1^2) * (1^2 + 3^2) * (1^2 + 3^2 + 5^2) * (1^2 + 3^2 + 5^2 + 7^2) = 1 * 10 * 35 * 84 = 29400.

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

Cf. A016754, A000447, A135438, A000108, A006472.

a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1)*(4*n^3-n)/3)
    end:
seq(a(n), n=0..15);  # Alois P. Heinz, Dec 03 2020

Array[((2 #)!*(2 # + 1)!)/(#!*12^#) &, 16, 0] (* Michael De Vlieger, Dec 10 2020 *)

for(n=0,9,print((2*n)!*(2*n+1)!/(n!*12^n)))

for(n=0,9,print(prod(i=1,n,sum(j=1,i,(2*j-1)^2))))

a(n) = Product_{i=1..n} (Sum_{j=1..i} (2*j - 1)^2).
a(n) = Product_{i=1..n} binomial(2*i + 1, 3).
a(n) = Product_{i=1..n} A000447(i).
a(n) = ((2*n)! * (2*n+1)!) / (n! * 12^n).
a(n) / A135438(n) = A000108(n).
a(n) = (Gamma(2*n + 2)*Gamma(n + 1/2))/(3^n*sqrt(Pi)). - Peter Luschny, Dec 11 2020
D-finite with recurrence 3*a(n) -n*(2*n-1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 25 2023

A375835 Triangle read by rows: T(n, k) is the number of chains of length k in the poset of permutations of an n-set.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 8, 3, 0, 24, 64, 59, 18, 0, 120, 574, 970, 695, 180, 0, 720, 5858, 16124, 20240, 11955, 2700, 0, 5040, 67752, 285264, 556591, 559895, 282555, 56700, 0, 40320, 880584, 5459712, 15519287, 23585870, 19879370, 8780940, 1587600, 0, 362880, 12746208, 113511982, 451541898, 971214825, 1213062690, 882179550, 347072040, 57153600

Offset: 0

Rajesh Kumar Mohapatra, Aug 31 2024

			The triangle T(n,k) begins:
  n\k 0    1      2      3      4       5      6      7 ...
  0   1
  1   0    1
  2   0    2      1
  3   0    6      8      3
  4   0   24     64     59     18
  5   0  120    574    970    695     180
  6   0  720   5858  16124  20240   11955   2700
  7   0 5040  67752 285264 556591  559895 282555  56700
  ...
The T(3, 2) = 8 chains in the poset of the permutations of {1, 2, 3} are:
{(1)(2)(3) < (1)(23), (1)(2)(3) < (2)(13), (1)(2)(3) < (3)(12), (1)(2)(3) < (123),(1)(2)(3) < (132), (1)(23) < (123), (2)(13) < (132), (3)(12) < (123)}.

Cf. A000007 (column k=0), A000142 (column k=1), A006472 (main diagonal), A375836 (row sums).

Cf. A048994, A331955, A330804, A331956, A331957.

b := proc(n, k, t) option remember; if k < 0 then return 0 fi; if {n, k} = {0} then return 1 fi; add(ifelse(k = 1, 1, b(v, k - 1, 1))*abs(Stirling1(n, v)), v = k..n-t) end: T := (n, k) -> b(n, k, 0): seq((seq(T(n, k), k=0..n)), n = 0..10);  # Peter Luschny, Sep 05 2024

b[n_, k_, t_] := b[n, k, t] = If[k < 0, 0, If[n == 0 && k == 0, 1,
Sum[If[k == 1, 1, b[v, k - 1, 1]] * Abs[StirlingS1[n, v]], {v, k, n - t}]]];
T[n_, k_] := b[n, k, 0]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]

Let Stirling1(n, k) denote the unsigned Stirling numbers of the first kind (A132393).
T(0, 0) = 1, T(0, k) = 0 for k > 0.
T(n, k) = Sum_{i_k=k..n} (Sum_{i_(k-1)=k-1..i_k - 1} (... (Sum_{i_2=2..i_3 - 1} (Sum_{i_1=1..i_2 - 1} Stirling1(n, i_k) * Stirling1(i_k, i_(k-1)) * ... * Stirling1(i_3, i_2) * Stirling1(i_2, i_1)))...)), where 1 <= k <= n.

