A220884 -id:A220884 - cp's OEIS frontend

A220883 Triangle read by rows: row n gives coefficients of expansion of Product_{k = 1..n-1} ((n + 1)*x + k), starting with lowest power.

1, 1, 3, 2, 12, 16, 6, 55, 150, 125, 24, 300, 1260, 2160, 1296, 120, 1918, 11025, 29155, 36015, 16807, 720, 14112, 103936, 376320, 716800, 688128, 262144, 5040, 117612, 1063692, 4934601, 12859560, 19013778, 14880348, 4782969, 40320, 1095840, 11812400, 67284000, 224490000, 453600000, 546000000, 360000000, 100000000, 362880, 11292336, 141896700, 963218080, 3943187325, 10190179923, 16741251450, 16953838770, 9646149645, 2357947691

Offset: 1

N. J. A. Sloane, Dec 29 2012

Related to Stirling numbers A008275, A008277.

			Triangle begins:
    1
    1     3
    2    12     16
    6    55    150    125
   24   300   1260   2160   1296
  120  1918  11025  29155  36015  16807
  720 14112 103936 376320 716800 688128 262144
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened).
Peter Bala, Fractional iteration of a series inversion operator
Jean-Christophe Novelli and Jean-Yves Thibon, Duplicial algebras and Lagrange inversion, arXiv preprint arXiv:1209.5959 [math.CO], 2012.

Cf. A220884, A008275, A008277, A056856, A260687.

seq(seq(coeff(mul((n+1)*t + k, k = 1..n-1), t, i), i = 0..n-1), n = 1 .. 10); # Peter Bala, Nov 16 2015
# Alternative:
T := (n, k) -> (-1)^(n-k)*(n+1)^(k-1)*Stirling1(n, k):
seq(print(seq(T(n, k), k=1..n)), n=1..8);
# Peter Luschny, Mar 20 2024

A220883[n_, k_] := (-1)^(n-k)*(n+1)^(k-1)*StirlingS1[n, k];
Table[A220883[n, k], {n, 10}, {k, n}] (* Paolo Xausa, Mar 19 2024 *)

From Peter Bala, Nov 16 2015: (Start)
E.g.f.: A(x,t) = x + (1 + 3*t)*x^2/2! + (1 + 4*t)*(2 + 4*t)*x^3/3! + ....
The function F(x,t) := 1 + t*A(x,t) has several nice properties:
F(x,t) = 1/x*Revert( x*(1 - x)^t ) = 1 + t*x + t*(1 + 3*t)*x^2/2! + t*(2 + 12*t + 16*t^2)*x^3/3! + ..., where Revert denotes the series reversion operator with respect to x.
F(x,t)*(1 - x*F(x,t))^t = 1.
F(x,t)^m = 1 + m*t*x + m*t*((m + 2)*t + 1)*x^2/2! + m*t*((m + 3)*t + 1)*((m + 3)*t + 2)*x^3/3! + m*t*((m + 4)*t + 1)*((m + 4)*t + 2)*((m + 4)*t + 3)*x^4/4! + ....
Log(F(x,t)) = t*x + t*(1 + 2*t)*x^2/2! + t*(1 + 3*t)*(2 + 3*t)*x^3/3! + t*(1 + 4*t)*(2 + 4*t)*(3 + 4*t)*x^4/4! + ... is the e.g.f for A056856.
F(x,t) = G(x,t)^t, where G(x,t) = 1 + x + (2 + 2*t)*x^2/2! + (2 + 3*t)*(3 + 3*t)*x^3/3! + (2 + 4*t)*(3 + 4*t)*(4 + 4*t)*x^4/4! + ... is the o.g.f. for A260687. (End)
T(n, k) = (-1)^(n-k)*(n+1)^(k-1)*Stirling1(n, k). - Peter Luschny, Mar 01 2021 [Corrected by Paolo Xausa, Mar 19 2024]

A370832 Triangle read by rows: T(n,k) gives the number of parking functions of size n with k lucky cars. 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 8, 6, 0, 6, 37, 58, 24, 0, 24, 204, 504, 444, 120, 0, 120, 1318, 4553, 6388, 3708, 720, 0, 720, 9792, 44176, 87296, 81136, 33984, 5040, 0, 5040, 82332, 463860, 1203921, 1582236, 1064124, 341136, 40320, 0, 40320, 773280, 5270480, 17164320, 29724000, 28328480, 14602320, 3733920, 362880

Offset: 0

Peter Kagey, Mar 02 2024

A car is called "lucky" if it gets its preferred parking spot.

Closely related to A220884.

			Table begins:
n\k|  0     1     2      3       4       5       6      7     8
---+-------------------------------------------------------------
 0 |  1
 1 |  0     1
 2 |  0     1     2
 3 |  0     2     8      6
 4 |  0     6    37     58      24
 5 |  0    24   204    504     444     120
 6 |  0   120  1318   4553    6388    3708     720
 7 |  0   720  9792  44176   87296   81136   33984   5040
 8 |  0  5040 82332 463860 1203921 1582236 1064124 341136 40320
      ...

Alois P. Heinz, Rows n = 0..140, flattened
Irfan Durmić, Alex Han, Pamela E. Harris, Rodrigo Ribeiro, and Mei Yin, Probabilistic Parking Functions, arXiv:2211.00536 [math.CO], 2022.
FindStat, St000135: The number of lucky cars of the parking function.

Cf. A002538, A067948, A220884.
Row sums give A000272(n+1).
Cf. A000142 (main diagonal and column k=1 shifted).

b:= proc(n) option remember; `if`(n=0, 1,
      expand(x*mul((n+1-k)+k*x, k=2..n)))
    end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 26 2024

row[n_] := (x (x - 1)^n Pochhammer[(n + x) / (x - 1), n]) / (n + x);
Table[CoefficientList[Series[row[n], {x, 0, n}], x], {n, 0, 8}] // Flatten
(* Peter Luschny, Jun 27 2024 *)

T(n, n) = n!.
T(n, 1) = (n-1)!.
Sum_{k=1..n} T(n, k) = (n+1)^(n-1).
T(n+1, n) = A002538(n).
G.f. for row n>0: x * Product_{j=2..n} (n + 1 + j*(x-1)).
T(n, k) = [x^k] (x*(x - 1)^n*Pochhammer((n + x) / (x - 1), n)) / (n + x). - Peter Luschny, Jun 27 2024

Edited by Alois P. Heinz, Jun 26 2024

cp's OEIS Frontend

A220883 Triangle read by rows: row n gives coefficients of expansion of Product_{k = 1..n-1} ((n + 1)*x + k), starting with lowest power.

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