A062145 -id:A062145 - cp's OEIS frontend

A008459 Square the entries of Pascal's triangle.

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 36, 16, 1, 1, 25, 100, 100, 25, 1, 1, 36, 225, 400, 225, 36, 1, 1, 49, 441, 1225, 1225, 441, 49, 1, 1, 64, 784, 3136, 4900, 3136, 784, 64, 1, 1, 81, 1296, 7056, 15876, 15876, 7056, 1296, 81, 1, 1, 100, 2025, 14400, 44100, 63504, 44100, 14400, 2025, 100, 1

Offset: 0

N. J. A. Sloane

Number of lattice paths from (0, 0) to (n, n) with steps (1, 0) and (0, 1), having k right turns. - Emeric Deutsch, Nov 23 2003
Product of A007318 and A105868. - Paul Barry, Nov 15 2005
Number of partitions that fit in an n X n box with Durfee square k. - Franklin T. Adams-Watters, Feb 20 2006
From Peter Bala, Oct 23 2008: (Start)
Narayana numbers of type B. Row n of this triangle is the h-vector of the simplicial complex dual to an associahedron of type B_n (a cyclohedron) [Fomin & Reading, p. 60]. See A063007 for the corresponding f-vectors for associahedra of type B_n. See A001263 for the h-vectors for associahedra of type A_n. The Hilbert transform of this triangular array is A108625 (see A145905 for the definition of this term).
Let A_n be the root lattice generated as a monoid by {e_i - e_j: 0 <= i, j <= n + 1}. Let P(A_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the h-vectors of a unimodular triangulation of P(A_n) [Ardila et al.]. A063007 is the corresponding array of f-vectors for these type A_n polytopes. See A086645 for the array of h-vectors for type C_n polytopes and A108558 for the array of h-vectors associated with type D_n polytopes.
(End)
The n-th row consists of the coefficients of the polynomial P_n(t) = Integral_{s = 0..2*Pi} (1 + t^2 - 2*t*cos(s))^n/Pi/2 ds. For example, when n = 3, we get P_3(t) = t^6 + 9*t^4 + 9*t^2 + 1; the coefficients are 1, 9, 9, 1. - Theodore Kolokolnikov, Oct 26 2010
Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0, 2*sqrt(-y)). Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence n!^2 as defined in Wang and Wang. - Peter Bala, Jul 24 2013
From Colin Defant, Sep 16 2018: (Start)
Let s denote West's stack-sorting map. T(n,k) is the number of permutations pi of [n+1] with k descents such that s(pi) avoids the patterns 132, 231, and 321. T(n,k) is also the number of permutations pi of [n+1] with k descents such that s(pi) avoids the patterns 132, 312, and 321.
T(n,k) is the number of permutations of [n+1] with k descents that avoid the patterns 1342, 3142, 3412, and 3421. (End)
The number of convex polyominoes whose smallest bounding rectangle has size (k+1)*(n+1-k) and which contain the lower left corner of the bounding rectangle (directed convex polyominoes). - Günter Rote, Feb 27 2019
Let P be the poset [n] X [n] ordered by the product order.  T(n,k) is the number of antichains in P containing exactly k elements. Cf. A063746. - Geoffrey Critzer, Mar 28 2020

			Pascal's triangle begins
  1
  1  1
  1  2   1
  1  3   3   1
  1  4   6   4   1
  1  5  10  10   5   1
  1  6  15  20  15   6   1
  1  7  21  35  35  21   7   1
...
so the present triangle begins
  1
  1   1
  1   4    1
  1   9    9     1
  1  16   36    16     1
  1  25  100   100    25    1
  1  36  225   400   225   36   1
  1  49  441  1225  1225  441  49   1
...

