A063746 -id:A063746 - 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

A083906 Table read by rows: T(n, k) is the number of length n binary words with exactly k inversions.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 2, 5, 3, 4, 3, 1, 6, 4, 6, 6, 6, 2, 2, 7, 5, 8, 9, 11, 9, 7, 4, 3, 1, 8, 6, 10, 12, 16, 16, 18, 12, 12, 8, 6, 2, 2, 9, 7, 12, 15, 21, 23, 29, 27, 26, 23, 21, 15, 13, 7, 4, 3, 1, 10, 8, 14, 18, 26, 30, 40, 42, 48, 44, 46, 40, 40, 30, 26, 18, 14, 8, 6, 2, 2

Offset: 0

Alford Arnold, Jun 19 2003

There are A033638(n) values in the n-th row, compliant with the order of the polynomial.
In the example for n=6 detailed below, the orders of [6, k]_q are 1, 6, 9, 10, 9, 6, 1 for k = 0..6,
the maximum order 10 defining the row length.
Note that 1 6 9 10 9 6 1 and related distributions are antidiagonals of A077028.
A083480 is a variation illustrating a relationship with numeric partitions, A000041.
The rows are formed by the nonzero entries of the columns of A049597.
The coefficient of q^j in the Gaussian polynomial [n, m]q is the number of binary words on alphabet {0,1} of length n having m 1's and j inversions. Hence T(n, k) is the number of length n binary words with exactly k inversions. - _Geoffrey Critzer, May 14 2017
If n is even the n-th row converges to n+1, n-1, n-4, ..., 19, 13, 7, 4, 3, 1 which is A029552 reversed, and if n is odd the sequence is twice A098613. - Michael Somos, Jun 25 2017

			When viewed as an array with A033638(r) entries per row, the table begins:
. 1 ............... : 1
. 2 ............... : 2
. 3 1 ............. : 3 + q = (1) + (1+q) + (1)
. 4 2 2 ........... : 4 + 2q + 2q^2 = 1 + (1+q+q^2) + (1+q+q^2) + 1
. 5 3 4 3 1 ....... : 5 + 3q + 4q^2 + 3q^3 + q^4
. 6 4 6 6 6 2 2
. 7 5 8 9 11 9 7 4 3 1
. 8 6 10 12 16 16 18 12 12 8 6 2 2
. 9 7 12 15 21 23 29 27 26 23 21 15 13 7 4 3 1
...
The second but last row is from the sum over 7 q-polynomials coefficients:
. 1 ....... : 1 = [6,0]_q
. 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,1]_q
. 1 1 2 2 3 2 2 1 1 ....... : 1+q+2q^2+2q^3+3q^4+2q^5+2q^6+q^7+q^8 = [6,2]_q
. 1 1 2 3 3 3 3 2 1 1 ....... : 1+q+2q^2+3q^3+3q^4+3q^5+3q^6+2q^7+q^8+q^9 = [6,3]_q
. 1 1 2 2 3 2 2 1 1 ....... : 1+q+2q^2+2q^3+3q^4+2q^5+2q^6+q^7+q^8 = [6,4]_q
. 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,5]_q
. 1 ....... : 1 = [6,6]_q

George E. Andrews, 'Theory of Partitions', 1976, page 242.

Seiichi Manyama, Rows n = 0..48, flattened
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Alexander Gruber, "The Egg:" Bizarre behavior of the roots of a family of polynomials Mathematics StackExchange Oct 04 2012
MathOverflow, Partition numbers and Gaussian binomial coefficient
Eric Weisstein, q-Binomial Coefficient, Mathworld.
Wikipedia, q-binomial
Index entries for sequences related to binary linear codes

Cf. A000025, A000034, A000041, A016116, A029552, A033638, A060546, A063746, A077028, A083479, A083480, A098613, A260460.

