A027789 -id:A027789 - cp's OEIS frontend

A006470 Number of tree-rooted planar maps with 3 faces and n vertices and no isthmuses.

2, 15, 60, 175, 420, 882, 1680, 2970, 4950, 7865, 12012, 17745, 25480, 35700, 48960, 65892, 87210, 113715, 146300, 185955, 233772, 290950, 358800, 438750, 532350, 641277, 767340, 912485, 1078800, 1268520, 1484032, 1727880, 2002770, 2311575, 2657340, 3043287, 3472820, 3949530, 4477200, 5059810, 5701542, 6406785, 7180140

Offset: 1

N. J. A. Sloane

a(n) is the number of ordered rooted trees with n+3 non-root nodes that have 3 leaves; see A108838. - Joerg Arndt, Aug 18 2014

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

Vincenzo Librandi, 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, Vol. 18, No. 3 (1975), pp. 222-259.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Column 3 of A342987.

Cf. A027789, A108838.

[(n+1)*Binomial(n+3, 4): n in [1..30]]; // Vincenzo Librandi, Jun 09 2013

A006470:=n->(n+1)*binomial(n+3,4): seq(A006470(n), n=1..50); # Wesley Ivan Hurt, May 02 2015

Table[n (n + 1)^2 (n + 2) (n + 3) / 24, {n, 50}] (* Vincenzo Librandi, May 03 2015 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{2,15,60,175,420,882},50] (* Harvey P. Dale, Jul 18 2024 *)

a(n) = (n+1)*binomial(n+3, 4).
a(n) = A027789(n)/2.
From Zerinvary Lajos, Dec 14 2005: (Start)
a(n) = binomial(n+2, 2)*binomial(n+4, 3)/2;
G.f.: x*(2+3*x)/(1-x)^6. (End)
From Wesley Ivan Hurt, May 02 2015: (Start)
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = n*(n+1)^2*(n+2)*(n+3)/24. (End)
Sum_{n>=1} 1/a(n) = 61/3 - 2*Pi^2. - Jaume Oliver Lafont, Jul 15 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2 - 16*log(2) + 5/3. - Amiram Eldar, Jan 28 2022

Name clarified by Andrew Howroyd, Apr 03 2021

A289792 Number of 4-cycles in the n-tetrahedral graph.

Original entry on oeis.org

0, 0, 0, 0, 90, 540, 1995, 5775, 14280, 31500, 63630, 119790, 212850, 360360, 585585, 918645, 1397760, 2070600, 2995740, 4244220, 5901210, 8067780, 10862775, 14424795, 18914280, 24515700, 31439850, 39926250, 50245650, 62702640, 77638365, 95433345, 116510400

Offset: 1

Eric W. Weisstein, Jul 12 2017

Extended to a(1)-a(5) using the formula.

Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Tetrahedral Graph
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A027789 (3-cycles), A289793 (5-cycles), A289794 (6-cycles).

Table[Binomial[n - 1, 4] (210 - 41 n + 7 n^2)/2, {n, 20}]
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 0, 0, 90, 540, 1995}, 20]
CoefficientList[Series[-((15 x^4 (6 - 6 x + 7 x^2))/(-1 + x)^7), {x, 0, 20}], x]

a(n) = binomial(n - 1, 4) * (210 - 41*n + 7*n^2)/2.
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7).
G.f.: (-15*x^5*(6 - 6*x + 7*x^2))/(-1 + x)^7.

A289793 Number of 5-cycles in the n-tetrahedral graph.

Original entry on oeis.org

0, 0, 0, 0, 312, 3024, 14868, 51744, 145152, 350784, 759528, 1511136, 2810808, 4948944, 8324316, 13470912, 21088704, 32078592, 47581776, 69023808, 98163576, 137147472, 188568996, 255534048, 341732160, 451513920, 589974840, 763045920, 977591160, 1241512272

Offset: 1

Eric W. Weisstein, Jul 12 2017

Extended to a(1)-a(5) using the formula.

Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Tetrahedral Graph
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A027789 (3-cycles), A289792 (4-cycles), A289794 (6-cycles).