T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 12.
J. Riordan, An introduction to combinatorial analysis, Dover Publications, Mineola, NY, 2002, page 191, Problem 15. MR1949650
P. G. Tait, On the Linear Differential Equation of the Second Order, Proceedings of the Royal Society of Edinburgh, 9 (1876), 93-98 (see p. 97) [From Tom Copeland, Sep 09 2010, vol number corrected Sep 10 2010]

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.
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.
F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices, arXiv:0809.5123 [math.CO], 2008.
Peter Bala, A commutative diagram of triangular arrays
E. Barcucci, A. Frosini and S. Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005), 62-78.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.
Carl M. Bender and Gerald V. Dunne, Polynomials and operator orderings, J. Math. Phys. 29 (1988), 1727-1731.
Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, and André Schulz, The Number of Convex Polyominoes with Given Height and Width, arXiv:1903.01095 [math.CO], 2019.
John H. Conway and N. J. A. Sloane, Low-dimensional lattices. VII Coordination sequences, Proc. R. Soc. Lond. A (1997) 453, 2369-2389.
R. Cori and G. Hetyei, Counting genus one partitions and permutations, arXiv preprint arXiv:1306.4628 [math.CO], 2013.
R. Cori and G. Hetyei, How to count genus one partitions, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Sergey Fomin and Nathan Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004, arXiv:math/0505518 [math.CO], 2005, 2008. [From _Peter Bala_, Oct 23 2008]
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
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. 10, 13.
Abdelkader Necer, Séries formelles et produit de Hadamard, Journal de théorie des nombres de Bordeaux, 9:2 (1997), pp. 319-335.
Weiping Wang and Tianming Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
Yi Wang and Arthur L.B. Yang, Total positivity of Narayana matrices, arXiv:1702.07822 [math.CO], 2017.
Harold R. L. Yang and Philip B. Zhang, Stable multivariate Narayana polynomials and labeled plane trees, arXiv:2403.15058 [math.CO], 2024. See p. 2.

Row sums are in A000984. Columns 0-3 are A000012, A000290, A000537, A001249.
Family of polynomials (see A062145): this sequence (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).
Cf. A007318, A055133, A116647, A001263, A086645, A063007, A108558, A108625 (Hilbert transform), A145903, A181543, A086645 (logarithmic derivative), A105868 (inverse binomial transform), A093118.

Flat(List([0..10],n->List([0..n],k->Binomial(n,k)^2))); # Muniru A Asiru, Mar 30 2018

/* As triangle */ [[Binomial(n, k)^2: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Dec 15 2016

seq(seq(binomial(n, k)^2, k=0..n), n=0..10);

Table[Binomial[n, k]^2, {n, 0, 11}, {k, 0, n}]//Flatten (* Alonso del Arte, Dec 08 2013 *)

create_list(binomial(n,k)^2,n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

T(n,k):=if n=k then 1 else if k=0 then 1 else T(n-1,k)*(n+k)/(n-k)+T(n-1,k-1); /* Vladimir Kruchinin, Oct 18 2014 */

A(x,y):=1/sqrt(1-2*x-2*x*y+x^2-2*x^2*y+x^2*y^2);
taylor(x*A(x,y)+x*y*A(x,y)+sqrt(1+4*x^2*y*A(x,y)^2),x,0,7,y,0,7); /* Vladimir Kruchinin, Oct 23 2020 */

{T(n, k) = if( k<0 || k>n, 0, binomial(n, k)^2)}; /* Michael Somos, May 03 2004 */

{T(n,k)=polcoeff(polcoeff(sum(m=0,n,(2*m)!/m!^2*x^(2*m)*y^m/(1-x-x*y+x*O(x^n))^(2*m+1)),n,x),k,y)} \\ Paul D. Hanna, Oct 31 2010

def A008459(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),n-comb(r+1,2))**2 # Chai Wah Wu, Nov 12 2024

