A114242 -id:A114242 - 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

A006857 a(n) = binomial(n+5,5) * binomial(n+5,4)/(n+5).

Original entry on oeis.org

1, 15, 105, 490, 1764, 5292, 13860, 32670, 70785, 143143, 273273, 496860, 866320, 1456560, 2372112, 3755844, 5799465, 8756055, 12954865, 18818646, 26883780, 37823500, 52474500, 71867250, 97260345, 130179231, 172459665, 226296280, 294296640, 379541184

Offset: 0

Simon Plouffe

Number of permutations of n+5 that avoid the pattern 132 and have exactly 4 descents.
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
Partial sums of A114242. - Peter Bala, Sep 21 2007
Dimensions of certain Lie algebra (see reference for precise definition).

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 167-169, Table 10.5/II/1).
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 239.
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 = 0..1000
Brandy Amanda Barnette, Counting Convex Sets on Products of Totally Ordered Sets, Masters Theses & Specialist Projects, Paper 1484, 2015.
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]
J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), 143-179. [Th. 7.3, case a=4]
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 p. 25.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

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.
5th column of the table of Narayana numbers A001263.
Cf. A002378, A114242.

A006857:= func< n | Binomial(n+4,3)*Binomial(n+5,5)/4 >;
[A006857(n): n in [0..40]]; // G. C. Greubel, Mar 12 2025

a:=n->(n+1)*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)/2880: seq(a(n),n=0..38); # Emeric Deutsch, Nov 18 2005

Table[Binomial[n+5,5] * Binomial[n+5,4]/(n+5), {n, 0, 50}] (* T. D. Noe, May 29 2012 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,15,105,490,1764,5292,13860,32670,70785},40] (* Harvey P. Dale, Oct 19 2024 *)

a(n) = binomial(n+5,5) * binomial(n+5,4)/(n+5) \\ Charles R Greathouse IV, Jun 11 2015

Vec((1+6*x+6*x^2+x^3)/(1-x)^9 + O(x^99)) \\ Altug Alkan, Sep 01 2016

def A006857(n): return binomial(n+4,3)*binomial(n+5,5)//4
print([A006857(n) for n in range(41)]) # G. C. Greubel, Mar 12 2025

From - Vladeta Jovovic, Jan 29 2003: (Start)
a(n) = (n+4)!*(n+5)!/(2880*n!*(n+1)!).
E.g.f.: 1/2880*(2880 + 40320*x + 109440*x^2 + 105120*x^3 + 45000*x^4 + 9504*x^5 + 1016*x^6 + 52*x^7 + x^8)*exp(x). (End)
From Mike Zabrocki, Aug 26 2004: (Start)
a(n) = C(n+5,8) + 6*C(n+6,8) + 6*C(n+7,8) + C(n+8,8).
a(n) = C(n+4,4)*C(n+5,4)/5.
O.g.f.: (1 + 6*x + 6*x^2 + x^3)/(1-x)^9. (End)
From Wolfdieter Lang, Nov 13 2007: (Start)
a(n) = A001263(n+5,5).
Numerator polynomial of the g.f is the fourth row polynomial of the Narayana triangle. (End)
a(n)= C(n+4,4)^2 - C(n+4,3)*C(n+4,5). - Gary Detlefs, Dec 05 2011
a(n) = Product_{i=1..4} A002378(n+i)/A002378(i). - Bruno Berselli, Sep 01 2016
From Amiram Eldar, Oct 19 2020: (Start)
Sum_{n>=0} 1/a(n) = 25 * (79 - 8*Pi^2).
Sum_{n>=0} (-1)^n/a(n) = 595/3 - 20*Pi^2. (End)

More terms from Vladeta Jovovic, Jan 29 2003
Better description from Mike Zabrocki, Aug 26 2004
New definition from N. J. A. Sloane, Aug 28 2010
Zabrocki formulas offset corrected by Gary Detlefs, Dec 05 2011

A107891 a(n) = (n+1)(n+2)^2(n+3)^2(n+4)(3n^2 + 15n + 20)/2880.

Original entry on oeis.org

1, 19, 155, 805, 3136, 9996, 27468, 67320, 150645, 313027, 611611, 1134497, 2012920, 3436720, 5673648, 9093096, 14194881, 21643755, 32310355, 47319349, 68105576, 96479020, 134699500, 185562000, 252493605, 339663051, 452103939

Offset: 0

Emeric Deutsch, Jun 12 2005

Kekulé numbers for certain benzenoids.

Partial sums of A114239. First differences of A047819. - Peter Bala, Sep 21 2007

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see pp. 167, 187 and p. 105 eq. (iii) for k=2 and m=5).

Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Paolo 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]."]
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A005585, A006542, A047819, A114239, A114242.

a:=n->(1/2880)*(n+1)*(n+2)^2*(n+3)^2*(n+4)*(3*n^2+15*n+20): seq(a(n),n=0..32);

Table[((1+n) (2+n)^2 (3+n)^2 (4+n) (20+3 n (5+n)))/2880,{n,0,40}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,19,155,805,3136,9996,27468,67320,150645},40] (* Harvey P. Dale, Dec 10 2021 *)

a(n-2) = (1/8) * Sum_{1 <= x_1, x_2 <= n} (x_1*x_2)^2*(det V(x_1,x_2))^2 = 1/8*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
G.f.: (1+10*x+20*x^2+10*x^3+x^4)/(1-x)^9. - Colin Barker, Feb 08 2012
a(n) = (A000330(n+2)*A000538(n+2) - (A000537(n+2))^2)/4. - J. M. Bergot, Sep 17 2013
Sum_{n>=0} 1/a(n) = 17095/4 - 240*Pi^2 - 162*sqrt(15)*Pi*tanh(sqrt(5/3)*Pi/2). - Amiram Eldar, May 29 2022

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

A004282 a(n) = n(n+1)^2(n+2)^2/12.

Original entry on oeis.org

0, 3, 24, 100, 300, 735, 1568, 3024, 5400, 9075, 14520, 22308, 33124, 47775, 67200, 92480, 124848, 165699, 216600, 279300, 355740, 448063, 558624, 690000, 845000, 1026675, 1238328, 1483524, 1766100, 2090175

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A006542, A107891, A114242.

[n*(n+1)^2*(n+2)^2/12: n in [0..50]]; // Vincenzo Librandi, Feb 09 2012

a:= n-> binomial(2+n, 2)*binomial(2+n, 3): seq(a(n), n=0..31); # Zerinvary Lajos, Apr 26 2007

Table[n*(n+1)^2*(n+2)^2/12,{n,0,40}] (* Vincenzo Librandi, Feb 09 2012 *)

a(n) = binomial(n+2,2)*binomial(n+2,3); \\ Charles R Greathouse IV, Feb 09 2012

a(n) = C(2+n, 2)*C(2+n, 3) = A000217(n+1)*A000292(n). - Zerinvary Lajos, Jan 10 2006
a(n-1) = Sum_{1 <= x_1, x_2 <= n} x_1*(det V(x_1,x_2))^2 = Sum_{1 <= i,j <= n} i*(i-j)^2, where V(x_1,x_2) is the Vandermonde matrix of order 2. - Peter Bala, Sep 21 2007
G.f.: x*(3+6*x+x^2)/(1-x)^6. - Colin Barker, Feb 09 2012
a(n) = Sum_{k=0..n} Sum_{i=0..n} (n-i+1) * C(k+1,k-1). - Wesley Ivan Hurt, Sep 21 2017
a(n) = A004302(n+1) - A000537(n+1). - J. M. Bergot, Mar 28 2018
From Amiram Eldar, May 29 2022: (Start)
Sum_{n>=1} 1/a(n) = 30 - 3*Pi^2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/2 - 24*log(2) + 12. (End)

A114244 a(n) = (n+1)(n+2)^2(n+3)*(7n^2 + 28n + 30)/360.

Original entry on oeis.org

1, 13, 76, 295, 889, 2254, 5040, 10242, 19305, 34243, 57772, 93457, 145873, 220780, 325312, 468180, 659889, 912969, 1242220, 1664971, 2201353, 2874586, 3711280, 4741750, 6000345, 7525791, 9361548, 11556181, 14163745, 17244184, 20863744, 25095400, 30019297

Offset: 0

Emeric Deutsch, Nov 18 2005

Kekulé numbers for certain benzenoids.

First differences of A114242. - Peter Bala, Sep 21 2007

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 167-169, Table 10.5/II/5).

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A114242.

a:=n->(n+1)*(n+2)^2*(n+3)*(7*n^2+28*n+30)/360: seq(a(n),n=0..35);

Table[(n + 1)*(n + 2)^2*(n + 3)*(7*n^2 + 28*n + 30)/360, {n, 0, 30}] (* Amiram Eldar, May 31 2022 *)
CoefficientList[Series[(1+x)(1+5x+x^2)/(1-x)^7,{x,0,40}],x] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{1,13,76,295,889,2254,5040},40] (* Harvey P. Dale, Mar 06 2023 *)

G.f.: (1+x)(1 + 5x + x^2)/(1-x)^7.
From Amiram Eldar, May 31 2022: (Start)
Sum_{n>=0} 1/a(n) = 5*Pi*(7*sqrt(14)*coth(sqrt(2/7)*Pi) - 6*Pi) - 1295/9.
Sum_{n>=0} (-1)^n/a(n) = 5*Pi*(7*sqrt(14)*cosech(sqrt(2/7)*Pi) + 3*Pi) - 2755/9. (End)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A006857 a(n) = binomial(n+5,5) * binomial(n+5,4)/(n+5).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

SageMath

Formula

Extensions

A107891 a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)*(3n^2 + 15n + 20)/2880.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A119308 Triangle for first differences of Catalan numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A004282 a(n) = n*(n+1)^2*(n+2)^2/12.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A114244 a(n) = (n+1)*(n+2)^2*(n+3)*(7n^2 + 28n + 30)/360.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

A107891 a(n) = (n+1)(n+2)^2(n+3)^2(n+4)(3n^2 + 15n + 20)/2880.

A004282 a(n) = n(n+1)^2(n+2)^2/12.

A114244 a(n) = (n+1)(n+2)^2(n+3)*(7n^2 + 28n + 30)/360.