A006414 -id:A006414 - cp's OEIS frontend

A006542 a(n) = binomial(n,3)*binomial(n-1,3)/4.

1, 10, 50, 175, 490, 1176, 2520, 4950, 9075, 15730, 26026, 41405, 63700, 95200, 138720, 197676, 276165, 379050, 512050, 681835, 896126, 1163800, 1495000, 1901250, 2395575, 2992626, 3708810, 4562425, 5573800, 6765440, 8162176, 9791320, 11682825, 13869450

Offset: 4

N. J. A. Sloane

Number of permutations of n+4 that avoid the pattern 132 and have exactly 3 descents. - Mike Zabrocki, Aug 26 2004
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 20 2005
a(n) = number of Dyck n-paths with exactly 4 peaks. - David Callan, Jul 03 2006
Six-dimensional figurate numbers for a hyperpyramid with pentagonal base. This corresponds to the sum(sum(sum(sum(1+sum(5*n))))) interpretation, see the Munafo webpage. - Robert Munafo, Jun 18 2009

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 166, no. 1).
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 238.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 4..200
Isaac Ahern and Sam Cook, Affine Symmetry Tensors in Minkowski Space, American Journal of Undergraduate Research, Volume 13, Issue 3, August 2016.
P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
Brandy Amanda Barnette, Counting Convex Sets on Products of Totally Ordered Sets, Masters Theses & Specialist Projects, Paper 1484, 2015.
V. E. Hoggatt, Jr., Letter to N. J. A. Sloane, Apr 1977
G. Kreweras, Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31.
G. Kreweras, Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31. [Annotated scanned copy]
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 6, 25.
Robert Munafo, C(n,3)C(n-1,3)/4
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

The expression binomial(m+n-1,n)^2-binomial(m+n,n+1)*binomial(m+n-2,n-1) for the values m = 2 through 14 produces the sequences A000012, A000217, A002415, A006542, A006857, A108679, A134288, A134289, A134290, A134291, A140925, A140935, A169937.
Cf. A000332, A000579, A001263, A002378, A004068, A005585, A005891, A006322, A006414, A047819, A107891, A114242.
Fourth column of the table of Narayana numbers A001263.
Apart from a scale factor, a column of A124428.

List([4..40], n-> n*(n-1)^2*(n-2)^2*(n-3)/144); # G. C. Greubel, Feb 24 2019

[ n*((n-1)*(n-2))^2*(n-3)/144 : n in [4..40] ]; // Wesley Ivan Hurt, Jun 17 2014

A006542:=-(1+3*z+z**2)/(z-1)**7; # conjectured by Simon Plouffe in his 1992 dissertation
A006542:=n->n*((n-1)*(n-2))^2*(n-3)/144; seq(A006542(n), n=4..40); # Wesley Ivan Hurt, Jun 17 2014

Table[Binomial[n, 3]*Binomial[n-1, 3]/4, {n, 4, 40}]

PARI
```
a(n)=n*((n-1)*(n-2))^2*(n-3)/144
```

[n*(n-1)^2*(n-2)^2*(n-3)/144 for n in (4..40)] # G. C. Greubel, Feb 24 2019

a(n) = C(n, 3)*C(n-1, 3)/4 = n*(n-1)^2*(n-2)^2*(n-3)/144.
a(n) = A000292(n-3)*A000292(n-2)/4.
E.g.f.: x^4*(6 + 6*x + x^2)*exp(x)/144. - Vladeta Jovovic, Jan 29 2003
a(n) = Sum(Sum(Sum(Sum(1 + Sum(5*n))))) = Sum (A006414). - Xavier Acloque, Oct 08 2003
a(n) = C(n, 6) + 3*C(n+1, 6) + C(n+2, 6). - Mike Zabrocki, Aug 26 2004
G.f.: x^4*(1 + 3*x + x^2)/(1-x)^7. - Emeric Deutsch, Jun 20 2005
a(n) = C(n-2, n-4)*C(n-1, n-3)*C(n, n-2)/18. - Zerinvary Lajos, Jul 29 2005
a(n) = C(n,4)*C(n,3)/n. - Mitch Harris, Jul 06 2006
a(n+2) = (1/4)*Sum_{1 <= x_1, x_2 <= n} x_1*x_2*(det V(x_1,x_2))^2 = (1/4)*Sum_{1 <= i,j <= n} i*j*(i-j)^2, where V(x_1,x_2) is the Vandermonde matrix of order 2. - Peter Bala, Sep 21 2007
a(n) = C(n-1,3)^2 - C(n-1,2)*C(n-1,4). - Gary Detlefs, Dec 05 2011
a(n) = A000292(A000217(n-1)) - A000217(A000292(n-1)). - Ivan N. Ianakiev, Jun 17 2014
a(n) = Product_{i=1..3} A002378(n-4+i)/A002378(i). - Bruno Berselli, Nov 12 2014 (Rewritten, Sep 01 2016.)
Sum_{n>=4} 1/a(n) = 238 - 24*Pi^2. - Jaume Oliver Lafont, Jul 10 2017
Sum_{n>=4} (-1)^n/a(n) = 134 - 192*log(2). - Amiram Eldar, Oct 19 2020
a(n) = A000332(n) + 5*A000579(n+1). - Yasser Arath Chavez Reyes, Aug 18 2024