R:=PowerSeriesRing(Rationals(), 100);
qBinom:= func< n,k,x | n eq 0 or k eq 0 select 1 else (&*[(1-x^(n-j))/(1-x^(j+1)): j in [0..k-1]]) >;
A083906:= func< n,k | Coefficient(R!((&+[qBinom(n,k,x): k in [0..n]]) ), k) >;
[A083906(n,k): k in [0..Floor(n^2/4)], n in [0..12]]; // G. C. Greubel, Feb 13 2024

QBinomial := proc(n,m,q) local i ; factor( mul((1-q^(n-i))/(1-q^(i+1)),i=0..m-1) ) ; expand(%) ; end:
A083906 := proc(n,k) add( QBinomial(n,m,q),m=0..n ) ; coeftayl(%,q=0,k) ; end:
for n from 0 to 10 do for k from 0 to A033638(n)-1 do printf("%d,",A083906(n,k)) ; od: od: # R. J. Mathar, May 28 2009
T := proc(n, k) if n < 0 or k < 0 or k > floor(n^2/4) then return 0 fi;
if n < 2 then return n + 1 fi; 2*T(n-1, k) - T(n-2, k) + T(n-2, k - n + 1) end:
seq(print(seq(T(n, k), k = 0..floor((n/2)^2))), n = 0..8);  # Peter Luschny, Feb 16 2024

Table[CoefficientList[Total[Table[FunctionExpand[QBinomial[n, k, q]], {k, 0, n}]],q], {n, 0, 10}] // Grid (* Geoffrey Critzer, May 14 2017 *)

{T(n, k) = polcoeff(sum(m=0, n, prod(k=0, m-1, (x^n - x^k) / (x^m - x^k))), k)}; /* Michael Somos, Jun 25 2017 */

def T(n,k): # T = A083906
    if k<0 or k> (n^2//4): return 0
    elif n<2 : return n+1
    else: return 2*T(n-1, k) - T(n-2, k) + T(n-2, k-n+1)
flatten([[T(n,k) for k in range(int(n^2//4)+1)] for n in range(13)]) # G. C. Greubel, Feb 13 2024

T(n, k) is the coefficient [q^k] of the Sum_{m=0..n} [n, m]_q over q-Binomial coefficients.
Row sums: Sum_{k=0..floor(n^2/4)} T(n,k) = 2^n.
For n >= k, T(n+1,k) = T(n, k) + A000041(k). - Geoffrey Critzer, Feb 12 2021
Sum_{k=0..floor(n^2/4)} (-1)^k*T(n, k) = A060546(n). - G. C. Greubel, Feb 13 2024
From Mikhail Kurkov, Feb 14 2024: (Start)
T(n, k) = 2*T(n-1, k) - T(n-2, k) + T(n-2, k - n + 1) for n >= 2 and 0 <= k <= floor(n^2/4).
Sum_{i=0..n} T(n-i, i) = A000041(n+1). Note that upper limit of the summation can be reduced to A083479(n) = (n+2) - ceiling(sqrt(4*n)).
Both results were proved (see MathOverflow link for details). (End)
From G. C. Greubel, Feb 17 2024: (Start)
T(n, floor(n^2/4)) = A000034(n).
Sum_{k=0..floor(n^2/4)} (-1)^k*T(n, k) = A016116(n+1).
Sum_{k=0..(n + 2) - ceiling(sqrt(4*n))} (-1)^k*T(n - k, k) = (-1)^n*A000025(n+1) = -A260460(n+1). (End)

Edited by R. J. Mathar, May 28 2009

New name using a comment from Geoffrey Critzer by Peter Luschny, Feb 17 2024

A063075 Number of partitions of 2n^2 whose Ferrers-plot fits within a 2n X 2n box and cover an n X n box; number of ways to cut a 2n X 2n chessboard into two equal-area pieces along a non-descending line from lower left to upper right and passing through the center.