Table[6 Binomial[n, 5] (-78 + 21 n + n^2), {n, 20}]
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 0, 0, 0, 312, 3024, 14868, 51744}, 20]
CoefficientList[Series[-((12 x^4 (-26 - 44 x + 49 x^2))/(-1 + x)^8), {x, 0, 20}], x]

a(n) = 6*binomial(n, 5)*(-78 + 21*n + n^2).
a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8).
G.f.: (-12*x^5*(-26 - 44*x + 49*x^2))/(-1 + x)^8.

A289794 Number of 6-cycles in the n-tetrahedral graph.

Original entry on oeis.org

0, 0, 0, 0, 920, 17760, 122640, 537040, 1794240, 4994640, 12178320, 26840880, 54620280, 104184080, 188348160, 325459680, 541078720, 869994720, 1358615520, 2067768480, 3075954840, 4483100160, 6414845360, 9027424560, 12513177600, 17106746800, 23092009200

Offset: 1

Eric W. Weisstein, Jul 12 2017

Extended to a(1)-a(5) using the formula.

Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Tetrahedral Graph
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A027789 (3-cycles), A289792 (4-cycles), A289793 (5-cycles).

Table[5 Binomial[n, 5] (454 - 409 n + 66 n^2 + n^3), {n, 20}]
LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {0, 0, 0, 0, 920, 17760, 122640, 537040, 1794240}, 20]
CoefficientList[Series[(40 x^4 (-23 - 237 x + 102 x^2 + 116 x^3))/(-1 + x)^9, {x, 0, 20}], x]

a(n) = 5*binomial(n, 5)*(454 - 409*n + 66*n^2 + n^3).
a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9).
G.f.: (40*x^5*(-23 - 237*x + 102*x^2 + 116*x^3))/(-1 + x)^9.

A293616 Array of generalized Eulerian number triangles read by ascending antidiagonals, with m >= 0, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 0, 1, 0, 1, 10, 0, 7, 1, 0, 1, 15, 0, 25, 4, 0, 0, 1, 21, 0, 65, 10, 0, 1, 0, 1, 28, 0, 140, 20, 0, 15, 4, 0, 1, 36, 0, 266, 35, 0, 90, 30, 1, 0, 1, 45, 0, 462, 56, 0, 350, 120, 5, 0, 0, 1, 55, 0, 750, 84, 0, 1050, 350, 15, 0, 1, 0

Offset: 0

Peter Luschny, Oct 14 2017

			Array starts:
m\j| 0   1  2     3    4  5       6       7    8  9      10      11      12
---|----------------------------------------------------------------------------
m=0| 1,  0, 0,    0,   0, 0,      0,      0,   0, 0,      0,      0,      0, ...
m=1| 1,  1, 0,    1,   1, 0,      1,      4,   1, 0,      1,     11,     11, ...
m=2| 1,  3, 0,    7,   4, 0,     15,     30,   5, 0,     31,    146,     91, ...
m=3| 1,  6, 0,   25,  10, 0,     90,    120,  15, 0,    301,    896,    406, ...
m=4| 1, 10, 0,   65,  20, 0,    350,    350,  35, 0,   1701,   3696,   1316, ...
m=5| 1, 15, 0,  140,  35, 0,   1050,    840,  70, 0,   6951,  11886,   3486, ...
m=6| 1, 21, 0,  266,  56, 0,   2646,   1764, 126, 0,  22827,  32172,   8022, ...
m=7| 1, 28, 0,  462,  84, 0,   5880,   3360, 210, 0,  63987,  76692,  16632, ...
m=8| 1, 36, 0,  750, 120, 0,  11880,   5940, 330, 0, 159027, 165792,  31812, ...
m=9| 1, 45, 0, 1155, 165, 0,  22275,   9900, 495, 0, 359502, 331617,  57057, ...
   A000217, A001296,A000292,A001297,A027789,A000332,A001298,A293610,A293611, ...
.
m\j| ...    13  14      15       16       17      18      19 20
---|----------------------------------------------------------------
m=0| ...,    0, 0,       0,       0,       0,      0,      0, 0, ...  [A000007]
m=1| ...,    1, 0,       1,      26,      66,     26,      1, 0, ...  [A173018]
m=2| ...,    6, 0,      63,     588,     868,    238,      7, 0, ...  [A062253]
m=3| ...,   21, 0,     966,    5376,    5586,   1176,     28, 0, ...  [A062254]
m=4| ...,   56, 0,    7770,   30660,   24570,   4200,     84, 0, ...  [A062255]
m=5| ...,  126, 0,   42525,  129780,   84630,  12180,    210, 0, ...
m=6| ...,  252, 0,  179487,  446292,  245322,  30492,    462, 0, ...
m=7| ...,  462, 0,  627396, 1315776,  625086,  68376,    924, 0, ...
m=8| ...,  792, 0, 1899612, 3444012, 1440582, 140712,   1716, 0, ...
m=9| ..., 1287, 0, 5135130, 8198190, 3063060, 270270,   3003, 0, ...
          A000389, A112494, A293612, A293613,A293614,A000579.
.
The parameter m runs over the triangles and j indexes the triangles by reading them by rows. Let T(m, n) denote the row [T(m, n, k) for 0 <= k <= n] and T(m) denote the triangle [T(m, n) for n >= 0]. Then for instance T(2) is the triangle A062253, T(4, 2) is row 2 of A062255 (which is [65, 20, 0]) and T(4, 2, 1) = 20.