Zabroki and Lajos formulas offset corrected by Gary Detlefs, Dec 05 2011

A006322 4-dimensional analog of centered polygonal numbers.

Original entry on oeis.org

1, 8, 31, 85, 190, 371, 658, 1086, 1695, 2530, 3641, 5083, 6916, 9205, 12020, 15436, 19533, 24396, 30115, 36785, 44506, 53383, 63526, 75050, 88075, 102726, 119133, 137431, 157760, 180265, 205096, 232408, 262361, 295120, 330855, 369741, 411958, 457691, 507130

Offset: 1

Albert Rich (Albert_Rich(AT)msn.com)

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
Partial sums give A006414. - L. Edson Jeffery, Dec 13 2011
Also the number of (w,x,y,z) with all terms in {1,...,n} and w<=x>=y<=z, see A211795. - Clark Kimberling, May 19 2012

			An illustration for a(5)=190: 5*(1+2+3+4+5)+4*(2+3+4+5)+3*(3+4+5)+2*(4+5)+1*(5) gives 75+56+36+18+5=190. - _J. M. Bergot_, Feb 13 2018

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 166, Table 10.4/I/4).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Plateau Polycubes and Lateral Area, arXiv:1811.05707 [math.CO], 2018. See Column 2 Table 2 p. 9.
Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]. See p. 31.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000330, A006414, A050446, A050447.

List([1..40], n->5*Binomial(n+2,4) + Binomial(n+1,2)); # Muniru A Asiru, Feb 13 2018

[n*(n+1)*(5*n^2 +5*n +2)/24: n in [1..40]]; // G. C. Greubel, Sep 02 2019

a:=n->5*binomial(n+2,4) + binomial(n+1,2): seq(a(n), n=1..40); # Muniru A Asiru, Feb 13 2018

Table[5*Binomial[n+2, 4] + Binomial[n+1, 2], {n, 40}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
CoefficientList[Series[(1+3x+x^2)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Jun 09 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,8,31,85,190},40] (* Harvey P. Dale, Sep 27 2016 *)

a(n)=n*(5*n^3+10*n^2+7*n+2)/24 \\ Charles R Greathouse IV, Dec 13 2011, corrected by Altug Alkan, Aug 15 2017

[n*(n+1)*(5*n^2 +5*n +2)/24 for n in (1..40)] # G. C. Greubel, Sep 02 2019

a(n) = 5*C(n+2,4) + C(n+1,2) = (C(5*n+4,4) - 1)/5^3 = n*(n+1)*(5*n^2 + 5*n + 2)/24.
a(n) = (((n+1)^5-n^5) - ((n+1)^3-n^3))/24. - Xavier Acloque, Jan 14 2003, corrected by Eric Rowland, Aug 15 2017
Partial sums of A004068. - Xavier Acloque, Jan 15 2003
G.f.: x*(1+3*x+x^2)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = Sum_{i=1..n} Sum_{j=1..n} i * min(i,j). - Enrique Pérez Herrero, Jan 30 2013
a(n) = A000537(n) - A000332(n+2). - J. M. Bergot, Jun 03 2017
Sum_{n>=1} 1/a(n) = 42 - 4*sqrt(15)*Pi*tanh(sqrt(3/5)*Pi/2). - Amiram Eldar, May 28 2022
From Elmo R. Oliveira, Aug 14 2025: (Start)
E.g.f.: exp(x)*x*(2 + x)*(12 + 30*x + 5*x^2)/24.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)

A241619 T(n,k)=Number of length n+2 0..k arrays with no consecutive three elements summing to more than k.