Original entry on oeis.org

1, 2, 8, 48, 390, 3656, 37834, 417540, 4836452, 58130756, 719541996, 9121965276, 117959864244, 1551101290792, 20689450250926, 279395018584860, 3813887739881184, 52557835511244660, 730403326965323706

Offset: 0

Wouter Meeussen, Aug 03 2001

nonn

			For a 6 X 6 board (n=3) the partition (6,6,2,2,2,0) represents a Ferrers plot that does not pass through the center of a 6*6 box.
From _Paul D. Hanna_, Dec 12 2006: (Start)
Central q-binomial coefficients begin:
  1;
  1 + q;
  1 + q + 2*q^2 + q^3 + q^4;
  1 + q + 2*q^2 + 3*q^3 + 3*q^4 + 3*q^5 + 3*q^6 + 2*q^7 + q^8 + q^9;
the coefficients of q in these polynomials form the rows of triangle A063746.
The sums of squared terms in rows of A063746 equal this sequence. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..450 (terms 0..80 from Paul D. Hanna)

Cf. A047993, A063074, A063746.

Table[(#.#)&@Table[T[k, n, n], {k, 0, n^2}], {n, 0, 24}] (* with T[m, a, b] as defined in A047993 *)

a(n)=polcoef((prod(j=1,n,(1-q^(n+j))/(1-q^j)))^2,n^2,q) \\ Tani Akinari, Jan 28 2022

a(n) = Sum_{k=0..n^2} A063746(n,k)^2; i.e., equals the sums of the squares of the coefficients of q in the central q-binomial coefficients. - Paul D. Hanna, Dec 12 2006
a(n) = [q^(n^2)](Product_{j=1..n} (1-q^(n+j))/(1-q^j))^2. - Tani Akinari, Jan 28 2022
a(n) ~ sqrt(3) * 2^(4*n - 1/2) / (Pi^(3/2) * n^(5/2)). - Vaclav Kotesovec, Feb 02 2022

A274888 Triangle read by rows: the q-analog of the swinging factorial which is defined as q-multinomial([floor(n/2), n mod 2, floor(n/2)]).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 4, 5, 6, 5, 4, 2, 1, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 1, 2, 4, 7, 10, 13, 16, 17, 17, 16, 13, 10, 7, 4, 2, 1, 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1, 1, 2, 4, 7, 12, 17, 24, 31, 39, 45, 51, 54, 56, 54, 51, 45, 39, 31, 24, 17, 12, 7, 4, 2, 1

Offset: 0

Peter Luschny, Jul 19 2016

The q-swing_factorial(n) is a univariate polynomial over the integers with degree floor((n+1)/2)^2 + ((n+1) mod 2) and at least floor(n/2) irreducible factors.
Evaluated at q=1 q-swing_factorial(n) gives the swinging factorial A056040(n).
Combinatorial interpretation: The definition of an orbital system is given in A232500 and in the link 'Orbitals'. The number of orbitals over n sectors is counted by the swinging factorial.
The major index of an orbital is the sum of the positions of steps which are immediately followed by a step with strictly smaller value. This statistic is an extension of the major index statistic given in A063746 which reappears in the even numbered rows here. This reflects the fact that the swinging factorial can be seen as an extension of the central binomial. As in the case of the central binomial also in the case of the swinging factorial the major index coincides with its q-analog.

			The polynomials start:
[0] 1
[1] 1
[2] q + 1
[3] (q + 1) * (q^2 + q + 1)
[4] (q^2 + 1) * (q^2 + q + 1)
[5] (q^2 + 1) * (q^2 + q + 1) * (q^4 + q^3 + q^2 + q + 1)
[6] (q + 1) * (q^2 - q + 1) * (q^2 + 1) * (q^4 + q^3 + q^2 + q + 1)
The coefficients of the polynomials start:
[n] [k=0,1,2,...] [row sum]
[0] [1] [1]
[1] [1] [1]
[2] [1, 1] [2]
[3] [1, 2, 2, 1] [6]
[4] [1, 1, 2, 1, 1] [6]
[5] [1, 2, 4, 5, 6, 5, 4, 2, 1] [30]
[6] [1, 1, 2, 3, 3, 3, 3, 2, 1, 1] [20]
[7] [1, 2, 4, 7, 10, 13, 16, 17, 17, 16, 13, 10, 7, 4, 2, 1] [140]
[8] [1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1] [70]
T(5, 4) = 6 because the 2 orbitals [-1,-1,1,1,0] and [-1,0,1,1,-1] have at position 4 and the 4 orbitals [0,-1,1,-1,1], [1,-1,0,-1,1], [1,-1,1,-1,0] and [1,0,1,-1,-1] at positions 1 and 3 a down step.