A378234 From higher-order arithmetic progressions: Corrected version of A259461.

Original entry on oeis.org

40, 5000, 472500, 43218000, 4148928000, 432081216000, 49509306000000, 6275893932000000, 881135508052800000, 136878615942868800000, 23474682634201999200000, 4432282735129048800000000, 918537831584839065600000000, 208281986149676045967360000000, 51516317681413623440962560000000

Offset: 0

Georg Fischer, Dec 16 2024

nonn

Only the first 5 terms of A259461 are correct. - R. J. Mathar, Jul 14 2015

"2 over n!" on page 13 in the Dienger article is A006472; A_3 is A001303.

Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]

Cf. A001303, A006472, A259459, A259460, A259461.

rV := proc(n,a,d)
    n*(n+1)/2*a+(n-1)*n*(n+1)/6*d;
end proc:
A259461 := proc(n)
    mul(rV(i,a,d),i=1..n+3) ;
    coeftayl(%,d=0,3) ;
    coeftayl(%,a=0,n) ;
end proc:
seq(A259461(n),n=1..5) ; # R. J. Mathar, Jul 14 2015

D-finite with recurrence: -2*n*(n+2)*a(n) + (n+4)^3*(n+5)*a(n-1) = 0.

a(n) = (n+5)!*(n+4)!^3 / (1296*2^(n+4)*n!^2*(n+2)*(n+1)).

A133401 Diagonal of polygorial array T(n,k) = n-th polygorial for k = n, for n > 2.

Original entry on oeis.org

18, 576, 46200, 7484400, 2137544640, 981562982400, 678245967907200, 670873729125600000, 913601739437346960000, 1660189302321994373529600, 3923769742187622047360640000, 11805614186177306251101945600000, 44403795869109177300313209696000000

Offset: 3

Jonathan Vos Post, Nov 25 2007

Array T(n,k) = k-th polygorial(n,k) begins:
k  |  polygorial(n,k)
| 1 1  3  18   180    2700     56700     1587600      57153600
| 1 1  4  36   576   14400    518400    25401600    1625702400
| 1 1  5  60  1320   46200   2356200   164934000   15173928000
| 1 1  6  90  2520  113400   7484400   681080400   81729648000
| 1 1  7 126  4284  235620  19085220  2137544640  316356606720
| 1 1  8 168  6720  436800  41932800  5577062400  981562982400
| 1 1  9 216  9936  745200  82717200 12738448800 2598643555200
| 1 1 10 270 14040 1193400 150368400 26314470000 6104957040000

			a(3) = polygorial(3,3) = A006472(3) = product of the first 3 triangular numbers = 1*3*6 = 18.
a(4) = polygorial(4,4) = A001044(4) = product of the first 4 squares = 1*4*9*16 = 576.
a(5) = polygorial(5,5) = A084939(5) = product of the first 5 pentagonal numbers = 1*5*12*22*35 = 46200.

Nathaniel Johnston, Table of n, a(n) for n = 3..100
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Cf. A006472, A001044, A000680, A084939, A084940, A084941, A084942, A084943, A084944, A085356.

A133401 := proc(n) return mul((n/2-1)*m^2-(n/2-2)*m,m=1..n): end: seq(A133401(n),n=3..15); # Nathaniel Johnston, May 05 2011

Table[Product[m*(4 - n + m*(n-2))/2, {m, 1, n}],{n, 3, 20}] (* Vaclav Kotesovec, Feb 20 2015 *)
Table[FullSimplify[(n-2)^n * Gamma[n+1] * Gamma[n+2/(n-2)] / (2^n*Gamma[2/(n-2)])],{n,3,15}] (* Vaclav Kotesovec, Feb 20 2015 *)
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k - 2), n]]; Array[ polygorial[#, #] &, 13, 3] (* Robert G. Wilson v, Dec 13 2016 *)

a(n) ~ Pi * n^(3*n-1) / (2^(n-2) * exp(2*n+2)). - Vaclav Kotesovec, Feb 20 2015

Edited by Nathaniel Johnston, May 05 2011

A140701 Partial products of A005448.

