A152291 -id:A152291 - cp's OEIS frontend

A152290 Coefficients in a q-analog of the LambertW function, as a triangle read by rows.

1, 1, 2, 1, 5, 5, 5, 1, 14, 21, 31, 30, 19, 9, 1, 42, 84, 154, 210, 245, 217, 175, 105, 49, 14, 1, 132, 330, 708, 1176, 1722, 2148, 2386, 2358, 2080, 1618, 1086, 644, 294, 104, 20, 1, 429, 1287, 3135, 6006, 10164, 15093, 20496, 25188, 28770, 30225, 29511, 26571

Offset: 0

Paul D. Hanna, Dec 02 2008

T(n,k) is the number of parking functions of length n with k inversions. - Kyle Celano, Aug 18 2025

			Triangle, with columns k=0..n*(n-1)/2 for row n>=0, begins:
  1;
  1;
  2, 1;
  5, 5, 5, 1;
  14, 21, 31, 30, 19, 9, 1;
  42, 84, 154, 210, 245, 217, 175, 105, 49, 14, 1;
  132, 330, 708, 1176, 1722, 2148, 2386, 2358, 2080, 1618, 1086, 644, 294, 104, 20, 1;
  429, 1287, 3135, 6006, 10164, 15093, 20496, 25188, 28770, 30225, 29511, 26571, 22161, 16926, 11832, 7392, 4089, 1932, 714, 195, 27, 1;...
where row sums = (n+1)^(n-1) and column 0 is A000108 (Catalan numbers).
Row sums at q=-1 = (n+1)^[(n-1)/2] (A152291): [1,1,1,4,5,36,49,512,729,...].
The generating function starts:
A(x,q) = 1 + x + (2 + q)*x^2/faq(2,q) + (5 + 5*q + 5*q^2 + q^3)*x^3/faq(3,q) + (14 + 21*q + 31*q^2 + 30*q^3 + 19*q^4 + 9*q^5 + q^6)*x^4/faq(4,q) + ...
G.f. satisfies: A(x,q) = e_q( x*A(x,q), q), where the q-exponential series e_q(x,q) begins:
e_q(x,q) = 1 + x + x^2/faq(2,q) + x^3/faq(3,q) +...+ x^n/faq(n,q) +...
The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1):
faq(0,q)=1, faq(1,q)=1, faq(2,q)=(1+q), faq(3,q)=(1+q)*(1+q+q^2),
faq(4,q)=(1+q)*(1+q+q^2)*(1+q+q^2+q^3), ...
Special cases of g.f.:
q=0: A(x,0) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 +... (Catalan)
q=1: A(x,1) = 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 +...= LambertW(-x)/(-x)
q=2: A(x,2) = 1 + x + 4/3*x^2 + 43/21*x^3 + 1076/315*x^4 + 58746/9765*x^5 +...
q=-1: Can A(x,-1) be defined? See A152291.

Paul D. Hanna, Rows 0 to 30 of the triangle, flattened.
Kyle Celano, Jennifer Elder, Kimberly P. Hadaway, Pamela E. Harris, Amanda Priestley, and Gabe Udell, Inversions in parking functions, arXiv:2508.11587 [math.CO], 2025.
Eric Weisstein's World of Mathematics, q-Exponential Function.
Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A152291 (q=-1), A000272 (row sums), A000108 (column 0), A002054 (column 1).
Cf. A152282 (q=2), A152283 (q=3).
Cf. A121774.

/* G.f.: LambertW_q(x,q) = (1/x)*Series_Reversion( x/e_q(x,q) ): */
{T(n,k)=local(e_q=1+sum(j=1,n,x^j/prod(i=1,j,(q^i-1)/(q-1))),LW_q=serreverse(x/e_q+x^2*O(x^n))/x); polcoeff(polcoeff(LW_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1))+q*O(q^k),k,q)}
for(n=0,8,for(k=0,n*(n-1)/2,print1(T(n,k),","));print(""))

G.f.: A(x,q) = Sum_{n>=0} Sum_{k=0..n*(n-1)/2} T(n,k)*q^k*x^n/faq(n,q), where faq(n,q) is the q-factorial of n.
G.f.: A(x,q) = (1/x)*Series_Reversion( x/e_q(x,q) ) where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential function.
G.f. satisfies: A(x,q) = e_q( x*A(x,q), q) and A( x/e_q(x,q), q) = e_q(x,q).
G.f. at q=1: A(x,1) = LambertW(-x)/(-x).
Row sums at q=+1: Sum_{k=0..n*(n-1)/2} T(n,k) = (n+1)^(n-1).
Row sums at q=-1: Sum_{k=0..n*(n-1)/2} T(n,k)*(-1)^k = (n+1)^[(n-1)/2] (A152291).
Sum_{k=0..n*(n-1)/2} T(n,k)*exp(2*Pi*I*k/n) = 1 for n>=1; i.e., the n-th row sum at q = exp(2*Pi*I/n), the n-th root of unity, equals 1 for n>=1. - Paul D. Hanna, Dec 04 2008
Sum_{k=0..n*(n-1)/2} T(n,k)*q^k = Sum_{pi} n!/(n-k+1)!*faq(n,q)/Product_{i=1..n} e(i)!*faq(i,q)^e(i), where pi runs through all nonnegative integer solutions of e(1)+2*e(2)+...+n*e(n) = n and k = e(1)+e(2)+...+e(n). - Vladeta Jovovic, Dec 05 2008
Sum_{k=0..[n/2]} T(n, n*k) = (1/n)*Sum_{d|n} phi(n/d)*(n+1)^(d-1), for n>0, with a(0)=1. - Paul D. Hanna, Jul 18 2013
Sum_{k=0..[n/2]} T(n, n*k) = A121774(n), the number of n-bead necklaces with n+1 colors, divided by (n+1). - Paul D. Hanna, Jul 18 2013

