A108647 -id:A108647 - cp's OEIS frontend

A103371 Number triangle T(n,k) = C(n,n-k)*C(n+1,n-k).

1, 2, 1, 3, 6, 1, 4, 18, 12, 1, 5, 40, 60, 20, 1, 6, 75, 200, 150, 30, 1, 7, 126, 525, 700, 315, 42, 1, 8, 196, 1176, 2450, 1960, 588, 56, 1, 9, 288, 2352, 7056, 8820, 4704, 1008, 72, 1, 10, 405, 4320, 17640, 31752, 26460, 10080, 1620, 90, 1, 11, 550, 7425, 39600, 97020

Offset: 0

Paul Barry, Feb 03 2005

Columns include A000027, A002411, A004302, A108647, A134287. Row sums are C(2n+1,n+1) or A001700.
T(n-1,k-1) is the number of ways to put n identical objects into k of altogether n distinguishable boxes. See the partition array A035206 from which this triangle arises after summing over all entries related to partitions with fixed part number k.
T(n, k) is also the number of order-preserving full transformations (of an n-chain) of height k (height(alpha) = |Im(alpha)|). - Abdullahi Umar, Oct 02 2008
The o.g.f. of the (n+1)-th diagonal is given by G(n, x) = (n+1)*Sum_{k=1..n} A001263(n, k)*x^(k-1) / (1 - x)^(2*n+1), for n >= 1 and for n = 0 it is G(0, x) = 1/(1-x). - Wolfdieter Lang, Jul 31 2017

			The triangle T(n, k) begins:
n\k  0   1    2     3     4     5     6    7  8 9 ...
0:   1
1:   2   1
2:   3   6    1
3:   4  18   12     1
4:   5  40   60    20     1
5:   6  75  200   150    30     1
6:   7 126  525   700   315    42     1
7:   8 196 1176  2450  1960   588    56    1
8:   9 288 2352  7056  8820  4704  1008   72  1
9:  10 405 4320 17640 31752 26460 10080 1620 90 1
...  reformatted. - _Wolfdieter Lang_, Jul 31 2017
From _R. J. Mathar_, Mar 29 2013: (Start)
The matrix inverse starts
       1;
      -2,       1;
       9,      -6,      1;
     -76,      54,    -12,      1;
    1055,    -760,    180,    -20,   1;
  -21906,   15825,  -3800,    450, -30,   1;
  636447, -460026, 110775, -13300, 945, -42, 1; (End)
O.g.f. of 4th diagonal [4, 40,200, ...] is G(3, x) = 4*(1 + 3*x + x^2)/(1 - x)^7, from the n = 3 row [1, 3, 1] of A001263. See a comment above. - _Wolfdieter Lang_, Jul 31 2017

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.
Paul Barry and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 16.
R. Cori and G. Hetyei, Counting genus one partitions and permutations, arXiv:1306.4628 [math.CO], 2013.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving full transformations, Semigroup Forum 72 (2006), 51-62.

Cf. A008459, A122899, A194595, A197653.
Cf. A007318, A000894 (central terms), A132813 (mirrored).
Cf. A000027, A002411, A004302, A108647, A134287, A135278, A001263.

a103371 n k = a103371_tabl !! n !! k
a103371_row n = a103371_tabl !! n
a103371_tabl = map reverse a132813_tabl
-- Reinhard Zumkeller, Apr 04 2014

/* As triangle */ [[Binomial(n,n-k)*Binomial(n+1,n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 01 2017

A103371 := (n,k) -> binomial(n,k)^2*(n+1)/(k+1);
seq(print(seq(A103371(n, k), k=0..n)), n=0..7); # Peter Luschny, Oct 19 2011

Flatten[Table[Binomial[n,n-k]Binomial[n+1,n-k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 26 2014 *)
CoefficientList[Series[Series[E^(x(1+y))(BesselI[0,2*x*Sqrt[y]]+BesselI[1,2*x*Sqrt[y]]/Sqrt[y]),{x,0,8}],{y,0,8}],{x,y}]*Range[0,8]! (* Natalia L. Skirrow, Apr 14 2025 *)

create_list(binomial(n,k)*binomial(n+1,k+1),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

for(n=0,10, for(k=0,n, print1(binomial(n,k)*binomial(n+1,k+1), ", "))) \\ G. C. Greubel, Nov 09 2018