Original entry on oeis.org

4, 10, 6, 20, 20, 9, 35, 50, 40, 13, 56, 105, 125, 76, 19, 84, 196, 315, 295, 147, 28, 120, 336, 686, 889, 711, 287, 41, 165, 540, 1344, 2254, 2567, 1730, 556, 60, 220, 825, 2430, 5040, 7586, 7483, 4175, 1077, 88, 286, 1210, 4125, 10242, 19374, 25774, 21631, 10077

Offset: 1

R. H. Hardin, Apr 26 2014

Table starts
...4...10....20.....35......56.......84......120.......165.......220........286
...6...20....50....105.....196......336......540.......825......1210.......1716
...9...40...125....315.....686.....1344.....2430......4125......6655......10296
..13...76...295....889....2254.....5040....10242.....19305.....34243......57772
..19..147...711...2567....7586....19374....44274.....92697....180829.....332761
..28..287..1730...7483...25774....75180...193194....449295....963886....1934647
..41..556..4175..21631...86828...289248...835812...2159025...5093737...11151140
..60.1077.10077..62547..292621..1113348..3617703..10380183..26932543...64309245
..88.2091.24377.181255..988303..4294574.15692003..50011289.142701909..371651553
.129.4057.58928.524877.3335451.16553380.68014233.240772037.755538278.2146210209

			Some solutions for n=5 k=4
..1....0....2....1....0....2....0....1....0....0....0....2....1....0....1....2
..0....0....1....3....0....0....4....2....1....0....1....1....3....0....0....1
..0....3....0....0....0....1....0....1....3....0....0....0....0....2....1....0
..0....0....0....1....2....0....0....0....0....0....1....0....1....1....1....2
..2....0....3....0....0....2....0....0....0....0....2....1....0....0....0....0
..0....1....0....1....2....1....2....0....1....0....0....1....1....0....3....1
..0....0....0....1....0....0....2....4....2....2....0....0....2....0....0....0

R. H. Hardin, Table of n, a(n) for n = 1..10011

Column 1 is A000930(n+4)
Row 1 is A000292(n+1)
Row 2 is A002415(n+2)
Row 3 is A006414
Row 4 is A114244

for m from 1 to 12 do
  r:= [seq(seq([i,j],j=0..m-i),i=0..m)];
  T[m]:= Matrix((m+1)*(m+2)/2,(m+1)*(m+2)/2, proc(i, j) if r[i][1]=r[j][2] and r[i][1]+r[i][2]+r[j][1]<=m then 1 else 0 fi end proc):
  U[m,0]:= Vector((m+1)*(m+2)/2,1);
od:
R:= NULL:
for i from 2 to 12 do
  for j from 1 to i-1 do
    U[i-j,j]:= T[i-j] . U[i-j,j-1];
    R:= R, convert(U[i-j,j],`+`)
od od:
R; # Robert Israel, Sep 04 2019

Empirical for column k, apparently a recurrence of order (k+1)*(k+2)/2:
k=1: a(n) = a(n-1) +a(n-3)
k=2: a(n) = a(n-1) +a(n-2) +2*a(n-3) -a(n-5) -a(n-6)
k=3: a(n) = 2*a(n-1) +4*a(n-3) -3*a(n-4) -a(n-5) -3*a(n-6) +2*a(n-7) +a(n-9) -a(n-10)
k=4: [order 15]
k=5: [order 21]
k=6: [order 28]
k=7: [order 36]
k=8: [order 45]
k=9: [order 55]
k=10: [order 66]
k=11: [order 78]
k=12: [order 91]
Empirical for row n, apparently a polynomial of degree n+2:
n=1: a(n) = (1/6)*n^3 + 1*n^2 + (11/6)*n + 1
n=2: a(n) = (1/12)*n^4 + (2/3)*n^3 + (23/12)*n^2 + (7/3)*n + 1
n=3: a(n) = (1/24)*n^5 + (5/12)*n^4 + (13/8)*n^3 + (37/12)*n^2 + (17/6)*n + 1
n=4: [polynomial of degree 6]
n=5: [polynomial of degree 7]
n=6: [polynomial of degree 8]
n=7: [polynomial of degree 9]
From Robert Israel, Sep 04 2019: (Start)
Column k satisfies a recurrence of order (k+1)*(k+2)/2, since a(n)=e^T T^n e where T is a (k+1)*(k+2)/2 matrix and e the vector of all 1's (see proofs at A241615 and A241618).
Row n is the Ehrhart polynomial of degree n+2 corresponding to the polytope {(x(1),...,x(n+2)): all x(i)>=0, x(i)+x(i+1)+x(i+2)<=1 for i=1..n}, whose vertices have all entries in {0,1}. (End)