T(n,k) = A007318(n,k)^2. - Sean A. Irvine, Mar 29 2018
E.g.f.: exp((1+y)*x)*BesselI(0, 2*sqrt(y)*x). - Vladeta Jovovic, Nov 17 2003
G.f.: 1/sqrt(1-2*x-2*x*y+x^2-2*x^2*y+x^2*y^2); g.f. for row n: (1-t)^n P_n[(1+t)/(1-t)] where the P_n's are the Legendre polynomials. - Emeric Deutsch, Nov 23 2003 [The original version of the bivariate g.f. has been modified with the roles of x and y interchanged so that now x corresponds to n and y to k. - Petros Hadjicostas, Oct 22 2017]
G.f. for column k is Sum_{j = 0..k} C(k, j)^2*x^(k+j)/(1 - x)^(2*k+1). - Paul Barry, Nov 15 2005
Column k has g.f. (x^k)*Legendre_P(k, (1+x)/(1-x))/(1 - x)^(k+1) = (x^k)*Sum_{j = 0..k} C(k, j)^2*x^j/(1 - x)^(2*k+1). - Paul Barry, Nov 19 2005
Let E be the operator D*x*D, where D denotes the derivative operator d/dx. Then (1/n!^2) * E^n(1/(1 - x)) = (row n generating polynomial)/(1 - x)^(2*n+1) = Sum_{k >= 0} binomial(n+k, k)^2*x^k. For example, when n = 3 we have (1/3!)^2*E^3(1/(1 - x)) = (1 + 9*x + 9*x^2 + x^3)/(1 - x)^7 = (1/3!)^2 * Sum_{k >= 0} ((k+1)*(k+2)*(k+3))^2*x^k. - Peter Bala, Oct 23 2008
G.f.: A(x, y) = Sum_{n >= 0} (2*n)!/n!^2 * x^(2*n)*y^n/(1 - x - x*y)^(2*n+1). - Paul D. Hanna, Oct 31 2010
From Peter Bala, Jul 24 2013: (Start)
Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0, 2*sqrt(-y)). Generating function: E(y)*E(x*y) = 1 + (1 + x)*y + (1 + 4*x + x^2)*y^2/2!^2 + (1 + 9*x + 9*x^2 + x^3)*y^3/3!^2 + .... Cf. the unsigned version of A021009 with generating function exp(y)*E(x*y).
The n-th power of this array has the generating function E(y)^n*E(x*y). In particular, the matrix inverse A055133 has the generating function E(x*y)/E(y). (End)
T(n,k) = T(n-1,k)*(n+k)/(n-k) + T(n-1,k-1), T(n,0) = T(n,n) = 1. - Vladimir Kruchinin, Oct 18 2014
Observe that the recurrence T(n,k) = T(n-1,k)*(n+k)/(n-k) - T(n-1,k-1), for n >= 2 and 1 <= k < n, with boundary conditions T(n,0) = T(n,n) = 1 gives Pascal's triangle A007318. - Peter Bala, Dec 21 2014
n-th row polynomial R(n, x) = [z^n] (1 + (1 + x)*z + x*z^2)^n. Note that 1/n*[z^(n-1)] (1 + (1 + x)*z + x*z^2)^n gives the row polynomials of A001263. - Peter Bala, Jun 24 2015
Binomial transform of A105868. If G(x,t) = 1/sqrt(1 - 2*(1 + t)*x + (1 - t)^2*x^2) denotes the o.g.f. of this array then 1 + x*d/dx log(G(x,t)) = 1 + (1 + t)*x + (1 + 6*t + t^2)*x^2 + ... is the o.g.f. for A086645. - Peter Bala, Sep 06 2015
T(n,k) = Sum_{i=0..n} C(n-i,k)*C(n,i)*C(n+i,i)*(-1)^(n-i-k). - Vladimir Kruchinin, Jan 14 2018
G.f. satisfies A(x,y) = x*A(x,y)+x*y*A(x,y)+sqrt(1+4*x^2*y*A(x,y)^2). - Vladimir Kruchinin, Oct 23 2020
G.f. satisfies the differential equation y * d^2(A(x,y))/dy^2 - x^2 * d^2(x*A(x,y))/dx^2 + 2*x^2* A(x,y)^3 = 0. - Sergii Voloshyn, Mar 07 2025
T(n,k) = Sum_{i=0..n} C(2*n+1,i)*C(n+k-i,n)^2*(-1)^i. - Natalia L. Skirrow, Apr 14 2025

A028552 a(n) = n*(n+3).

