A006411 -id:A006411 - cp's OEIS frontend

A342984 Triangle read by rows: T(n,k) is the number of nonseparable tree-rooted planar maps with n edges and k faces, n >= 0, k = 1..n+1.

1, 1, 1, 0, 2, 0, 0, 3, 3, 0, 0, 4, 20, 4, 0, 0, 5, 75, 75, 5, 0, 0, 6, 210, 604, 210, 6, 0, 0, 7, 490, 3150, 3150, 490, 7, 0, 0, 8, 1008, 12480, 27556, 12480, 1008, 8, 0, 0, 9, 1890, 40788, 170793, 170793, 40788, 1890, 9, 0, 0, 10, 3300, 115500, 829920, 1565844, 829920, 115500, 3300, 10, 0

Offset: 0

Andrew Howroyd, Apr 03 2021

The number of vertices is n + 2 - k.
A tree-rooted planar map is a planar map with a distinguished spanning tree.
For k >= 2, column k is a polynomial of degree 4*(k-2)+1.

			Triangle begins:
  1;
  1, 1;
  0, 2,    0;
  0, 3,    3,     0;
  0, 4,   20,     4,     0;
  0, 5,   75,    75,     5,     0;
  0, 6,  210,   604,   210,     6,    0;
  0, 7,  490,  3150,  3150,   490,    7, 0;
  0, 8, 1008, 12480, 27556, 12480, 1008, 8, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, Table IVa.

Columns (and diagonals) 3..5 are A006411, A006412, A006413.
Row sums are A004304.
Cf. A342982, A342985, A342987.

\\ here F(n,y) gives A342982 as g.f.
F(n,y)={sum(n=0, n, x^n*sum(i=0, n, my(j=n-i); y^i*(2*i+2*j)!/(i!*(i+1)!*j!*(j+1)!))) + O(x*x^n)}
H(n)={my(g=F(n,y), v=Vec(subst(g, x, serreverse(x*g^2)))); vector(#v, n, Vecrev(v[n], n))}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }

T(n,n+2-k) = T(n,k).

G.f. A(x,y) satisfies F(x,y) = A(x*F(x,y)^2,y) where F(x,y) is the g.f. of A342982.

A006412 Number of nonseparable tree-rooted planar maps with n + 3 edges and 4 vertices.

Original entry on oeis.org

4, 75, 604, 3150, 12480, 40788, 115500, 292578, 677820, 1459315, 2954952, 5679700, 10438272, 18449760, 31511880, 52213596, 84206100, 132543411, 204105220, 308116050, 456776320, 666022500, 956435220, 1354315950, 1892954700, 2614113099, 3569749200, 4824012424

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Column 4 of A342984.

Cf. A006411, A006413.

A006412[n_] := Binomial[n + 5, 6]*(n + 3)*(n*(13*n + 57) + 14)/84;
Array[A006412, 30] (* Paolo Xausa, Aug 20 2025 *)

a(n) = {binomial(n+5,6)*(n + 3)*(13*n^2 + 57*n + 14)/84} \\ Andrew Howroyd, Apr 05 2021

a(n) = 4 * binomial(n + 4, 5) + 51 * binomial(n + 4, 6) + 163 * binomial(n + 4, 7) + 194 * binomial(n + 4, 8) + 78 * binomial(n + 4, 9). - Sean A. Irvine, Apr 03 2017
a(n) = binomial(n+5,6)*(n + 3)*(13*n^2 + 57*n + 14)/84. - Andrew Howroyd, Apr 05 2021
G.f.: x*(4 + 35*x + 34*x^2 + 5*x^3)/(1 - x)^10. - Stefano Spezia, Aug 19 2025

Terms a(11) and beyond from Andrew Howroyd, Apr 05 2021

A006413 Number of nonseparable tree-rooted planar maps with n + 4 edges and 5 vertices.

Original entry on oeis.org

5, 210, 3150, 27556, 170793, 829920, 3359356, 11786190, 36845718, 104719524, 274707420, 672982128, 1554007910, 3407724936, 7139933088, 14366348780, 27878652291, 52364814150, 95497666810, 169546939380, 293722986375, 497527759560, 825473130300, 1343631834090

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259.
Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Column 5 of A342984.

Cf. A006411, A006412.