A143945 Wiener index of the grid P_n x P_n, where P_n is the path graph on n vertices.

Original entry on oeis.org

0, 8, 72, 320, 1000, 2520, 5488, 10752, 19440, 33000, 53240, 82368, 123032, 178360, 252000, 348160, 471648, 627912, 823080, 1064000, 1358280, 1714328, 2141392, 2649600, 3250000, 3954600, 4776408, 5729472, 6828920, 8091000, 9533120, 11173888, 13033152, 15132040

Offset: 1

Emeric Deutsch, Sep 20 2008

The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.

			a(2)=8 because in P_2 x P_2 (a square) there are 4 distances equal to 1 and 2 distances equal to 2 (4*1 + 2*2 = 8).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000 (corrected by Ray Chandler, Jan 19 2019)
Dragan Stevanovic, Hosoya polynomial of composite graphs, Discrete Math., Vol. 235, No. 1-3 (2001), pp. 237-244.
B.-Y. Yang and Y.-N. Yeh, Wiener polynomials of some chemically interesting graphs, International Journal of Quantum Chemistry, Vol. 99 (2004), pp. 80-91.
Y.-N. Yeh and I. Gutman, On the sum of all distances in composite graphs, Discrete Math., Vol. 135, No. 1-3 (1994), pp. 359-365.
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Main diagonal of A143368.

Cf. A006414, A143944, A245828, A192828.

[n^3*(n^2-1)/3: n in [1..40]]; // Vincenzo Librandi, Feb 08 2014

Maple
```
seq((1/3)*n^3*(n^2-1),n=1..33);
```

Table[n^3 (n^2 - 1)/3, {n, 40}] (* Harvey P. Dale, Feb 07 2014 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 8, 72, 320, 1000, 2520}, 30] (* Harvey P. Dale, Feb 07 2014 *)
CoefficientList[Series[8 x (1 + 3 x + x^2)/(x - 1)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 08 2014 *)

a(n)=n^3*(n^2-1)/3 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = Sum_{k=1..2n-2} k*A143944(n,k).
a(n) = n^3*(n^2-1)/3.
a(n) = 8*A006414(n-2). G.f.: 8*x^2*(1+3*x+x^2)/(x-1)^6. - R. J. Mathar, Sep 15 2010
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(2)=8, a(3)=72, a(4)=320, a(5)=1000, a(6)=2520, a(7)=5488. - Harvey P. Dale, Feb 07 2014
From Amiram Eldar, Jan 09 2022: (Start)
Sum_{n>=2} 1/a(n) = 15/4 - 3*zeta(3).
Sum_{n>=2} (-1)^n/a(n) = 9*zeta(3)/4 + 6*log(2) - 27/4. (End)

A119308 Triangle for first differences of Catalan numbers.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 14, 9, 1, 5, 30, 40, 14, 1, 6, 55, 125, 90, 20, 1, 7, 91, 315, 385, 175, 27, 1, 8, 140, 686, 1274, 980, 308, 35, 1, 9, 204, 1344, 3528, 4116, 2184, 504, 44, 1, 10, 285, 2430, 8568, 14112, 11340, 4410, 780, 54, 1, 11, 385, 4125

Offset: 0

Paul Barry, May 13 2006

Row sums are A000245(n+1). Columns include A000330, A006414, as well as certain Kekulé numbers (A114242, A108647, ...).
Diagonal sums are A188460.
Coefficient array of the second column of the inverse of the Riordan array ((1+r*x)/(1+(r+1)x+r*x^2), x/(1+(r+1)x+r*x^2)). - Paul Barry, Apr 01 2011

			Triangle begins:
1;
2,   1;
3,   5,    1;
4,  14,    9,    1;
5,  30,   40,   14,    1;
6,  55,  125,   90,   20,    1;
7,  91,  315,  385,  175,   27,   1;
8, 140,  686, 1274,  980,  308,  35,  1;
9, 204, 1344, 3528, 4116, 2184, 504, 44, 1;

Indranil Ghosh, Rows 0..100, flattened
Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study (2023) Vol. 56, No. 1, 1-17.