Original entry on oeis.org

1, 1, 4, 40, 760, 23560, 1083760, 69360640, 5895654400, 642626329600, 87397180825600, 14507932017049600, 2887078471392870400, 678463440777324544000, 185898982772986925056000, 58744078556263868317696000, 21206612358811256462688256000

Offset: 0

Jonathan Vos Post, May 24 2008

			a(10) = 87397180825600 = 1 * 4 * 10 * 19 * 31 * 46 * 64 * 85 * 109 * 136.

Harvey P. Dale, Table of n, a(n) for n = 0..244 [a(0) added by Robert C. Lyons].
Eric Weisstein's World of Mathematics, Centered Triangular Number

Cf. A005448.
For analog with centered n-gonal numbers see A140702.
For analog with regular triangular numbers see A006472.
For the analog with a partial sum instead of a partial product see A006003.

Table[Product[(3*k^2-3*k+2)/2,{k,1,n}],{n,1,20}] (* Vaclav Kotesovec, Jul 11 2015 *)
FoldList[Times,3*Accumulate[Range[0,20]]+1] (* Harvey P. Dale, Aug 05 2018 *)

a(n) = prod(k=1, n, 3*k*(k-1)/2 + 1); \\ Michel Marcus, Mar 02 2023

a(n) = Product_{k=1..n} A005448(k).
a(n) ~ cosh(Pi*sqrt(5/3)/2) * 3^n * n^(2*n) / (exp(2*n) * 2^(n-1)). - Vaclav Kotesovec, Jul 11 2015
a(n) = (2/3)^(1 - n) * Pochhammer(1 + (3 - i*sqrt(15))/6, n - 1) * Pochhammer(1 + (3 + i*sqrt(15))/6, n - 1), for n>=1. - Antonio Graciá Llorente, Sep 10 2023

A140702 Main diagonal of array A(k,n) = product of first n centered n-gonal numbers.

Original entry on oeis.org

40, 1625, 151776, 27316471, 8429601664, 4108830350625, 2977546171600000, 3062351613203813051, 4308809606735976861696, 8050856986181775515023417, 19490752185922086291273856000, 59888297825402713913058605859375, 229474927848540723655596345639141376

Offset: 3

Jonathan Vos Post, May 24 2008

For analog with regular (not centered) n-gonal numbers, see A133401.
Array A(k,n) = k-th polygorial(n,k) begins:
k  |  CenteredPolygorial(n,k)
---+-------------------------
3  | 1 4   40   760   23560    1083760      69360640      5895654400   A140701
4  | 1 5   65  1625   66625    4064125     345450625     39035920625
5  | 1 6   96  2976   151776   11534976   1222707456    172401751296
6  | 1 7  133  4921   300181   27316471   3469191817    586293417073
7  | 1 8  176  7568   537328   56956768   8429601664   1660631527808
8  | 1 9  225 11025   893025  108056025  18261468225   4108830350625
9  | 1 10 280 15400  1401400  190590400  36212176000   9161680528000

			a(3) = 3rd centered polygorial number polygorial(3,3) = A140701(3) = product of the first 3 centered triangular numbers = 1 * 4 * 10 = 40.
a(4) = 4th centered polygorial number centered polygorial(4,4) = product of the first 4 centered square numbers A001844 = 1 * 5 * 13 * 25 = 1625.
a(5) = 5th centered pentagorial number centered polygorial(5,5) = product of the first 5 centered pentagonal numbers A005891 = 1 * 5 * 12 * 22 * 35 = 151776.
a(6) = 6th centered hexagorial number centered polygorial(6,6) = product of the first 6 centered hexagonal numbers A003215 = 1 * 7 * 19 * 37 * 61 * 91 = 27316471.

Nathaniel Johnston, Table of n, a(n) for n = 3..100
Eric W. Weisstein, Centered Triangular Number.

Cf. A005448, A006003, A006472, A133401, A140701.