Eric Weisstein's World of Mathematics, Nielsen Generalized Polylogarithm.

A000217(n) = T(n, 1, 0), A001296(n) = T(n, 2, 0), A000292(n) = T(n, 2, 1),
A001297(n) = T(n, 3, 0), A027789(n) = T(n, 3, 1), A000332(n) = T(n, 3, 2),
A001298(n) = T(n, 4, 0), A293610(n) = T(n, 4, 1), A293611(n) = T(n, 4, 2),
A000389(n) = T(n, 4, 3), A112494(n) = T(n, 5, 0), A293612(n) = T(n, 5, 1),
A293613(n) = T(n, 5, 2), A293614(n) = T(n, 5, 3), A000579(n) = T(n, 5, 4),
A144969(n) = T(n, 6, 0), A000580(n) = T(n, 6, 5), A000295(n) = T(1, n, 1),
A000460(n) = T(1, n, 2), A000498(n) = T(1, n, 3), A000505(n) = T(1, n, 4),
A000514(n) = T(1, n, 5), A001243(n) = T(1, n, 6), A001244(n) = T(1, n, 7),
A126646(n) = T(2, n, 0), A007820(n) = T(n, n, 0).
Cf. A173018, A062253, A062254, A062255, A293609.

A293616 := proc(m, n, k) option remember:
if m = 0 then m^n elif k < 0 or k > n then 0 elif n = 0 then 1 else
(k+m)*A293616(m,n-1,k) + (n-k)*A293616(m,n-1,k-1) + A293616(m-1,n,k) fi end:
for m in [$0..4] do for n in [$0..6] do print(seq(A293616(m, n, k), k=0..n)) od od;
# Sample uses:
A001298 := n -> A293616(n, 4, 0): A293614 := n -> A293616(n, 5, 3):
# Flatten:
a := proc(n) local w; w := proc(k) local t, s; t := 1; s := 1;
while t <= k do s := s + 1; t := t + s od; [s - 1, s - t + k] end:
seq(A293616(n - k, w(k)[1], w(k)[2]), k=0..n) end: seq(a(n), n = 0..11);

GenEulerianRow[0, n_] := Table[If[n==0 && j==0,1,0], {j,0,n}];
GenEulerianRow[m_, n_] := If[n==0,{1},Join[CoefficientList[x^(-m) (1 - x)^(n+m)
    PolyLog[-n-m, m, x] /. Log[1-x] -> 0, x], {0}]];
(* Sample use: *)
A173018Row[n_] := GenEulerianRow[1, n]; Table[A173018Row[n], {n, 0, 6}]

T(m, n, k) = (k + m)*T(m, n-1, k) + (n - k)*T(m, n-1, k-1) + T(m-1, n, k) with boundary conditions T(0, n, k) = 0^n; T(m, n, k) = 0 if k < 0 or k > n; and T(m, 0, k) = 0^k.