Number triangle T(n, k) = C(n, n-k)*C(n+1, n-k) = C(n, k)*C(n+1, k+1); Column k of this triangle has g.f. Sum_{j=0..k} (C(k, j)*C(k+1, j) * x^(k+j))/(1-x)^(2*k+2); coefficients of the numerators are the rows of the reverse triangle C(n, k)*C(n+1, k).
T(n,k) = C(n, k)*Sum_{j=0..(n-k)} C(n-j, k). - Paul Barry, Jan 12 2006
T(n,k) = (n+1-k)*N(n+1,k+1), with N(n,k):=A001263(n,k), the Narayana triangle (with offset [1,1]).
O.g.f.: ((1-(1-y)*x)/sqrt((1-(1+y)*x)^2-4*x^2*y) -1)/2, (from o.g.f. of A001263, Narayana triangle). - Wolfdieter Lang, Nov 13 2007
From Peter Bala, Jan 24 2008: (Start)
Matrix product of A007318 and A122899.
O.g.f. for row n: (1-x)^n*P(n,1,0,(1+x)/(1-x)) = 1/(2*x)*(1-x)^(n+1)*( Legendre_P(n+1,(1+x)/(1-x)) - Legendre_P(n,(1+x)/(1-x)) ), where P(n,a,b,x) denotes the Jacobi polynomial.
O.g.f. for column k: x^k/(1-x)^(k+2)*P(k,0,1,(1+x)/(1-x)). Compare with A008459. (End)
Let S(n,k) = binomial(2*n,n)^(k+1)*((n+1)^(k+1)-n^(k+1))/(n+1)^k. Then T(2*n,n) = S(n,1). (Cf. A194595, A197653, A197654). - Peter Luschny, Oct 20 2011
T(n,k) = A003056(n+1,k+1)*C(n,k)^2/(k+1). - Peter Luschny, Oct 29 2011
T(n,k) = A007318(n, k)*A135278(n, k), n >= k >= 0. - Wolfdieter Lang, Jul 31 2017
From Natalia L. Skirrow, Apr 14 2025: (Start)
T(n,k) = A008459(n,k) + n*N(n,k+1).
E.g.f.: e^(x*(1+y))*(I_0(2*x*sqrt(y)) + I_1(2*x*sqrt(y))/sqrt(y)), where I_n is the modified Bessel function of the first kind. (The I_0 contributes A008459(n,k), the I_1 contributes n*N(n,k+1))
O.g.f. for row n: (n+1)*2F1(-n,-n;2;y) = (n+1)*2F1(2+n,2+n;2;y)*(1-y)^(2*(n+1)) (by Euler's hypergeometric transformation); (n+1)*2F1(2+n,2+n;2;y) is the o.g.f. for row n of (k+n+1)!^2/(k!*(k+1)!*n!*(n+1)!), which is column n+1 of A132812.
O.g.f. for column k: 2F1(1+k,2+k;1;x)*x^k = 2F1(-k,-1-k;1;x)*x^k/(1-x)^(2+2*k). 2F1(-k,-1-k;1;x) is the kth row of A132813, the reflection of the kth row of this triangle.
O.g.f. for diagonal d (beginning at a(d,0)): (d+1)*x^d*2F1(d+1,d+2;2;x*y). 2F1(d+1,d+2;2;x) = 2F1(1-d,-d;2;x)/(1-x)^(2*d+1), numerator being the o.g.f. of row d of the Narayana triangle.
These respectively yield:
T(n,k) = Sum_{i=0..n+k} C(2*(n+1),i)*(-1)^i*A132812(n+1+k-i,n+1),
T(d+k,k) = Sum_{i=0..k} C(d-i+1+2*k,d-i)*T(k,k-i),
T(d+k,k) = Sum_{i=0..d} C(k-i + 2*d,k-i)*N(d,i+1)*(d+1).
E.g.f. for column k: 1F1(2+k;1;x)*x^k/k!.
E.g.f. for diagonal d: (d+1)*x^d*1F1(d+2;2;x*y)/d!. (End)