Original entry on oeis.org

0, 4, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180, 208, 238, 270, 304, 340, 378, 418, 460, 504, 550, 598, 648, 700, 754, 810, 868, 928, 990, 1054, 1120, 1188, 1258, 1330, 1404, 1480, 1558, 1638, 1720, 1804, 1890, 1978, 2068, 2160, 2254, 2350, 2448, 2548, 2650

Offset: 0

Patrick De Geest

n*(n-3), for n >= 3, is the number of [body] diagonals of an n-gonal prism. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
a(n) = A028387(n)-1. Half of the difference between n(n+1)(n+2)(n+3) and the largest square less than it. Calling this difference "SquareMod": a(n) = (1/2)*SquareMod(n(n+1)(n+2)(n+3)). - Rainer Rosenthal, Sep 04 2004
n != -2 such that x^4 + x^3 - n*x^2 + x + 1 is reducible over the integers. Starting at 10: n such that x^4 + x^3 - n*x^2 + x + 1 is a product of irreducible quadratic factors over the integers. - James R. Buddenhagen, Apr 19 2005
If a 3-set Y and a 3-set Z, having two element in common, are subsets of an n-set X then a(n-4) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007
Starting with offset 1 = binomial transform of [4, 6, 2, 0, 0, 0, ...]. - Gary W. Adamson, Jan 09 2009
The sequence provides all nonnegative integers m such that 4*m + 9 is a square. - Vincenzo Librandi, Mar 03 2013
The second-order linear recurrence relations b(n)=3*b(n-1) + a(m-3)*b(n-2), n>=2, b(0)=0, b(1)=1, have closed form solutions involving only powers of m and 3-m where m>=4 is a positive integer; and lim_{n->infinity} b(n+1)/b(n) = 4. - Felix P. Muga II, Mar 18 2014
If a rook is placed at a corner of an n X n chessboard, the expected number of moves for it to reach the opposite corner is a(n-1). (See Mathematics Stack Exchange link.) - Eric M. Schmidt, Oct 29 2014
Partial sums of the even composites (which are A005843 without the 2). - R. J. Mathar, Sep 09 2015
a(n) is the number of segments necessary to represent n squares of area 1, 4, ..., n^2 having the upper and left sides overlapped:
         __      __      __  __
  ||   ||  |   ||  |  |  ||  |  |  |
         |___|   | __|  |  | __|  |  |
                   | __ |  | __ |  |
                               | __  __|
    4       10         18            28       - Stefano Spezia, May 29 2023

			G.f. = 4*x + 10*x^2 + 18*x^3 + 28*x^4 + 40*x^5 + 54*x^6 + 70*x^7 + 88*x^8 + ...

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
Milan Janjic, Two Enumerative Functions.
Mathematics Stack Exchange, Expected number of turns for a rook to move to top right-most corner?.
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Aleksandar Petojević, A Note about the Pochhammer Symbol, Mathematica Moravica, Vol. 12-1 (2008), 37-42.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A002522, A005843, A028387, A062145, A156798.

[n*(n+3): n in [0..150]]; // Vincenzo Librandi, Apr 21 2011

A028552 := proc(n) n*(n+3); end proc: # R. J. Mathar, Jan 29 2011

LinearRecurrence[{3,-3,1},{0,4,10},50] (* Harvey P. Dale, Feb 05 2012 *)
Table[ChineseRemainder[{n, n + 1}, {n + 2, n + 3}], {n, -1, 80}] (* Zak Seidov, Oct 25 2014 *)
Table[ChineseRemainder@@TakeDrop[Range[n,n+3],2],{n,-1,50}] (* Harvey P. Dale, Mar 14 2025 *)

makelist(n*(n+3),n,0,20); /* Martin Ettl, Jan 22 2013 */

a(n)=n*(n+3) \\ Charles R Greathouse IV, Mar 16 2012

a(n) = 2*A000096(n).
a(A002522(n)) = A156798(n). - Reinhard Zumkeller, Feb 16 2009
a(n) = a(n-1) + 2*(n+1) for n>0, with a(0)=0. - Vincenzo Librandi, Aug 05 2010
Sum_{n>=1} 1/a(n) = 11/18 via Sum_{n>=0} 1/((n+x)*(n+y)) = (psi(x)-psi(y))/ (x-y). - R. J. Mathar, Mar 22 2011
G.f.: 2*x*(2 - x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 31 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=0, a(1)=4, a(2)=10. - Harvey P. Dale, Feb 05 2012
a(n) = 4*C(n+1,2) - 2*C(n,2) for n>=0. - Felix P. Muga II, Mar 11 2014
a(-3 - n) = a(n) for all n in Z. - Michael Somos, Mar 18 2014
E.g.f.: (x^3 + 4*x)*exp(x). - G. C. Greubel, Jul 20 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/3 - 5/18. - Amiram Eldar, Jan 15 2021
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = 2*cos(sqrt(13)*Pi/2)/Pi.
Product_{n>=1} (1 + 1/a(n)) = -6*cos(sqrt(5)*Pi/2)/Pi. (End)

A062190 Coefficient triangle of certain polynomials N(5; m,x).