A006413[n_] := Binomial[n + 7, 8]*(n + 4)*(n*(n*(n*(23*n + 279) + 941) + 599) + 138)/1980;
Array[A006413, 25] (* Paolo Xausa, Aug 20 2025 *)

a(n) = {binomial(n+7, 8)*(n + 4)*(23*n^4 + 279*n^3 + 941*n^2 + 599*n + 138)/1980} \\ Andrew Howroyd, Apr 05 2021

a(n) = 5 * binomial(n + 6, 7) + 170 * binomial(n + 6, 8) + 1440 * binomial(n + 6, 9) + 4906 * binomial(n + 6, 10) + 7927 * binomial(n + 6, 11) + 6090 * binomial(n + 6, 12) + 1794 * binomial(n + 6, 13). - Sean A. Irvine, Apr 03 2017
a(n) = binomial(n+7,8)*(n + 4)*(23*n^4 + 279*n^3 + 941*n^2 + 599*n + 138)/1980. - Andrew Howroyd, Apr 05 2021
G.f.: x*(5 + 140*x + 665*x^2 + 746*x^3 + 224*x^4 + 14*x^5)/(1 - x)^14. - Stefano Spezia, Aug 19 2025

Terms a(10) and beyond from Andrew Howroyd, Apr 05 2021

A027777 a(n) = 2(n+1)binomial(n+2,4).

Original entry on oeis.org

6, 40, 150, 420, 980, 2016, 3780, 6600, 10890, 17160, 26026, 38220, 54600, 76160, 104040, 139536, 184110, 239400, 307230, 389620, 488796, 607200, 747500, 912600, 1105650, 1330056, 1589490, 1887900, 2229520, 2618880, 3060816, 3560480, 4123350, 4755240, 5462310

Offset: 2

Thi Ngoc Dinh (via R. K. Guy)

Number of 7-subsequences of [ 1, n ] with just 2 contiguous pairs.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Equals second right hand column of A163934. - Johannes W. Meijer, Oct 16 2009

Cf. A006411.

Table[2(n+1)Binomial[n+2,4],{n,2,35}]  (* Harvey P. Dale, Feb 03 2011 *)

G.f.: 2*(3+2x)*x^2/(1-x)^6.
a(n) = 2*A006411(n+1).
a(n) = C(n+1, 3)*C(n+2, 2) - Zerinvary Lajos, May 13 2005, corrected by R. J. Mathar, Feb 13 2016
From Amiram Eldar, Jan 28 2022: (Start)
Sum_{n>=2} 1/a(n) = Pi^2 - 29/3.
Sum_{n>=2} (-1)^n/a(n) = Pi^2/2 + 8*log(2) - 31/3. (End)

A214457 Table read by antidiagonals in which entry T(n,k) in row n and column k gives the number of possible rhombus tilings of an octagon with interior angles of 135 degrees and sequences of side lengths {n, k, 1, 1, n, k, 1, 1} (as the octagon is traversed), n,k in {1,2,3,...}.

Original entry on oeis.org

8, 20, 20, 40, 75, 40, 70, 210, 210, 70, 112, 490, 784, 490, 112, 168, 1008, 2352, 2352, 1008, 168, 240, 1890, 6048, 8820, 6048, 1890, 240, 330, 3300, 13860, 27720, 27720, 13860, 3300, 330, 440, 5445, 29040, 76230, 104544, 76230, 29040, 5445, 440

Offset: 1

L. Edson Jeffery, Jul 18 2012

Proof of the formula for T(n,k) is given in [Elnitsky].

So-called "generalized Narayana numbers" (see A145596), linking rhombus tilings of polygons to certain walks or paths through the square lattice.

			See [Jeffery]. T(1,1) = 8 because there are eight ways to tile the proposed octagon with rhombuses.
Table begins as
    8    20    40     70    112  ...
   20    75   210    490   1008  ...
   40   210   784   2352   6048  ...
   70   490  2352   8820  27720  ...
  112  1008  6048  27720  76230  ...
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)
Serge Elnitsky, Rhombic tilings of polygons and classes of reduced words in Coxeter groups (preprint), J. Combin. Theory Ser. A, Vol. 77, Issue 2, 193-221 (1997).
L. E. Jeffery, Worked out example for A214457(1,1)=8
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.

