A090223 -id:A090223 - 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.

A090222 Array used for numerators of g.f.s for column sequences of array A090216 ((5,5)-Stirling2).

Original entry on oeis.org

1, 600, 600, 648000, 200, 2592000, 1270080000, 25, 2871000, 13592880000, 4267468800000, 1, 1294920, 36462182400, 100221504768000, 23228686172160000, 284800, 38559024000, 551224880640000, 1056582600192000000

Offset: 5

Wolfdieter Lang, Dec 01 2003

The row length sequence for this array is A090223(k-5)+1= floor(4*(k-5)/5)+1, k>=5: [1, 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, ...].

The g.f. G(k,x) for the k-th column (with leading zeros) of array A090216 is given there. The recurrence is G(k,x) = x*sum(binomial(k-r,5-r)*fallfac(5,5-r)*G(k-r,x),r=1..5))/(1-fallfac(k,5)*x), k>=5, with inputs G(k,x)=0 for k=1,2,3,4 and G(5,x)=x/(1-5!*x); where fallfac(n,m) := A008279(n,m) (falling factorials with fallfac(n,0) := 1). Computed from the Blasiak et al. reference, eqs. (20) and (21) with r=5: recurrence for S_{5,5}(n,k).

			[1]; [600]; [648000,200]; [2592000,1270080000,25]; ...
G(6,x)/x^2 = 600/((1-5!*x)*(1-6*5*4*3*2*x)). kmax(6)=0, hence P(6,x)=a(6,0)=600; x^2 from x^ceiling(6/5).

W. Lang, First 7 rows.

a(k, n) from: sum(a(k, n)*x^n, n=0..kmax(k)) = G(k, x)* product(1-fallfac(p, 5)*x, p=5..k)/x^ceiling(k/5), k>=5, with G(k, x) defined from the recurrence given above and kmax(k) := floor(4*(k-5)/5)= A090223(k-5).

A226096 Squares with doubled (4*n+2)^2.

Original entry on oeis.org

1, 4, 4, 9, 16, 25, 36, 36, 49, 64, 81, 100, 100, 121, 144, 169, 196, 196, 225, 256, 289, 324, 324, 361, 400, 441, 484, 484, 529, 576, 625, 676, 676, 729, 784, 841, 900, 900, 961, 1024, 1089, 1156, 1156, 1225, 1296, 1369

Offset: 0

Paul Curtz, May 26 2013

Also nondecreasing ordered values of A226008 (except 0).
Consider A225948/A226008 ordered according to a(n): 0/1, -15/4, -3/4, 2/9, 3/16, 6/25, -7/36, 5/36, 12/49, 15/64, 20/81, ... = b(n)/a(n), and consider the sequence with period 5: 1, 64, 16, 1, 4, ... = t(n); then a(n) = 4*b(n) + t(n).
The recurrences in Formula lines are also valid for b(n).
Note that the fractions b(n)/a(n) of rank 0, 3,4,5, 8,9,10, ... = A047205:
0,  2/9, 3/16, 6/25, 12/49, 15/64, 20/81, ... are all in A226023(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,2,-2,0,0,0,-1,1).

Cf. A016826, A226008.

MapIndexed[ If [Mod[First[#2], 4] == 2, Sequence @@ {#1, #1}, #1] &, Range[40]]^2 (* Jean-François Alcover, May 28 2013 *)

a(n+5) - a(n) = 8*A090223(n+4).
a(n) = 1 followed by (A090223(n) + 2)^2.
a(n) = 3*a(n-5) -3*a(n-10) +a(n-15).
G.f.: (x^9 + 3*x^8 + 5*x^6 + 7*x^5 + 7*x^4 + 5*x^3 + 3*x + 1)/((1 - x)*(1 - x^5)^2). [Ralf Stephan, May 30 2013]
a(n) = a(n-1) +2*a(n-5) -2*a(n-6) -a(n-10) +a(n-11). [Bruno Berselli, May 30 2013]
a(n) = (24*(16*floor(n/5)^2 + 8*floor(n/5) + 1) - (11 + 24*floor(n/5))*(n - 5*floor(n/5))^4 + 2*(49 + 104*floor(n/5))*(n - 5*floor(n/5))^3 - 23*(11 + 24*floor(n/5))*(n - 5*floor(n/5))^2 + 2*(119 + 280*floor(n/5))*(n - 5*floor(n/5)))/24. - Luce ETIENNE, May 08 2017

A279169 a(n) = floor( 4*n^2/5 ).