Cf. A000108, A001263.

a[k_,j_]:=If[k<=j,Binomial[j+1,2(j-k)]*CatalanNumber[j-k],0];
Flatten[Table[Sum[Binomial[n,j]*a[k,j],{j,0,n}],{n,0,10},{k,0,n}]] (* Indranil Ghosh, Mar 03 2017 *)

catalan(n)=binomial(2*n,n)/(n+1);
a(k,j)=if (k<=j,binomial(j+1,2*(j-k))*catalan(j-k),0);
tabl(nn)={for (n=0, nn, for (k=0, n, print1(sum(j=0, n, binomial(n,j)*a(k,j)),", "););print(););};
tabl(10); \\ Indranil Ghosh, Mar 03 2017

T(n,k) = Sum_{j=0..n} C(n,j)*[k<=j]*C(j+1,k+1)*C(k+1,j-k)/(j-k+1).
Column k has g.f.: sum{j=0..k, C(k,j)*C(k+1,j)x^j/(j+1)}*x^k/(1-x)^(2(k+1)).
T(n,k) = Sum_{j=0..n} C(n,j)*if(k<=j, C(j+1,2(j-k))*A000108(j-k), 0).
G.f.: (((x-1)*sqrt(x^2*y^2+(-2*x^2-2*x)*y+x^2-2*x+1)+(-x^2-x)*y+x^2-2*x+1)/(2*x^3*y^2)). - Vladimir Kruchinin, Nov 15 2020
T(n,k) = C(n+1,k)*(2*C(n+1,k+2)+C(n+1,k+1))/(n+1). - Vladimir Kruchinin, Nov 16 2020

A006415 Number of nonseparable toroidal tree-rooted maps with n + 3 edges and n + 1 vertices.

Original entry on oeis.org

4, 104, 1020, 6092, 26670, 94128, 283338, 754380, 1821534, 4061200

Offset: 0

N. J. A. Sloane

The number of faces is 2. - Andrew Howroyd, Apr 05 2021

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

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.

Cf. A006414, A006441.

Conjecture: a(n) = 4 * binomial(n + 4, n) + 84 * binomial(n + 4, n - 1) + 456 * binomial(n + 4, n - 2) + 996 * binomial(n + 4, n - 3) + 950 * binomial(n + 4, n - 4) + 330 * binomial(n + 5, n - 5). - Sean A. Irvine, Apr 03 2017

A006441 Number of nonseparable toroidal tree-rooted maps with n + 4 edges and n + 1 vertices.

Original entry on oeis.org

10, 595, 11010, 111650, 773640, 4104225, 17838730, 66390610, 218140650

Offset: 0

N. J. A. Sloane

The number of faces is 3. - Andrew Howroyd, Apr 05 2021

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

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.

Cf. A006414, A006415.

Name clarified by Andrew Howroyd, Apr 05 2021

A071910 a(n) = t(n)t(n+1)t(n+2), where t() are the triangular numbers.

Original entry on oeis.org

0, 18, 180, 900, 3150, 8820, 21168, 45360, 89100, 163350, 283140, 468468, 745290, 1146600, 1713600, 2496960, 3558168, 4970970, 6822900, 9216900, 12273030, 16130268, 20948400, 26910000, 34222500, 43120350, 53867268, 66758580, 82123650, 100328400, 121777920

Offset: 0

N. J. A. Sloane, Jun 13 2002

nonn

a(n) is also the number of three-dimensional cage assemblies such that the assembly is not a cube. See also A052149 for the two-dimensional version and to A059827 for the non-exclusive version. - Alejandro Rodriguez, Oct 20 2020

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A006542, (first differences of a(n) /18) A006414, (second differences of a(n) /18) A006322, (third differences of a(n) /18) A004068, (fourth differences of a(n) /18) A005891, (fifth differences of a(n) /18) A008706.