Original entry on oeis.org

1, 1, 6, 1, 14, 21, 1, 24, 84, 56, 1, 36, 216, 336, 126, 1, 50, 450, 1200, 1050, 252, 1, 66, 825, 3300, 4950, 2772, 462, 1, 84, 1386, 7700, 17325, 16632, 6468, 792, 1, 104, 2184, 16016, 50050, 72072, 48048, 13728, 1287, 1, 126, 3276, 30576, 126126, 252252, 252252, 123552, 27027, 2002

Offset: 0

Wolfdieter Lang, Jun 19 2001

The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=5) Laguerre triangle L(5; n+m,m)= A062138(n+m,m), n >= 0, is N(5; m,x)/(1-x)^(2*(m+3)), with the row polynomials N(5; m,x) := Sum_{k=0..m} a(m,k)*x^k.

			Triangle begins as:
  1;
  1,   6;
  1,  14,   21;
  1,  24,   84,    56;
  1,  36,  216,   336,    126;
  1,  50,  450,  1200,   1050,    252;
  1,  66,  825,  3300,   4950,   2772,     462;
  1,  84, 1386,  7700,  17325,  16632,    6468,    792;
  1, 104, 2184, 16016,  50050,  72072,   48048,  13728,   1287;
  1, 126, 3276, 30576, 126126, 252252,  252252, 123552,  27027,  2002;
  1, 150, 4725, 54600, 286650, 756756, 1051050, 772200, 289575, 50050, 3003;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), this sequence (c=6).
Columns k: A028557 (k=1), A104676 (k=2), A104677 (k=3), A104678 (k=4), A104679 (k=5), A104680 (k=6).
Diagonals: A000389 (k=n), A027818 (k=n-1), A104670 (k=n-2), A104671 (k=n-3), A104672 (k=n-4), A104673 (k=n-5), A104674 (k=n-6).
Cf. A003516 (row sums), A113894 (main diagonal).

A062190:= func< n,k | Binomial(n,k)*Binomial(n+5,k) >;
[A062190(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 28 2025

A062190 := proc(m,k)
    add( (binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j,j=0..m) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Nov 29 2015

NN[5, m_, x_] := x^m*(2*m+5)!*Hypergeometric2F1[-m, -m, -2*m-5, (x-1)/x]/((m+5)!*m!); Table[CoefficientList[NN[5, m, x], x], {m, 0, 8}] // Flatten (* Jean-François Alcover, Sep 18 2013 *)
A062190[n_,k_]:= Binomial[n,k]*Binomial[n+5,k];
Table[A062190[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 28 2025 *)

def A062190(n,k): return binomial(n,k)*binomial(n+5,k)
print(flatten([[A062190(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Feb 28 2025

T(m, k) = [x^k]N(5; m, x), with N(5; m, x) = ((1-x)^(2*(m+3)))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+6))).
N(5; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j).
N(5; m, x)= x^m*(2*m+5)! * 2F1(-m, -m; -2*m-5; (x-1)/x)/((m+5)!*m!). - Jean-François Alcover, Sep 18 2013
T(n, k) = binomial(n, k)*binomial(n+5, k). - G. C. Greubel, Feb 28 2025

A062196 Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).

Original entry on oeis.org

1, 1, 3, 1, 8, 6, 1, 15, 30, 10, 1, 24, 90, 80, 15, 1, 35, 210, 350, 175, 21, 1, 48, 420, 1120, 1050, 336, 28, 1, 63, 756, 2940, 4410, 2646, 588, 36, 1, 80, 1260, 6720, 14700, 14112, 5880, 960, 45, 1, 99, 1980, 13860, 41580, 58212, 38808, 11880, 1485, 55

Offset: 0

Wolfdieter Lang, Jun 19 2001

Also the coefficient triangle of certain polynomials N(2; m,x) := Sum_{k=0..m} T(m,k)*x^k. The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=2) Laguerre triangle L(2; n+m,m) = A062139(n+m,m), n >= 0, is N(2; m,x)/(1-x)^(3+2*m), with the row polynomials N(2; m,x).