A132813 Triangle read by rows: A001263 * A127648 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 12, 18, 4, 1, 20, 60, 40, 5, 1, 30, 150, 200, 75, 6, 1, 42, 315, 700, 525, 126, 7, 1, 56, 588, 1960, 2450, 1176, 196, 8, 1, 72, 1008, 4704, 8820, 7056, 2352, 288, 9, 1, 90, 1620, 10080, 26460, 31752, 17640, 4320, 405, 10

Offset: 0

Gary W. Adamson, Sep 01 2007

Also T(n,k) = binomial(n-1, k-1)*binomial(n, k-1), related to Narayana polynomials (see Sulanke reference). - Roger L. Bagula, Apr 09 2008

h-vector for cluster complex associated to the root system B_n. See p. 8, Athanasiadis and C. Savvidou. - Tom Copeland, Oct 19 2014

			First few rows of the triangle are:
  1;
  1,  2;
  1,  6,   3;
  1, 12,  18,   4;
  1, 20,  60,  40,   5;
  1, 30, 150, 200,  75,   6;
  1, 42, 315, 700, 525, 126, 7;
  ...

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
N. Alexeev and A. Tikhomirov, Singular Values Distribution of Squares of Elliptic Random Matrices and type-B Narayana Polynomials, arXiv preprint arXiv:1501.04615 [math.PR], 2015.
C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
Robert. A. Sulanke, Counting Lattice Paths by Narayana Polynomials Electronic J. Combinatorics 7, No. 1, R40, 1-9, 2000.

Family of polynomials (see A062145): A008459 (c=1), this sequence (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).
Columns: A000012 (k=0), A002378 (k=1), A006011 (k=2), 4*A006542 (k=3), 5*A006857 (k=4), 6*A108679 (k=5), 7*A134288 (k=6), 8*A134289 (k=7), 9*A134290 (k=8), 10*A134291 (k=9).
Diagonals: A000027 (k=n), A002411 (k=n-1), A004302 (k=n-2), A108647 (k=n-3), A134287 (k=n-4).
Main diagonal: A000894.
Sums: (-1)^floor((n+1)/2)*A001405 (signed row), A001700 (row), A203611 (diagonal).
Cf. A001263, A007318, A127648, A281260.
Cf. A103371 (mirrored).

Flat(List([0..10],n->List([0..n], k->(k+1)*Binomial(n+1,k+1)*Binomial(n+1,k)/(n+1)))); # Muniru A Asiru, Feb 26 2019

a132813 n k = a132813_tabl !! n !! k
a132813_row n = a132813_tabl !! n
a132813_tabl = zipWith (zipWith (*)) a007318_tabl $ tail a007318_tabl
-- Reinhard Zumkeller, Apr 04 2014

/* triangle */ [[(k+1)*Binomial(n+1,k+1)*Binomial(n+1,k)/(n+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 19 2014

P := (n, x) -> hypergeom([1-n, -n], [1], x): for n from 1 to 9 do PolynomialTools:-CoefficientList(simplify(P(n,x)),x) od; # Peter Luschny, Nov 26 2014

T[n_,k_]=Binomial[n-1,k-1]*Binomial[n,k-1]; Table[Table[T[n,k],{k,1,n}],{n,1,11}]; Flatten[%] (* Roger L. Bagula, Apr 09 2008 *)
P[n_, x_] := HypergeometricPFQ[{1-n, -n}, {1}, x]; Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten (* Jean-François Alcover, Nov 27 2014, after Peter Luschny *)

tabl(nn) = {for (n = 1, nn, for (k = 1, n, print1(binomial(n-1, k-1)*binomial(n, k-1) , ", ");););} \\ Michel Marcus, Feb 12 2014

def A132813(n,k): return binomial(n,k)*binomial(n+1,k)
print(flatten([[A132813(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 12 2025