Join[{0},Times@@@Partition[Accumulate[Range[40]],3,1]] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,18,180,900,3150,8820,21168},40] (* Harvey P. Dale, Aug 08 2025 *)

t(n) = n*(n+1)/2;
a(n) = t(n)*t(n+1)*t(n+2); \\ Michel Marcus, Oct 21 2015

a(n) = 18*A006542(n+3). - Vladeta Jovovic, Jun 14 2002
G.f.: 18*x*(1+3*x+x^2)/(1-x)^7. - Vladeta Jovovic, Jun 14 2002
a(n) = ((n+1)*(n+2))^3/8 - Sum_{i=1..n+1} i^3. - Jon Perry, Feb 13 2004
a(n) = C(2+n, n)*C(3+n, 1+n)*C(4+n, 2+n). - Zerinvary Lajos, Jul 29 2005
a(n) = A059827(n+1) - A000537(n+1). - Michel Marcus, Oct 21 2015

A107984 Triangle read by rows: T(n,k) = (k+1)(n+2)(2n-k+3)*(n-k+1)/6 for 0 <= k <= n.

Original entry on oeis.org

1, 5, 4, 14, 16, 10, 30, 40, 35, 20, 55, 80, 81, 64, 35, 91, 140, 154, 140, 105, 56, 140, 224, 260, 256, 220, 160, 84, 204, 336, 405, 420, 390, 324, 231, 120, 285, 480, 595, 640, 625, 560, 455, 320, 165, 385, 660, 836, 924, 935, 880, 770, 616, 429, 220, 506, 880

Offset: 0

Emeric Deutsch, Jun 12 2005

Kekulé numbers for certain benzenoids. Column 0 yields A000330. Main diagonal yields A000292. Row sums yield A006414.

			Triangle begins:
   1;
   5,  4;
  14, 16, 10;
  30, 40, 35, 20;

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 237, K{B(n,3,-l)}).

Cf. A000330, A000292, A006414.

T:=proc(n,k) if k<=n then (k+1)*(n+2)*(2*n-k+3)*(n-k+1)/6 else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

A107984_row(n)=vector(n+1,k,k*(2*n-k+4)*(n-k+2))*(n+2)/6 \\ M. F. Hasler, Dec 26 2016

T(n-2,k-1) = n*(2*n-k)*(n-k)*k/6. - M. F. Hasler, Dec 26 2016

G.f.: (1 + x - 4*x^2*y + x^3*y^2 + x^4*y^2)/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Jul 11 2025

A107972 Triangle read by rows: T(n,k) = (k+1)(k+2)(n+2)(3n-2k+3)/12 for 0<=k<=n.

Original entry on oeis.org

1, 3, 6, 6, 14, 20, 10, 25, 40, 50, 15, 39, 66, 90, 105, 21, 56, 98, 140, 175, 196, 28, 76, 136, 200, 260, 308, 336, 36, 99, 180, 270, 360, 441, 504, 540, 45, 125, 230, 350, 475, 595, 700, 780, 825, 55, 154, 286, 440, 605, 770, 924, 1056, 1155, 1210, 66, 186, 348

Offset: 0

Emeric Deutsch, Jun 12 2005

Kekulé numbers for certain benzenoids. Column 0 yields the triangular numbers (A000217). Row sums yield A006414. T(n,n) = A002415(n+2).

			Triangle begins:
1;
3,6;
6,14,20;
10,25,40,50;

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 237; K{B(n,2,l)}).

Cf. A000217, A006414, A002415.

T:=proc(n,k) if k<=n then (k+1)*(k+2)*(n+2)*(3*n-2*k+3)/12 else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

cp's OEIS Frontend

A006542 a(n) = binomial(n,3)*binomial(n-1,3)/4.

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A006322 4-dimensional analog of centered polygonal numbers.

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A241619 T(n,k)=Number of length n+2 0..k arrays with no consecutive three elements summing to more than k.

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A143945 Wiener index of the grid P_n x P_n, where P_n is the path graph on n vertices.

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A119308 Triangle for first differences of Catalan numbers.

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A006415 Number of nonseparable toroidal tree-rooted maps with n + 3 edges and n + 1 vertices.

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A071910 a(n) = t(n)t(n+1)t(n+2), where t() are the triangular numbers.

A107984 Triangle read by rows: T(n,k) = (k+1)(n+2)(2n-k+3)*(n-k+1)/6 for 0 <= k <= n.