			Triangle starts:
  n\k 0...1.....2......3..... 4.....;
  [0] 1;
  [1] 1,  3;
  [2] 1,  8,    6;
  [3] 1, 15,   30,    10;
  [4] 1, 24,   90,    80,    15;
  [5] 1, 35,  210,   350,   175,    21;
  [6] 1, 48,  420,  1120,  1050,   336,    28;
  [7] 1, 63,  756,  2940,  4410,  2646,   588,    36;
  [8] 1, 80, 1260,  6720, 14700, 14112,  5880,   960,   45;
  [9] 1, 99, 1980, 13860, 41580, 58212, 38808, 11880, 1485, 55.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), this sequence (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).
Sums include: A001791 (row), (-1)^n*A089849(n+1) (alternating sign row).
Diagonals: A000217 (k=n), A002417 (k=n-1), A001297 (k=n-2), A105946 (k=n-3), A105947 (k=n-4), A105948 (k=n-5), A107319 (k=n-6).
Columns: A005563 (k=1), A033487 (k=2), A027790 (k=3), A107395 (k=4), A107396 (k=5), A107397 (k=6), A107398 (k=7), A107399 (k=8).
Cf. A000108, A089849.

A062196:= func;
[A062196(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 21 2025

T := (n, k) -> binomial(n, k)*binomial(n + 2, k);
seq(seq(T(n, k), k=0..n), n=0..9); # Peter Luschny, Sep 30 2021

A062196[n_, k_]:= Binomial[n, k]*Binomial[n+2, k];
Table[A062196[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 21 2025 *)

def A062196(n,k): return binomial(n,k)*binomial(n+2,k)
print(flatten([[A062196(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Feb 21 2025

T(m, k) = [x^k] N(2; m, x), where N(2; m, x) = ((1-x)^(3+2*m))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+3))).
N(2; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+2-j)!/((m+2)!*(m-j)!)*(x^(m-j)))*(1-x)^j).
T(n,m) = binomial(n, m)*(binomial(n+1, m) + binomial(n+1, m-1)). - Vladimir Kruchinin, Apr 06 2018
From G. C. Greubel, Feb 21 2025: (Start)
T(2*n, n) = (n+1)^2*A000108(n)*A000108(n+1).
T(2*n-1, n) = (4*n^2 - 1)*A000108(n-1)*A000108(n), n >= 1.
T(2*n+1, n) = (1/2)*binomial(n+2,2)*A000108(n+1)*A000108(n+2). (End)

New name by Peter Luschny, Sep 30 2021

A062264 Coefficient triangle of certain polynomials N(4; m,x).

Original entry on oeis.org

1, 1, 5, 1, 12, 15, 1, 21, 63, 35, 1, 32, 168, 224, 70, 1, 45, 360, 840, 630, 126, 1, 60, 675, 2400, 3150, 1512, 210, 1, 77, 1155, 5775, 11550, 9702, 3234, 330, 1, 96, 1848, 12320, 34650, 44352, 25872, 6336, 495, 1, 117, 2808, 24024, 90090, 162162, 144144, 61776, 11583, 715

Offset: 0

Wolfdieter Lang, Jun 19 2001

The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=4) Laguerre triangle L(4; n+m,m) = A062140(n+m,m), n >= 0, is N(4; m,x)/(1-x)^(5+2*m), with the row polynomials N(4; m,x) := Sum_{k=0..m} T(m,k)*x^k.

			Triangle begins as:
  1;
  1,   5;
  1,  12,   15;
  1,  21,   63,    35;
  1,  32,  168,   224,     70;
  1,  45,  360,   840,    630,    126;
  1,  60,  675,  2400,   3150,   1512,    210;
  1,  77, 1155,  5775,  11550,   9702,   3234,    330;
  1,  96, 1848, 12320,  34650,  44352,  25872,   6336,    495;
  1, 117, 2808, 24024,  90090, 162162, 144144,  61776,  11583,   715;
  1, 140, 4095, 43680, 210210, 504504, 630630, 411840, 135135, 20020, 1001;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), this sequence (c=5), A062190 (c=6).
Columns: A028347 (k=2), A104473 (k=3), A104474 (k=4), A104475 (k=5), A027814 (k=6), A103604 (k=7), A104476 (k=8), A104478 (k=9).
Diagonals: A000332 (k=n), A027810 (k=n-1), A105249 (k=n-2), A105250 (k=n-3), A105251 (k=n-4), A105252 (k=n-5), A105253 (k=n-6), A105254 (k=n-7).
Sums: A002694 (row).