T(n,k) = (k+1)*binomial(n+1,k+1)*binomial(n+1,k)/(n+1), n >= k >= 0.
From Roger L. Bagula, May 14 2010: (Start)
T(n, m) = coefficients(p(x,n)), where
p(x,n) = (1-x)^(2*n)*Sum_{k >= 0} binomial(k+n-1, k)*binomial(n+k, k)*x^k,
 or p(x,n) = (1-x)^(2*n)*Hypergeometric2F1([n, n+1], [1], x). (End)
T(n,k) = binomial(n,k) * binomial(n+1,k). - Reinhard Zumkeller, Apr 04 2014
These are the coefficients of the polynomials Hypergeometric2F1([1-n,-n], [1], x). - Peter Luschny, Nov 26 2014
G.f.: A(x,y) = A281260(x,y)/(1-A281260(x,y))/x. - Vladimir Kruchinin, Oct 10 2020

A108648 a(n) = (n+1)^2(n+2)^3(n+3)/24.

Original entry on oeis.org

1, 18, 120, 500, 1575, 4116, 9408, 19440, 37125, 66550, 113256, 184548, 289835, 441000, 652800, 943296, 1334313, 1851930, 2527000, 3395700, 4500111, 5888828, 7617600, 9750000, 12358125, 15523326, 19336968, 23901220, 29329875, 35749200

Offset: 0

Emeric Deutsch, Jun 13 2005

Kekulé numbers for certain benzenoids.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, no. 24).

Colin Barker, Table of n, a(n) for n = 0..1000
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, no. 24).
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A108647.

[(n+1)^2*(n+2)^3*(n+3)/24: n in [0..30]]; // G. C. Greubel, Oct 28 2022

a:=(n+1)^2*(n+2)^3*(n+3)/24: seq(a(n),n=0..36);

Table[(n+1)^2*(n+2)^3*(n+3)/24, {n,0,30}] (* G. C. Greubel, Oct 28 2022 *)

Vec((1 + 11*x + 15*x^2 + 3*x^3) / (1 - x)^7 + O(x^30)) \\ Colin Barker, Apr 22 2020

[(n+1)^2*(n+2)^3*(n+3)/24 for n in (0..30)] # G. C. Greubel, Oct 28 2022

From Colin Barker, Apr 22 2020: (Start)
G.f.: (1 + 11*x + 15*x^2 + 3*x^3) / (1 - x)^7.
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) for n>6.
(End)
a(n) = A000217(n+1) * A002415(n+2). - J. M. Bergot, May 21 2022
From Amiram Eldar, May 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 24*zeta(3) + 6*Pi^2 - 87.
Sum_{n>=0} (-1)^n/a(n) = 99 - Pi^2 - 96*log(2) - 18*zeta(3). (End)
E.g.f.: (24 + 408*x + 1020*x^2 + 772*x^3 + 224*x^4 + 26*x^5 + x^6)*exp(x)/4!. - G. C. Greubel, Oct 28 2022

A108650 a(n) = (n+1)^2(n+2)(n+3)(3n+4)/24.

Original entry on oeis.org

1, 14, 75, 260, 700, 1596, 3234, 6000, 10395, 17050, 26741, 40404, 59150, 84280, 117300, 159936, 214149, 282150, 366415, 469700, 595056, 745844, 925750, 1138800, 1389375, 1682226, 2022489, 2415700, 2867810, 3385200, 3974696, 4643584

Offset: 0

Emeric Deutsch, Jun 13 2005

Kekulé numbers for certain benzenoids.

Bo Gyu Jeong, Table of n, a(n) for n = 0..5000
Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, Mixed restricted Stirling numbers, arXiv:1812.02955 [math.CO], 2018.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, no. 26).
C. Krishnamachari, The operator (xD)^n, J. Indian Math. Soc., 15 (1923),3-4. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A108645, A108646, A108647, A108648, A108649.

Cf. A001296, A001477.