Let h(m, n) = x^(-m)*(1 - x)^(n + m)*PolyLog(-n - m, m, x) and p(m, n) the polynomial given by the expansion of h(m, n) after replacing log(1 - x) by 0. Then T(m, n, k) is the k-th coefficient of p(m, n) for 0 <= k < n.

A118963 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that have k double rises (n >= 1, k >= 0).

Original entry on oeis.org

2, 3, 3, 4, 12, 4, 5, 30, 30, 5, 6, 60, 120, 60, 6, 7, 105, 350, 350, 105, 7, 8, 168, 840, 1400, 840, 168, 8, 9, 252, 1764, 4410, 4410, 1764, 252, 9, 10, 360, 3360, 11760, 17640, 11760, 3360, 360, 10, 11, 495, 5940, 27720, 58212, 58212, 27720, 5940, 495, 11, 12

Offset: 1

Emeric Deutsch, May 07 2006

A Grand Dyck path of semilength n is a path in the half-plane x >= 0, starting at (0,0), ending at (2n,0) and consisting of steps u = (1,1) and d = (1,-1); a double rise in a Grand Dyck path is an occurrence of uu in the path.
For double rises only above the x-axis see A118964.
This is the triangle of Narayana with row n multiplied by n + 1. - Peter Luschny, May 02 2022

			T(3,2)=4 because we have uuuddd, duuudd, dduuud and ddduuu.
Triangle begins:
  2;
  3,    3;
  4,   12,    4;
  5,   30,   30,    5;
  6,   60,  120,   60,    6;
  7,  105,  350,  350,  105,    7;
  8,  168,  840, 1400,  840,  168,    8;
  9,  252, 1764, 4410, 4410, 1764,  252,    9;

Cf. A000917, A000984 (row sums), A027480, A027789, A118964, A001263.

r:=(1-z-t*z-sqrt(z^2*t^2-2*z^2*t-2*z*t+z^2-2*z+1))/2/t/z: G:=(1+r)^2/(1-t*r^2)-1: Gser:=simplify(series(G,z=0,15)): for n from 1 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form
for n from 0 to 10 do seq(binomial(n,i)*binomial(n+2,n+1-i), i=0..n ); od; # Zerinvary Lajos, Nov 03 2006

T(n,1) = n(n^2 - 1)/2 (A027480).
T(n,2) = (n+1)n(n-1)^2*(n-2)/12 (A027789).
T(n,k) = ((n+1)/n)*binomial(n,k)*binomial(n,k+1).
Sum_{k>=0} k*T(n,k) = (2n-1)!/(n!(n-2)!) (A000917).
G.f.: G(t,z) = (1+r)^2/(1 - tr^2) - 1, where r = r(t,z) is the Narayana function, defined by (1+r)(1+tr)z = r, r(t,0) = 0. More generally, the g.f. H = H(t,s,u,z), where t,s and u mark double rises above, below and on the x-axis, respectively, is H = (1 + r(s,z))/(1 - z(1 + tr(t,z))(1 + ur(s,z))).
Row n is given by seq(binomial(n, k)*binomial(n+2, n+1-k), k=0..n). - Zerinvary Lajos, Nov 03 2006
T(n,k)/(n+1) = A001263(n,k). - Peter Luschny, May 02 2022

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A006470 Number of tree-rooted planar maps with 3 faces and n vertices and no isthmuses.

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A289792 Number of 4-cycles in the n-tetrahedral graph.

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A289793 Number of 5-cycles in the n-tetrahedral graph.

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A289794 Number of 6-cycles in the n-tetrahedral graph.

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A293616 Array of generalized Eulerian number triangles read by ascending antidiagonals, with m >= 0, n >= 0 and 0 <= k <= n.

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A118963 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that have k double rises (n >= 1, k >= 0).

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