A078707 Number of vectors of length n that are symmetric about the middle, where each element is drawn from a set of n distinct elements.

Original entry on oeis.org

1, 1, 2, 9, 16, 125, 216, 2401, 4096, 59049, 100000, 1771561, 2985984, 62748517, 105413504, 2562890625, 4294967296, 118587876497, 198359290368, 6131066257801, 10240000000000, 350277500542221, 584318301411328, 21914624432020321, 36520347436056576

Offset: 0

Mark Sterling, Dec 18 2002

			Examples added by _N. J. A. Sloane_, Jun 17 2014:
n=1: 1 (1).
n=2: 11, 22 (2).
n=3: 111X3, 121X6 (9).
n=4: 1111X4, 1221X12 (16).
n=5: 11111X5, 11211X20, 12221X20, 12121X20, 12321X60 (125).

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A004526, A243520.
Cf. A168658, A275549.
This is for Coxeter type B what A152291 is for Coxeter type A.

a:= n-> n^ceil(n/2): seq(a(n), n=0..30);  # Alois P. Heinz, Jul 23 2014

Join[{1}, Table[n^Ceiling[n/2], {n, 30}]] (* Wesley Ivan Hurt, Jan 15 2017 *)

PARI
```
for(n=1,22,print1(n^((n+n%2)/2),","))
```

a(n) = n^(floor((n+1)/2)) = n^ceiling(n/2).

Extended by Klaus Brockhaus, Dec 19 2002

a(0)=1 inserted by Alois P. Heinz, Jul 23 2014

A203411 Discriminant of the cyclotomic binomial period polynomial for an odd prime.

Original entry on oeis.org

1, 5, 49, 14641, 371293, 410338673, 16983563041, 41426511213649, 10260628712958602189, 756943935220796320321, 456487940826035155404146917, 4394336169668803158610484050361, 467056167777397914441056671494001, 6111571184724799803076702357055363809

Offset: 2

Franz Vrabec, Jan 01 2012

nonn

			a(5) = 11^4 = 14641, because prime(5) = 11.

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
J. Brillhart, Note on the discriminant of certain cyclotomic period polynomials, Pacific Journal of Mathematics, 152/1(1992), 15-19.
L. Carlitz and F. R. Olson, Maillet's determinant, Proceedings of the American Mathematical Society, 6/2 (1955), 265-269.
L. Carlitz, A special determinant, Proceedings of the American Mathematical Society, 6/2 (1955), 270-272.

Cf. A152291.

#^((#-3)/2)&/@Prime[Range[2,20]] (* Harvey P. Dale, Aug 11 2023 *)

a(n) = prime(n)^((prime(n)-3)/2); \\ Michel Marcus, Apr 15 2017

a(n) = prime(n)^((prime(n)-3)/2).

More terms from Franklin T. Adams-Watters, Jan 24 2012

A363861 Sequence related to chains in type D noncrossing partitions.

Original entry on oeis.org

4, 6, 64, 100, 1296, 2058, 32768, 52488, 1000000, 1610510, 35831808, 57921708, 1475789056, 2392031250, 68719476736, 111612119056, 3570467226624, 5808378560022, 204800000000000, 333597619564020, 12855002631049216, 20961814674106394, 876488338465357824, 1430511474609375000, 64509974703297150976

Offset: 3

F. Chapoton, Jun 25 2023

This is counting chains in the noncrossing partition lattices of type D_n that proceed by steps of type A2, except at most one step of type A1 at the end. This is a decomposition number in the terminology of Krattenthaler and Müller.

C. Krattenthaler and T. W. Müller, Decomposition Numbers For Finite Coxeter Groups And Generalised Non-Crossing Partitions, TAMS, vol. 362, 2010.

This is for Coxeter type D what A078707 is for Coxeter type B and A152291 is for Coxeter type A.

print([(n-2)*(n-1)**(n/2-1) if not n % 2 else (n-1)**((n+1)/2) for n in range(3,28)])

a(n) = (n-2)*(n-1)^(n/2-1) if n is even else a(n) = (n-1)^((n+1)/2).

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A152290 Coefficients in a q-analog of the LambertW function, as a triangle read by rows.

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A078707 Number of vectors of length n that are symmetric about the middle, where each element is drawn from a set of n distinct elements.

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A203411 Discriminant of the cyclotomic binomial period polynomial for an odd prime.

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Mathematica

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A363861 Sequence related to chains in type D noncrossing partitions.

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