[(n+1)*StirlingSecond(n+3,n+1): n in [0..40]]; // G. C. Greubel, Oct 19 2023

a:= n-> (n+1)^2*(n+2)*(n+3)*(3*n+4)/24: seq(a(n),n=0..36);
seq((n+1)*stirling2(n+3,n+1), n=0..32); # Zerinvary Lajos, Jan 20 2007

Table[((n+1)^2 (n+2)(n+3)(3n+4))/24,{n,0,40}] (* or *) Table[n StirlingS2[n+2,n],{n,40}] (* Harvey P. Dale, Dec 01 2013 *)

Vec((1 + 8*x + 6*x^2) / (1 - x)^6 + O(x^30)) \\ Colin Barker, Apr 22 2020

[(n+1)*stirling_number2(n+3,n+1) for n in range(41)] # G. C. Greubel, Oct 19 2023

From Zerinvary Lajos, Jan 20 2007: (Start)
a(n) = A001477(n+1)*A001296(n+1) = (n+1)*A001296(n+1).
a(n) = (n+1)*Stirling2(n+3,n+1). (End)
From Colin Barker, Apr 22 2020: (Start)
G.f.: (1 + 8*x + 6*x^2) / (1 - x)^6.
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) for n>5.
(End)
From Amiram Eldar, May 29 2022: (Start)
Sum_{n>=0} 1/a(n) 2*Pi^2 + 54*sqrt(3)*Pi/5 + 486*log(3)/5 - 921/5.
Sum_{n>=0} (-1)^n/a(n) = Pi^2 - 108*sqrt(3)*Pi/5 - 528*log(2)/5 + 909/5. (End)
E.g.f.: (1/24)*(24 +312*x +576*x^2 +304*x^3 +55*x^4 +3*x^5)*exp(x). - G. C. Greubel, Oct 19 2023

A108645 a(n) = (n+1)(n+2)^2(n+3)^2(n+4)(2n^2 + 6n + 5)/720.

Original entry on oeis.org

1, 26, 250, 1435, 5978, 19992, 56952, 143550, 328515, 695266, 1379378, 2591953, 4650100, 8015840, 13344864, 21546684, 33857829, 51929850, 77934010, 114684647, 165783310, 235785880, 330395000, 456680250, 623328615, 840927906, 1122285906

Offset: 0

Emeric Deutsch, Jun 13 2005

nonn

Kekulé numbers for certain benzenoids.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, no. 21).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A108646, A108647, A108648, A108649, A108650.

B:=Binomial; [(2*n^2+6*n+5)*B(n+4,4)*B(n+3,2)/15: n in [0..40]]; // G. C. Greubel, Oct 19 2023

a:=(n+1)*(n+2)^2*(n+3)^2*(n+4)*(2*n^2+6*n+5)/720: seq(a(n),n=0..30);

Table[(n+1)(n+2)^2(n+3)^2(n+4)(2n^2+6n+5)/720,{n,0,30}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,26,250,1435, 5978,19992,56952,143550,328515},30] (* Harvey P. Dale, Sep 05 2016 *)

b=binomial; [(2*n^2+6*n+5)*b(n+4,4)*b(n+3,2)/15 for n in range(41)] # G. C. Greubel, Oct 19 2023

G.f.: (1+17*x+52*x^2+37*x^3+5*x^4)/(1-x)^9. - Harvey P. Dale, Sep 05 2016

E.g.f.: (1/6!)*(720 + 18000*x + 71640*x^2 + 91440*x^3 + 49050*x^4 + 12486*x^5 + 1565*x^6 + 92*x^7 + 2*x^8)*exp(x). - G. C. Greubel, Oct 19 2023

A108646 a(n) = (n+1)(n+2)^2(n+3)(11n^3 + 58n^2 + 101n + 60)/720.

Original entry on oeis.org

1, 23, 194, 985, 3668, 11074, 28728, 66438, 140415, 276001, 511082, 900263, 1519882, 2473940, 3901024, 5982300, 8950653, 13101051, 18802210, 26509637, 36780128, 50287798, 67841720, 90405250, 119117115, 155314341, 200557098

Offset: 0

Emeric Deutsch, Jun 13 2005

Kekulé numbers for certain benzenoids.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, no. 22).

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A108645, A108647, A108648, A108649, A108650.