Peter Luschny, Orbitals

Cf. A056040 (row sums), A274887 (q-factorial), A063746 (q-central_binomial).

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

QSwingFactorial_coeffs := proc(n) local P,a,b;
a := mul((p^(n-i)-1)/(p^(i+1)-1),i=0..iquo(n,2)-1);
b := ((p^(iquo(n,2)+1)-1)/(p-1))^((1-(-1)^n)/2);
P := simplify(a*b); seq(coeff(P,p,j),j=0..degree(P)) end:
for n from 0 to 9 do print(QSwingFactorial_coeffs(n)) od;
# Alternatively (recursive):
with(QDifferenceEquations):
QSwingRec := proc(n,q) local r; if n = 0 then return 1 fi:
if irem(n,2) = 0 then r := (1+q^(n/2))/QBrackets(n/2,q)
else r := QBrackets(n,q) fi; r*QSwingRec(n-1,q) end:
Trow := proc(n) expand(QSimplify(QSwingRec(n,q)));
seq(coeff(%,q,j),j=0..degree(%)) end: seq(Trow(n),n=0..10);

p[n_] := QFactorial[n, q] / QFactorial[Quotient[n, 2], q]^2
Table[CoefficientList[p[n] // FunctionExpand, q], {n,0,9}] // Flatten

from sage.combinat.q_analogues import q_factorial
def q_swing_factorial(n, q=None):
    return q_factorial(n)//q_factorial(n//2)^2
for n in (0..8): print(q_swing_factorial(n).list())

# uses[unit_orbitals from A274709]
# Brute force counting
def orbital_major_index(n):
    S = [0]*(((n+1)//2)^2 + ((n+1) % 2))
    for u in unit_orbitals(n):
        L = [i+1 if u[i+1] < u[i] else 0 for i in (0..n-2)]
        # i+1 because u is 0-based whereas convention assumes 1-base.
        S[sum(L)] += 1
    return S
for n in (0..9): print(orbital_major_index(n))

q_swing_factorial(n) = q_factorial(n)/q_factorial(floor(n/2))^2.
q_swing_factorial(n) = q_binomial(n-eta(n),floor((n-eta(n))/2))*q_int(n)^eta(n) with eta(n) = (1-(-1)^n)/2.
Recurrence: q_swing_factorial(0,q) = 1 and for n>0 q_swing_factorial(n,q) = r*q_swing_factorial(n-1,q) with r = (1+q^(n/2))/[n/2;q] if n is even else r = [n;q]. Here [a;q] are the q_brackets.
The generating polynomial for row n is P_n(p) = ((p^(floor(n/2)+1)-1)/(p-1))^((1-(-1)^n)/2)*Product_{i=0..floor(n/2)-1}((p^(n-i)-1)/(p^(i+1)-1)).

A136621 Transpose T(n,k) of Parker's partition triangle A047812 (n >= 1 and 0 <= k <= n-1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 5, 1, 1, 11, 20, 9, 1, 1, 18, 51, 48, 13, 1, 1, 26, 112, 169, 100, 20, 1, 1, 38, 221, 486, 461, 194, 28, 1, 1, 52, 411, 1210, 1667, 1128, 352, 40, 1, 1, 73, 720, 2761, 5095, 4959, 2517, 615, 54, 1, 1, 97, 1221, 5850, 13894, 18084, 13241, 5288, 1034, 75, 1

Offset: 1

Alford Arnold, Jan 26 2008

Parker's triangle is closely associated with q-binomial coefficients and Gaussian polynomials; cf. A063746. For example, row 4 of A063746 is 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1, the coefficients of [8, 4], while the entries in row 4 of A047812 are the coefficients of q^(k*(4+1)) = q^(5*k) in [8, 4] where k runs from 0 to n-1 = 3. Likewise, by symmetry, "1 7 5 1" is embedded also because they are the coefficients of q^(5*(3-k)), where k runs from 0 to n-1 = 3. [Edited by Petros Hadjicostas, May 30 2020]