Empirical: T(1,n) = T(n,1) = 2*A000292(n+1); T(2,n) = T(n,2) = A006411(n+1); T(n,n) = A145600(n+1).

Table[2*(# + k + 1)!*(# + k + 2)!/(#!*k!*(# + 2)!*(k + 2)!) &[n - k + 1], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Feb 26 2024 *)

T(n,k) = 2*(n+k+1)!*(n+k+2)!/[n!*k!*(n+2)!*(k+2)!].

A347056 Triangle read by rows: T(n,k) = (n+1)(n+2)(k+3)*binomial(n,k)/6, 0 <= k <= n.

Original entry on oeis.org

1, 3, 4, 6, 16, 10, 10, 40, 50, 20, 15, 80, 150, 120, 35, 21, 140, 350, 420, 245, 56, 28, 224, 700, 1120, 980, 448, 84, 36, 336, 1260, 2520, 2940, 2016, 756, 120, 45, 480, 2100, 5040, 7350, 6720, 3780, 1200, 165, 55, 660, 3300, 9240, 16170, 18480, 13860, 6600, 1815, 220

Offset: 0

Luc Rousseau, Aug 14 2021

This triangle is T[3] in the sequence (T[p]) of triangles defined by: T[p](n,k) = (k+p)*(n+p-1)!/(k!*(n-k)!*p!) and T[0](0,0)=1.

Riordan triangle (1/(1-x)^3, x/(1-x)) with column k scaled with A000292(k+1) = binomial(k+3, 3), for k >= 0. - Wolfdieter Lang, Sep 30 2021

			T(6,2) = (6+1)*(6+2)*(2+3)*binomial(6,2)/6 = 7*8*5*15/6 = 700.
The triangle T begins:
n \ k  0   1    2     3     4     5     6     7     8    9  10 ...
0:     1
1:     3   4
2:     6  16   10
3:    10  40   50    20
4:    15  80  150   120    35
5:    21 140  350   420   245    56
6:    28 224  700  1120   980   448    84
7:    36 336 1260  2520  2940  2016   756   120
8:    45 480 2100  5040  7350  6720  3780  1200   165
9:    55 660 3300  9240 16170 18480 13860  6600  1815  220
10:   66 880 4950 15840 32340 44352 41580 26400 10890 2640 286
... - _Wolfdieter Lang_, Sep 30 2021

Luc Rousseau, Illustration: the A347056 triangle in a sequence of contiguous triangles.

Cf. A097805 (p=0), A103406 (p=1), A124932 (essentially p=2).
From Wolfdieter Lang, Sep 30 2021: (Start)
Columns (with leading zeros): A000217(n+1), 4*A000294, 10*A000332(n+2), 20*A000389(n+2), 35*A000579(n+2), 56*A000580(n+2), 84*A000581(n+2), 120*A000582(n+2), ...
Diagonals: A000292(k+1), A004320(k+1), 2*A006411(k+1), 10*A040977, ... (End)

T(p,n,k)=if(n==0&&p==0,1,((k+p)*(n+p-1)!)/(k!*(n-k)!*p!))
for(n=0,9,for(k=0,n,print1(T(3,n,k),", ")))

T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6.

G.f. column k: x^k*binomial(k+3, 3)/(1 - x)^(k+3), for k >= 0. - Wolfdieter Lang, Sep 30 2021

cp's OEIS Frontend

A342984 Triangle read by rows: T(n,k) is the number of nonseparable tree-rooted planar maps with n edges and k faces, n >= 0, k = 1..n+1.

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A006412 Number of nonseparable tree-rooted planar maps with n + 3 edges and 4 vertices.

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A006413 Number of nonseparable tree-rooted planar maps with n + 4 edges and 5 vertices.

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A027777 a(n) = 2*(n+1)*binomial(n+2,4).

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A214457 Table read by antidiagonals in which entry T(n,k) in row n and column k gives the number of possible rhombus tilings of an octagon with interior angles of 135 degrees and sequences of side lengths {n, k, 1, 1, n, k, 1, 1} (as the octagon is traversed), n,k in {1,2,3,...}.

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Mathematica

Formula

A347056 Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.

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Examples

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PARI

Formula

A027777 a(n) = 2(n+1)binomial(n+2,4).

A347056 Triangle read by rows: T(n,k) = (n+1)(n+2)(k+3)*binomial(n,k)/6, 0 <= k <= n.