[(n+2)*(11*n^3+58*n^2+101*n+60)*Binomial(n+3,3)/120: n in [0..40]]; // G. C. Greubel, Oct 19 2023

a:=(n+1)*(n+2)^2*(n+3)*(11*n^3+58*n^2+101*n+60)/720: seq(a(n),n=0..30);

Table[(n+2)*(n+3)!*(11*n^3+58*n^2+101*n+60)/(6!*n!), {n,0,40}] (* G. C. Greubel, Oct 19 2023 *)

A108646_list, m = [], [77, -85, 28, -1, 1, 1, 1, 1]
for _ in range(10001):
    A108646_list.append(m[-1])
    for i in range(7):
        m[i+1] += m[i] # Chai Wah Wu, Jun 12 2016

[(n+2)*(11*n^3+58*n^2+101*n+60)*binomial(n+3,3)/120 for n in range(41)] # G. C. Greubel, Oct 19 2023

From Chai Wah Wu, Jun 12 2016: (Start)
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) for n > 7.
G.f.: (1 + 15*x + 38*x^2 + 21*x^3 + 2*x^4)/(1 - x)^8. (End)
E.g.f.: (1/6!)*(720 + 15840*x + 53640*x^2 + 56520*x^3 + 24030*x^4 + 4548*x^5 + 377*x^6 + 11*x^7)*exp(x). - G. C. Greubel, Oct 19 2023

A108649 a(n) = (n+1)(n+2)(n+3)(13n^3 + 69n^2 + 113n + 60)/360.

Original entry on oeis.org

1, 17, 111, 457, 1428, 3710, 8442, 17382, 33099, 59191, 100529, 163527, 256438, 389676, 576164, 831708, 1175397, 1630029, 2222563, 2984597, 3952872, 5169802, 6684030, 8551010, 10833615, 13602771, 16938117, 20928691, 25673642, 31282968

Offset: 0

Emeric Deutsch, Jun 13 2005

Kekulé numbers for certain benzenoids.

Colin Barker, Table of n, a(n) for n = 0..1000
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, no. 25).
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A108645, A108646, A108647, A108648, A108650.

[(13*n^3+69*n^2+113*n+60)*Binomial(n+3,3)/60: n in [0..40]]; // G. C. Greubel, Oct 19 2023

a:=(n+1)*(n+2)*(n+3)*(13*n^3+69*n^2+113*n+60)/360: seq(a(n),n=0..36);

Table[(n+1)(n+2)(n+3)(13n^3+69n^2+113n+60)/360,{n,0,30}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1}, {1,17,111,457,1428,3710, 8442},30] (* Harvey P. Dale, Jul 01 2012 *)

Vec((1+10*x+13*x^2+2*x^3)/(1-x)^7 + O(x^40)) \\ Colin Barker, Apr 22 2020

[(13*n^3+69*n^2+113*n+60)*binomial(n+3,3)/60 for n in range(41)] # G. C. Greubel, Oct 19 2023

a(0)=1, a(1)=17, a(2)=111, a(3)=457, a(4)=1428, a(5)=3710, a(6)=8442, 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). - Harvey P. Dale, Jul 01 2012
G.f.: (1 + 10*x + 13*x^2 + 2*x^3) / (1 - x)^7. - Colin Barker, Apr 22 2020
E.g.f.: (1/360)*(360 + 5760*x + 14040*x^2 + 10440*x^3 + 2985*x^4 + 342*x^5 + 13*x^6)*exp(x). - G. C. Greubel, Oct 19 2023

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

A134287 Fifth column of triangle A103371 (without leading zeros).

Original entry on oeis.org

1, 30, 315, 1960, 8820, 31752, 97020, 261360, 637065, 1431430, 3006003, 5962320, 11262160, 20391840, 35581680, 60093504, 98590905, 157608990, 246142435, 376372920, 564559380, 832117000, 1206913500, 1724814000, 2431508625