			Row four of A047812 is 1 5 7 1, so row four of the present entry is 1 7 5 1.
From _Petros Hadjicostas_, May 30 2020: (Start)
Triangle T(n,k) (with rows n >= 1 and columns k = 0..n-1) begins:
  1;
  1,  1;
  1,  3,   1;
  1,  7,   5,    1;
  1, 11,  20,    9,    1;
  1, 18,  51,   48,   13,    1;
  1, 26, 112,  169,  100,   20,   1;
  1, 38, 221,  486,  461,  194,  28,  1;
  1, 52, 411, 1210, 1667, 1128, 352, 40, 1;
  ... (End)

Alois P. Heinz, Rows n = 1..141, flattened
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.
Wikipedia, E. T. Parker.

Cf. A000108 (Catalan row sums), A047812, A063746.

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i
       b((n-k-1)*(n+1), n$2):
seq(seq(T(n, k), k=0..n-1), n=1..12);  # Alois P. Heinz, May 30 2020

T[n_, k_]:= SeriesCoefficient[QBinomial[2*n, n, q], {q, 0, k*(n+1)}];
Table[T[n, n-k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, May 31 2020 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[n < 0 || t i < n, 0, b[n, i - 1, t] + b[n - i, Min[i, n - i], t - 1]]];
T[n_, k_] := b[(n-k-1)(n+1), n, n];
Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

T(n, k) = #partitions(k*(n+1), n, n);
for (n=1, 10, for (k=0, n-1, print1(T(n, n-1-k), ", "); ); print(); ); \\ Petros Hadjicostas, May 30 2020
/* Second program, courtesy of G. C. Greubel */
T(n,k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
vector(12, n, vector(n, k, T(n,n-k))) \\ Petros Hadjicostas, May 31 2020

def T(n,k):
    P. = PowerSeriesRing(ZZ, k*(n+1)+1)
    return P( q_binomial(2*n, n, x) ).list()[k*(n+1)]
[[ T(n,n-k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, May 31 2020

Name edited by Petros Hadjicostas, May 30 2020

A102422 Number of partitions of n with k <= 5 parts and each part p <= 5.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 9, 11, 14, 16, 18, 19, 20, 20, 19, 18, 16, 14, 11, 9, 7, 5, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Thomas Wieder, Jan 09 2005

There are only 26 nonzero terms.
Contribution from Toby Gottfried, Feb 19 2009: (Start)
a(n) is the number of partitions of n+5 into exactly 5 parts with each part p: 1 <= p <= 6
i.e. the number of different ways to get a total of n+5 with 5 (normal, 6-sided) dice in any order (End)

			a(7)=11 because we can write 7=1+2+2+2 or 5+2 or 1+2+4 or 3+4 or 1+3+3 or 1+1+1+1+3 or 1+1+2+3 or 2+2+3 or 1+1+1+2+2 1+1+1+4 or 1+1+5.
A total of 8 comes from 1+1+1+1+4, 1+1+1+2+3, 1+1+2+2+2 and a(3) = 3 [8 = 3+5] [From _Toby Gottfried_, Feb 19 2009]

See A102420 for k=5 and p<=5.
Cf. A000041, A102420, A063746.
Contribution from Toby Gottfried, Feb 19 2009: (Start)
A102420 has the numbers for 4 dice
A063260 gives the number of permuted rolls of each possible total for any number of dice. (End)

G.f.: = 1+z+2*z^2+3*z^3+5*z^4+7*z^5+9*z^6+11*z^7+14*z^8+16*z^9+18*z^10+19*z^11+20*z^12+20*z^13+19*z^14+18*z^15+16*z^16+14*z^17+11*z^18+9*z^19 +7*z^20+5*z^21+3*z^22+2*z^23+z^24+z^25.

A274887 Triangle read by rows: coefficients of the q-factorial.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 5, 3, 1, 1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1, 1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1, 1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1

Offset: 0

Peter Luschny, Jul 19 2016

The main entry for this sequence is A008302 (Mahonian numbers).
q-factorial(n) is a univariate polynomial over the integers with degree n*(n-1)/2.
Evaluated at q=1 the q-factorial(n) gives the factorial A000142(n).

