A090222 -id:A090222 - cp's OEIS frontend

A090216 Generalized Stirling2 array S_{5,5}(n,k).

1, 120, 600, 600, 200, 25, 1, 14400, 504000, 2664000, 4608000, 3501000, 1350360, 284800, 33800, 2225, 75, 1, 1728000, 371520000, 7629120000, 42762240000, 97388280000, 110386900800, 70137648000, 26920728000, 6548346000, 1039382000

Offset: 1

Wolfdieter Lang, Dec 01 2003

The row length sequence for this array is [1, 6, 11, 16, 21, 26, 31,...]= A016861(n-1), n>=1.

The g.f. for the k-th column, (with leading zeros and k>=5) is G(k,x)= x^ceiling(k/5)*P(k,x)/product(1-fallfac(p,5)*x,p=5..k), with fallfac(n,m) := A008279(n,m) (falling factorials) and P(k,x) := sum(A090222(k,m)*x^m,m=0..kmax(k)), k>=5, with kmax(k) := floor(4*(k-5)/5)= A090223(k-5). For the recurrence of the G(k,x) see A090222.

			Triangle begins:
  [1];
  [120,600,600,200,25,1];
  [14400,504000,2664000,4608000,3501000,1350360,284800,33800,2225,75,1];
  ...

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004; Phys. Lett. A 309 (3-4) (2003) 198-205.
Wolfdieter Lang, First 5 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A090217, A090209 (row sums), A090218 (alternating row sums).

fallfac[n_, k_] := Pochhammer[n-k+1, k]; a[n_, k_] := (((-1)^k)/k!)*Sum[((-1)^p)*Binomial[k, p]*fallfac[p, 5]^n, {p, 5, k}]; Table[a[n, k], {n, 1, 5}, {k, 5, 5*n}] // Flatten (* Jean-François Alcover, Mar 05 2014 *)

from sympy import binomial, factorial, ff
def a(n, k): return sum((-1)**p * binomial(k, p) * ff(p, 5)**n for p in range(5, k+1)) * (-1)**k / factorial(k) # David Radcliffe, Jul 01 2025

a(n, k) = (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*fallfac(p, 5)^n, p=5..k), with fallfac(p, 5) := A008279(p, 5)=product(p+1-q, q=1..5); 5<= k <= 5*n, n>=1, else 0. From eq.(19) with r=5 of the Blasiak et al. reference.

E^n = Sum_{k=5..5n) a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^5d^5/dx^5.

A090223 Nonnegative integers with doubled multiples of 4.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 48, 49, 50, 51, 52, 52, 53, 54, 55, 56, 56, 57, 58

Offset: 0

Wolfdieter Lang, Dec 01 2003

Degrees of row-polynomials of array A090222.

a(n) is the number of full orbits completed by body A for n full orbits completed by body B in a celestial system with two orbiting bodies A and B with orbital resonance A:B equal to 4:5. This resonance is exhibited by the planets Kepler-90b and Kepler-90c in the planetary system of the star Kepler-90. - Felix Fröhlich, May 03 2021

Felix Fröhlich, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A057353 and other floors of ratios references there.

Cf. A090222.

[4*n div 5: n in [0..80]]; // Bruno Berselli, Dec 07 2016

A090223:=n->floor(4*n/5); seq(A090223(n), n=0..100); # Wesley Ivan Hurt, Nov 15 2013

Table[Floor[4n/5], {n,0,100}] (* Wesley Ivan Hurt, Nov 15 2013 *)

a(n)=4*n\5 \\ Charles R Greathouse IV, Sep 02 2015

a(n) = floor(4*n/5).
G.f.:  x^2 *(1+x^2)*(1+x)/((1-x^5)*(1-x)) = (x^2)*(1-x^4)/((1-x^5)*(1-x)^2).
a(n) = n - 1 - A002266(n - 1). - Wesley Ivan Hurt, Nov 15 2013
a(n) = A057354(2*n). - R. J. Mathar, Jul 21 2020
5*a(n) = 4*n-2+A117444(n+2) . - R. J. Mathar, Jul 21 2020
Sum_{n>=2} (-1)^n/a(n) = (2*sqrt(2)-1)*Pi/8. - Amiram Eldar, Sep 30 2022

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A090216 Generalized Stirling2 array S_{5,5}(n,k).

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