Original entry on oeis.org

0, 0, 3, 7, 12, 20, 28, 39, 51, 64, 80, 96, 115, 135, 156, 180, 204, 231, 259, 288, 320, 352, 387, 423, 460, 500, 540, 583, 627, 672, 720, 768, 819, 871, 924, 980, 1036, 1095, 1155, 1216, 1280, 1344, 1411, 1479, 1548, 1620, 1692, 1767, 1843, 1920, 2000, 2080, 2163, 2247

Offset: 0

Bruno Berselli, Dec 07 2016

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A090223: floor(4*n/5).
Subsequence of A008728, A014601, A118015, A131242.
Cf. similar sequences with closed form floor(k*n^2/5): A118015 (k=1), A033437 (k=2), A184535 (k=3).

Magma
```
[4*n^2 div 5: n in [0..60]];
```

Table[Floor[4 n^2/5], {n, 0, 60}]
LinearRecurrence[{2,-1,0,0,1,-2,1},{0,0,3,7,12,20,28},60] (* Harvey P. Dale, Nov 07 2020 *)

PARI
```
vector(60, n, n--; floor(4*n^2/5))
```
Python
```
[int(4*n**2/5) for n in range(60)]
```
Sage
```
[floor(4*n^2/5) for n in range(60)]
```

O.g.f.: x^2*(3 + x + x^2 + 3*x^3)/((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(-n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7).
a(5*m+r) = 4*m*(5*m + 2*r) + a(r), where m >= 0 and 0 <= r < 5. Example: for m=4 and r=3, a(5*4+3) = a(23) = 4*4*(5*4 + 2*3) + a(3) = 416 + 7 = 423.
a(n) = A118015(2*n) = A008728(4*n+2) = A131242(4*n+4) = A014601(floor(2*n^2/5)).
Sum_{n>=2} 1/a(n) = Pi^2/120 + sqrt(29 - 62/sqrt(5))*Pi/8 + 5/16. - Amiram Eldar, Sep 26 2022

A034115 Fractional part of square root of a(n) starts with 9: first term of runs.

Original entry on oeis.org

35, 48, 63, 80, 99, 119, 142, 167, 194, 223, 253, 286, 321, 358, 397, 437, 480, 525, 572, 621, 671, 724, 779, 836, 895, 955, 1018, 1083, 1150, 1219, 1289, 1362, 1437, 1514, 1593, 1673, 1756, 1841, 1928, 2017, 2107, 2200, 2295, 2392, 2491, 2591, 2694

Offset: 1

Patrick De Geest, Sep 15 1998

How is this different from A034105? - N. J. A. Sloane, Mar 30 2007

Answer: A034115 has the starts of runs of consecutive values of A034105. That is, frac{sqrt[a(n)]} >= 0.9, but frac{sqrt[a(n)-1]} < 0.9. - Don Reble, Jul 17 2020

			358, 359 and 360 are a run of 3 numbers in A034105, so 358 is in this sequence, but 359 and 360 are not. - _R. J. Mathar_, Jul 21 2020

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A027690, A034105, A090223.

Join[{35},Select[Partition[Select[Range[3000],NumberDigit[Sqrt[#],-1] == 9&],2,1],(#[[2]]-#[[1]]!=1&)][[All,2]]] (* or *) LinearRecurrence[{2,-1,0,0,1,-2,1},{35,48,63,80,99,119,142},50] (* Harvey P. Dale, Aug 14 2021 *)

a(n) = n^2 + 9*n + 25 + floor(4*n / 5) = A027690(n+4)+A090223(n). - Don Reble, Jul 17 2020

A327440 a(n) = floor(3*n/10).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23

Offset: 0

Bruno Berselli, Sep 11 2019

The sequence can be obtained from A008585 by deleting the last digit of each term.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A008585.

Similar sequences with the formula floor(k*n/10): A059995 (k=1); A002266 (k=2); A057354 (k=4); A004526 (k=5); A057355 (k=6); A188511 (k=7); A090223 (k=8).

[div(3*n, 10) for n in 0:80] |> println

Table[Floor[3 n/10], {n, 0, 80}]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3}, 80]

PARI
```
vector(80, n, n--; floor(3*n/10))
```

O.g.f.: x^4*(1 + x^3 + x^6)/((1 + x)*(1 - x)^2*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) = (x^4 + x^7 + x^10)/(1 - x - x^10 + x^11).

a(n) = a(n-1) + a(n-10) - a(n-11) for n > 10.

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

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A090222 Array used for numerators of g.f.s for column sequences of array A090216 ((5,5)-Stirling2).

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A226096 Squares with doubled (4*n+2)^2.

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A279169 a(n) = floor( 4*n^2/5 ).

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A034115 Fractional part of square root of a(n) starts with 9: first term of runs.

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A327440 a(n) = floor(3*n/10).

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