			The polynomials start:
[0] 1
[1] 1
[2] q + 1
[3] (q + 1) * (q^2 + q + 1)
[4] (q + 1)^2 * (q^2 + 1) * (q^2 + q + 1)
[5] (q + 1)^2 * (q^2 + 1) * (q^2 + q + 1) * (q^4 + q^3 + q^2 + q + 1)
The triangle starts:
[1]
[1]
[1, 1]
[1, 2, 2, 1]
[1, 3, 5, 6, 5, 3, 1]
[1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1]
[1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1]

G. C. Greubel, Rows n = 0..30 of triangle, flattened
NIST Digital Library of Mathematical Functions, q-Factorials. (Release 1.0.11 of 2016-06-08)

Cf. A008302 (the same for all n > 0), A000142 (row sums), A063746 (q-central_binomial), A129175 (q-Catalan), A274886 (q-extended_Catalan), A274888 (q-swing_factorial), A275216 (q-binomial), A275215 (q-Narayana).

B:= func< n,x | n eq 0 select 1 else (&*[1-x^j: j in [1..n]])/(1-x)^n >;
R:=PowerSeriesRing(Integers(), 30);
[Coefficients(R!( B(n,x) )): n in [0..9]]; // G. C. Greubel, May 22 2019

Table[CoefficientList[QFactorial[n,q]//FunctionExpand, q], {n,0,9} ]//Flatten

for(n=0, 8, print1(Vec(if(n==0, 1, prod(j=1, n, 1-x^j)/(1-x)^n)), ", "); print(); ) \\ G. C. Greubel, May 23 2019

from sage.combinat.q_analogues import q_factorial
for n in (0..5): print(q_factorial(n).list())

a(n) = A008302(n) for all n > 0. - M. F. Hasler, Jan 06 2024

A138774 Triangle read by rows: T(n,k) is the number of partitions of k that fit into a 2n by n box (n>=0; 0<=k<=2n^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 1, 1, 1, 1, 2, 3, 4, 5, 7, 7, 8, 8, 8, 7, 7, 5, 4, 3, 2, 1, 1, 1, 1, 2, 3, 5, 6, 9, 11, 15, 17, 21, 23, 27, 28, 31, 31, 33, 31, 31, 28, 27, 23, 21, 17, 15, 11, 9, 6, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 36, 45, 53, 63, 72, 83, 92, 103, 111, 121

Offset: 0

Emeric Deutsch, May 03 2008

Row n contains 1+2n^2 terms.
Sum of entries in row n is binomial(3n,n) (=A005809(n)).
Also, number of n-multisets in [0..2n] whose elements sum up to k. M. F. Hasler, Apr 15 2012

			T(2,4)=3 because we have 4, 31 and 22.
T(3,13)=5 because we have 661,652,643,553 and 544.
Triangle starts:
1;
1,1,1;
1,1,2,2,3,2,2,1,1;
1,1,2,3,4,5,7,7,8,8,8,7,7,5,4,3,2,1,1;

G. E. Andrews and K. Eriksson, Integer partitions, Cambridge Univ. Press, 2004, pp. 67-69.

Alois P. Heinz, Rows n = 0..25, flattened

Cf. A005809, A063746.

br:=proc(n) options operator, arrow: sum(q^i,i=0..n-1) end proc: f:= proc(n) options operator, arrow: mul(br(j),j=1..n) end proc: cbr:=proc(n,k) options operator, arrow: simplify(f(n)/(f(k)*f(n-k))) end proc: for n from 0 to 5 do P[n]:=sort(expand(cbr(3*n,n))) end do: for n from 0 to 5 do seq(coeff(P[n],q, j),j=0..2*n^2) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i, k) option remember;
      `if`(n=0, 1, `if`(i<1 or k<1, 0, b(n, i-1, k)+
      `if`(i>n, 0, b(n-i, i, k-1))))
    end:
T:= n-> seq(b(k, min(n, k), 2*n), k=0..2*n^2):
seq(T(n), n=0..6);  # Alois P. Heinz, Apr 05 2012