A140702 := proc(n) mul(n*k*(k-1)/2+1,k=1..n): end: seq(A140702(n),n=3..15); # Nathaniel Johnston, Oct 01 2011

Table[Product[n*k*(k-1)/2+1,{k,1,n}],{n,3,20}] (* Vaclav Kotesovec, Jul 11 2015 *)

a(n) ~ Pi * n^(3*n-1) / (exp(2*n) * 2^(n-2)). - Vaclav Kotesovec, Jul 11 2015

a(9) corrected and more terms from Nathaniel Johnston, Oct 01 2011

A330778 Triangle read by rows: T(n,k) is the number of balanced reduced multisystems of weight n with maximum depth and atoms colored using exactly k colors.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 2, 17, 33, 18, 5, 86, 321, 420, 180, 16, 520, 3306, 7752, 7650, 2700, 61, 3682, 37533, 140172, 238560, 189000, 56700, 272, 30050, 473604, 2644356, 6899070, 9196740, 6085800, 1587600, 1385, 278414, 6630909, 53244180, 199775820, 398328480, 435954960, 247665600, 57153600

Offset: 1

Andrew Howroyd, Dec 30 2019

			Triangle begins:
    1;
    1,     1;
    1,     4,      3;
    2,    17,     33,      18;
    5,    86,    321,     420,     180;
   16,   520,   3306,    7752,    7650,    2700;
   61,  3682,  37533,  140172,  238560,  189000,   56700;
  272, 30050, 473604, 2644356, 6899070, 9196740, 6085800, 1587600;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Column 1 is A000111.
Main diagonal is A006472.
Row sums are A330676.
Cf. A330776.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=vector(n), u=vector(n)); v[1]=k; for(n=1, #v, for(i=n, #v, u[i] += v[i]*(-1)^(i-n)*binomial(i-1, n-1)); v=EulerT(v)); u}
M(n)={my(v=vector(n, k, R(n, k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k, i)*v[i])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, 1..n]))}

A334129 Numbers that can be written as a product of one or more consecutive triangular numbers.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 18, 21, 28, 36, 45, 55, 60, 66, 78, 91, 105, 120, 136, 150, 153, 171, 180, 190, 210, 231, 253, 276, 300, 315, 325, 351, 378, 406, 435, 465, 496, 528, 561, 588, 595, 630, 666, 703, 741, 780, 820, 861, 900, 903, 946, 990, 1008, 1035

Offset: 1

Ilya Gutkovskiy, Apr 14 2020

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triangular Number
Index to sequences related to polygonal numbers

Cf. A000217, A006472, A085780, A085782, A334130.

lmt = 1050; t = PolygonalNumber[3, #] & /@ Range[0, Sqrt[ 2lmt]]; f[n_] := Select[ Times @@@ Partition[t, n +1, 1], # < lmt &]; lst = {}; k = 0; While[f@k != {}, lst = Join[lst, f@k]; k++]; Union@lst (* Robert G. Wilson v, Apr 16 2020 *)

list(lim)=if(lim<1, return(if(lim<0,[],[0]))); my(v=List([0,1]),t=1,m=2); lim\=1; while(t<=lim, listput(v,t); t=m*m++/2); for(e=1,m, for(i=3,m-e, t=factorback(Vec(v[i..i+e])); if(t>lim, break); listput(v,t))); Set(v) \\ Charles R Greathouse IV, Apr 16 2020

cp's OEIS Frontend

A306266 Number of reciprocally monophyletic coalescence sequences for 2n lineages, n each in 2 species.

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A328299 Number of n-step walks on cubic lattice starting at (0,0,0), ending at (floor(n/3), floor((n+1)/3), floor((n+2)/3)), remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).

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A339411 Product of partial sums of odd squares.

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A375835 Triangle read by rows: T(n, k) is the number of chains of length k in the poset of permutations of an n-set.

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A378234 From higher-order arithmetic progressions: Corrected version of A259461.

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A133401 Diagonal of polygorial array T(n,k) = n-th polygorial for k = n, for n > 2.

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A140701 Partial products of A005448.

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