A062264:= func< n,k | Binomial(n,k)*Binomial(n+4,k) >;
[A062264(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 03 2025

A062264[n_, k_]:= Binomial[n,k]*Binomial[n+4,k];
Table[A062264[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 03 2025 *)

def A062264(n,k): return binomial(n,k)*binomial(n+4,k)
print(flatten([[A062264(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 03 2025

T(m, k) = [x^k] N(4; m, x), with N(4; m, x) = ((1-x)^(2*m+5))*(d^m/dx^m)((x^m)/(m!*(1-x)^(m+5))).
N(4; m, x) = Sum_{j=0..m} (binomial(m, j)*(2*m+4-j)!/((m+4)!*(m-j)!)*(x^(m-j))*(1-x)^j).
From G. C. Greubel, Mar 03 2025: (Start)
T(n, k) = binomial(n,k)*binomial(n+4,k).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/4)*( (1+(-1)^n)*(-1)^((n+2)/2)*(n^2 + 5*n - 2)*Catalan((n+2)/2)/(n+1) + 8*(1-(-1)^n)*(-1)^((n+1)/2)*Catalan((n+1)/2) ). (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

A027803 a(n) = 35(n+1)binomial(n+4, 7)/4.

Original entry on oeis.org

35, 350, 1890, 7350, 23100, 62370, 150150, 330330, 675675, 1301300, 2382380, 4176900, 7054320, 11531100, 18314100, 28352940, 42902475, 63596610, 92534750, 132382250, 186486300, 259008750, 355077450, 480957750, 644245875

Offset: 3

Thi Ngoc Dinh (via R. K. Guy)

Number of 12-subsequences of [ 1, n ] with just 4 contiguous pairs.

G. C. Greubel, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A053347, A062145.

A027803:= func< n | 5*(n+1)*(n+4)*Binomial(n+3,6)/4 >;
[A027803(n): n in [3..45]]; // G. C. Greubel, Mar 11 2025

Table[35 (n+1) Binomial[n+4,7]/4, {n,3,30}] (* or *) Table[Binomial[n+1, 4] Binomial[n+4,4], {n,3,30}] (* Michael De Vlieger, Mar 16 2016 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1}, {35,350,1890,7350,23100, 62370,150150,330330,675675}, 30] (* Harvey P. Dale, May 07 2022 *)

def A027803(n): return binomial(n+1,4)*binomial(n+4,4)
print([A027803(n) for n in range(3,46)]) # G. C. Greubel, Mar 11 2025

a(n) = 35*A053347(n-3).
G.f.: 35*x^3*(1+x)/(1-x)^9.
a(n) = C(n+1, 4)*C(n+4, 4). - Zerinvary Lajos, May 10 2005, corrected by R. J. Mathar, Mar 16 2016
From Amiram Eldar, Jan 25 2022: (Start)
Sum_{n>=3} 1/a(n) = 5929/225 - 8*Pi^2/3.
Sum_{n>=3} (-1)^(n+1)/a(n) = 4*Pi^2/3 - 197/15. (End)
E.g.f.: (1/576)*x^3*(3360 + 5040*x + 2352*x^2 + 448*x^3 + 36*x^4 + x^5 )*exp(x). - G. C. Greubel, Mar 11 2025

A062144 Sixth (unsigned) column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

Original entry on oeis.org

1, 54, 1890, 55440, 1496880, 38918880, 998917920, 25686460800, 667847980800, 17660868825600, 476843458291200, 13178219210956800, 373382877643776000, 10856825211488256000, 324153781314435072000

Offset: 0

Wolfdieter Lang, Jun 19 2001

			a(2) = (2+5)! * binomial(2+8,8)/ 5! = (5040 * 45) / 120 = 1890. - _Indranil Ghosh_, Feb 24 2017

Indranil Ghosh, Table of n, a(n) for n = 0..400
Index entries for sequences related to Laguerre polynomials

Cf. A062143.