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1 || k<1, 0, b[n, i-1, k] + If[i>n, 0, b[n-i, i, k-1]]]]; T[n_] := Table[b[k, Min[n, k], 2*n], {k, 0, 2 n^2}]; Table[T[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)

T138774(n,k)=polcoeff(prod(i=0,2*n,sum(j=0,n,x^(j*(i*(2*n^2+n+1)+1)),O(x^(k*(2*n^2+n+1)+n+1)))),k*(2*n^2+n+1)+n) /* replacing the inner sum by the expression for the geometric series seems less efficient */
for(n=0,5,for(k=0,2*n^2,print1(T138774(n,k)","))) \\ M. F. Hasler, Apr 15 2012

G.f. of row n = the q-binomial coefficient [3n,n].

A260894 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 - x^(n+k))/(1 - x^k).

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 10, 15, 20, 28, 36, 50, 64, 86, 110, 145, 184, 238, 300, 384, 481, 608, 756, 948, 1172, 1456, 1790, 2208, 2700, 3310, 4026, 4906, 5941, 7200, 8680, 10470, 12570, 15092, 18050, 21583, 25718, 30634, 36376, 43174, 51102, 60446, 71324, 84102, 98948

Offset: 0

Paul D. Hanna, Aug 03 2015

nonn

Antidiagonal sums of irregular triangle A063746 yields this sequence at offset 1.
From Jeremy Lovejoy, Oct 13 2022: (Start)
a(n) is also the number of partitions of n whose rank is at least -1, the rank of a partition being the largest part minus the number of parts.   For example, the 7 partitions of 5 are (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), and (1,1,1,1,1), having ranks 4,2,1,0,-1,-2, and -4, respectively, and so a(5) = 5.
a(n) is also the number of partitions of n such that the smallest positive multiple of 3 not occurring in the partition is not a multiple of 6.   For example, in the 7 partitions of 5 listed above, the smallest positive multiples of 3 not occurring are 3,3,6,6,3,3, and 3, respectively, and so a(5) = 5.  (End)

			G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 8*x^6 + 10*x^7 +...
where
A(x) = 1 + x*(1-x^2)/(1-x) + x^2*(1-x^3)*(1-x^4)/((1-x)*(1-x^2)) + x^3*(1-x^4)*(1-x^5)*(1-x^6)/((1-x)*(1-x^2)*(1-x^3)) + x^4*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^8)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) +...

George E. Andrews, David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
Rupam Barman, Ajit Singh, On Mex-related partition functions of Andrews and Newman, arXiv:2009.11602 [math.NT], 2020.

Cf. A064174, A063746.

Cf. A000041, A064173, A064174.

Table[SeriesCoefficient[Sum[x^k * Product[(1-x^(k+j))/(1-x^j), {j,1,k}], {k,0,n}], {x,0,n}], {n,0,50}] (* Vaclav Kotesovec, Aug 08 2015 *)

{a(n) = local(A=1); A = sum(m=0,n,x^m*prod(k=1,m,(1-x^(m+k))/(1-x^k +x*O(x^n)))); polcoeff(A,n)}
for(n=0,60,print1(a(n),", "))

G.f.: (1/Product_{n>=1}(1-x^n))*Sum_{n>=0}(-1)^n*x^((3*n^2+3*n)/2). - Jeremy Lovejoy, Oct 13 2022.

cp's OEIS Frontend

A107110 Square array by antidiagonals where T(n,k) is the number of partitions of k into no more than n parts each no more than n. Visible version of A063746.

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A008459 Square the entries of Pascal's triangle.

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A083906 Table read by rows: T(n, k) is the number of length n binary words with exactly k inversions.

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A063075 Number of partitions of 2n^2 whose Ferrers-plot fits within a 2n X 2n box and cover an n X n box; number of ways to cut a 2n X 2n chessboard into two equal-area pieces along a non-descending line from lower left to upper right and passing through the center.

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A274888 Triangle read by rows: the q-analog of the swinging factorial which is defined as q-multinomial([floor(n/2), n mod 2, floor(n/2)]).

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A136621 Transpose T(n,k) of Parker's partition triangle A047812 (n >= 1 and 0 <= k <= n-1).

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