Offset: 0

Wolfdieter Lang, Nov 13 2007

Kekulé numbers for certain benzenoids.
a(n) = K(L(n))*K(O(2,4,n)) with the Cyvin and Gutman Kekulé number notation. See p. 62 for the L(n) structure with K(L(n))=n+1 and p. 105 (i) for the O(k,m,n) structure and its Kekulé number. This corresponds to an essentially disconnected 7-tier benzenoid structure similar to the 6-tier structure shown on p. 230, nr. 23 (see A108647).
a(n-5), n >= 5, is the number of ways to put n identical objects into m=5 of altogether n distinguishable boxes (n-5 boxes stay empty).

			a(2)=315 because n=7 identical balls can be put into m=5 of n=7 distinguishable boxes in binomial(7,5)*(5!/(4!*1!)+ 5!/(3!*2!)) = 21*(5+10) = 315 ways. The m=5 part partitions of 7, namely (1^4,3) and (1^3,2^2) specify the filling of each of the 21 possible five box choices. - _Wolfdieter Lang_, Nov 13 2007

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A108647 (fourth column of triangle A103371).

a134287 = flip a103371 4 . (+ 4)  -- Reinhard Zumkeller, Apr 04 2014

[5*Binomial(n+5, 5)^2/(n+5): n in [0..30]]; // G. C. Greubel, Oct 28 2022

seq(binomial(n+4,4)^2*(n+5)/5, n=0..24); # Peter Luschny, Jan 13 2014

CoefficientList[Series[(1 + 20 x + 60 x^2 + 40 x^3 + 5 x^4)/(1 - x)^10, {x, 0, 24}], x]

5*binomial(n+5,5)^2/(n+5) $ n = 0..35; // Zerinvary Lajos, May 09 2008

a(n) = 5*binomial(n+5, 5)^2/(n+5); \\ Michel Marcus, Jan 07 2014

[5*binomial(n+5,5)^2/(n+5) for n in range(31)] # G. C. Greubel, Oct 28 2022

a(n) = A103371(n+4,4), n >= 0.
a(n) = ((n+1)*(n+2)*(n+3)*(n+4))^2*(n+5)/2880, n >= 0. 2880 = 4!*5! = A010790(4).
G.f.: (1+20*x+60*x^2+40*x^3+5*x^4)/(1-x)^10. Numerator polynomial from fifth row of triangle A132813.
a(n) = 5*C(n+5,5)^2/(n+5), n >= 0. - Zerinvary Lajos, May 09 2008
a(n) = (C(n+6,6)*C(n+5,4)+5*C(n+5,6)*C(n+5,4))/(n+5). - Gary Detlefs, Jan 06 2014
From Amiram Eldar, May 31 2022: (Start)
Sum_{n>=0} 1/a(n) = 350*Pi^2/3 - 13805/12.
Sum_{n>=0} (-1)^n/a(n) = 5*Pi^2 + 640*log(2)/3 - 785/4. (End)
E.g.f.: (2880 + 83520*x + 368640*x^2 + 529920*x^3 + 330120*x^4 + 102024*x^5 + 16616*x^6 + 1432*x^7 + 61*x^8 + x^9)*exp(x)/2880. - G. C. Greubel, Oct 28 2022

cp's OEIS Frontend

A103371 Number triangle T(n,k) = C(n,n-k)*C(n+1,n-k).

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A132813 Triangle read by rows: A001263 * A127648 as infinite lower triangular matrices.

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A108648 a(n) = (n+1)^2*(n+2)^3*(n+3)/24.

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A108650 a(n) = (n+1)^2*(n+2)*(n+3)*(3*n+4)/24.

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A108645 a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)*(2*n^2 + 6*n + 5)/720.

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A108646 a(n) = (n+1)*(n+2)^2*(n+3)*(11*n^3 + 58*n^2 + 101*n + 60)/720.

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A108648 a(n) = (n+1)^2(n+2)^3(n+3)/24.

A108650 a(n) = (n+1)^2(n+2)(n+3)(3n+4)/24.

A108645 a(n) = (n+1)(n+2)^2(n+3)^2(n+4)(2n^2 + 6n + 5)/720.

A108646 a(n) = (n+1)(n+2)^2(n+3)(11n^3 + 58n^2 + 101n + 60)/720.

A108649 a(n) = (n+1)(n+2)(n+3)(13n^3 + 69n^2 + 113n + 60)/360.