[Factorial(n+5)*Binomial(n+8, 8)/Factorial(5): n in [0..20]]; // G. C. Greubel, May 11 2018

Table[(n+5)!*Binomial[n+8,8]/5!,{n,0,14}] (* Indranil Ghosh, Feb 24 2017 *)

a(n)=(n+5)!*binomial(n+8, 8)/5! \\ Indranil Ghosh, Feb 24 2017

import math
f=math.factorial
def C(n, r):return f(n)/f(r)/f(n-r)
def A062144(n): return f(n+5)*C(n+8, 8)/f(5) # Indranil Ghosh, Feb 24 2017

a(n) = (n+5)!*binomial(n+8, 8)/5!.
E.g.f.: N(3;5, x)/(1-x)^14 with N(3;5, x) := Sum_{k=0..5} A062145(5, k) *x^k = 1 +40*x +280*x^2 +560*x^3 +350*x^4 +56*x^5.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j) then a(n-5) = (-1)^(n-1)*f(n,5, -9), (n>=5). - Milan Janjic, Mar 01 2009

A105939 a(n) = binomial(n+3,3)*binomial(n+6,3).

Original entry on oeis.org

20, 140, 560, 1680, 4200, 9240, 18480, 34320, 60060, 100100, 160160, 247520, 371280, 542640, 775200, 1085280, 1492260, 2018940, 2691920, 3542000, 4604600, 5920200, 7534800, 9500400, 11875500, 14725620, 18123840, 22151360, 26898080, 32463200, 38955840, 46495680

Offset: 0

Zerinvary Lajos, Apr 27 2005

a(n) is the number of ordered pairs (A,B) of size 3 subsets of {1,2,...,n+6} such that A and B are disjoint. - Geoffrey Critzer, Sep 03 2013

			If n=0 then C(0+3,0)*C(0+6,3) = C(3,0)*C(6,3) = 1*20 = 20.
If n=8 then C(8+3,8)*C(8+6,3) = C(11,8)*C(14,3) = 165*364 = 60060.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000292, A062145.

A105939:= func< n | 20*Binomial(n+6,6) >;
[A105939(n): n in [0..40]]; // G. C. Greubel, Mar 11 2025

nn=25; f[x_]:=Exp[x](x^3/3!)^2;Range[0,nn]! CoefficientList[Series[ a=f''''''[x],{x,0,nn}],x] (* Geoffrey Critzer, Sep 03 2013 *)
Table[Binomial[n+3,3]Binomial[n+6,3],{n,0,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{20,140,560,1680,4200,9240,18480},30] (* Harvey P. Dale, Mar 09 2022 *)
20*Binomial[Range[0,40] +6,6] (* G. C. Greubel, Mar 11 2025 *)

def A105939(n): return 20*binomial(n+6,6)
print([A105939(n) for n in range(41)]) # G. C. Greubel, Mar 11 2025

G.f.: 20/(1-x)^7. - Colin Barker, Jun 06 2012
E.g.f.: (d/dx)^6 (x^3/3!)^2 * exp(x). - Geoffrey Critzer, Sep 03 2013
a(n) = A000292(n+1)*A000292(n+4) = 20*A000579(n+6). - R. J. Mathar, Nov 30 2015
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=0} 1/a(n) = 3/50.
Sum_{n>=0} (-1)^n/a(n) = 48*log(2)/5 - 661/100. (End)
E.g.f.: (1/36)*(720 + 4320*x + 5400*x^2 + 2400*x^3 + 450*x^4 + 36*x^5 + x^6)*exp(x). - G. C. Greubel, Mar 11 2025

More terms from Geoffrey Critzer, Sep 03 2013

cp's OEIS Frontend

A008459 Square the entries of Pascal's triangle.

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A028552 a(n) = n*(n+3).

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A062190 Coefficient triangle of certain polynomials N(5; m,x).

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A062196 Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).

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A062264 Coefficient triangle of certain polynomials N(4; m,x).

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

A027803 a(n) = 35(n+1)binomial(n+4, 7)/4.