A008621 -id:A008621 - cp's OEIS frontend

A348388 Irregular triangle read by rows: T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

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

Offset: 2

Wolfdieter Lang, Oct 31 2021

This irregular triangle T(n, k) gives the number of multiples of number k, larger than k and not exceeding n, for k = 1, 2, ..., floor(n/2), for n >= 2. See A348389 for the array of these multiples.
The length of row n is floor(n/2) = A004526(n), for n >= 2.
The row sums give A002541(n). See the formula given there by Wesley Ivan Hurt, May 08 2016.
The columns give the k-fold repeated positive integers k, for k >= 1.

			The irregular triangle T(n, k) begins:
n\k   1 2 3 4 5 6 7 8 9 10 ...
------------------------------
2:    1
3:    2
4:    3 1
5:    4 1
6:    5 2 1
7:    6 2 1
8:    7 3 1 1
9:    8 3 2 1
10:   9 4 2 1 1
11:  10 4 2 1 1
12:  11 5 3 2 1 1
13:  12 5 3 2 1 1
14:  13 6 3 2 1 1 1
15:  14 6 4 2 2 1 1
16:  15 7 4 3 2 1 1 1
17:  16 7 4 3 2 1 1 1
18:  17 8 5 3 2 2 1 1 1
19:  18 8 5 3 2 2 1 1 1
20:  19 9 5 4 3 2 1 1 1  1
...

Karl-Heinz Hofmann, Table of n, a(n) for n = 2..10101

Columns k (with varying offsets): A000027, A004526, A008620, A008621, A002266, A097992, ...

Cf. A002541, A004526, A010766, A348389.

T[n_, k_] := Floor[(n - k)/k]; Table[T[n, k], {n, 2, 20}, {k, 1, Floor[n/2]}] // Flatten (* Amiram Eldar, Nov 02 2021 *)

def A348388row(n): return [(n - k) // k for k in range(1, 1 + n // 2)]
for n in range(2, 21): print(A348388row(n))  # Peter Luschny, Nov 05 2021

T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

G.f. of column k: G(k, x) = x^(2*k)/((1 - x)*(1 - x^k)).

A110868 Molien series for real 32-dimensional Clifford group of genus 5 and order 96253116206284800.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 7, 9, 16, 23, 46, 74

Offset: 0

G. Nebe, Sep 21 2005

nonn

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for Molien series

Cf. A008621, A008718, A024186, A110160, this sequence, A110869, A110876, A110880. See also A001309.

A193771 Expansion of 1 / (1 - x - x^3 + x^6) in powers of x.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 17, 23, 31, 41, 54, 72, 96, 127, 168, 223, 296, 392, 519, 688, 912, 1208, 1600, 2120, 2809, 3721, 4929, 6530, 8651, 11460, 15181, 20111, 26642, 35293, 46753, 61935, 82047, 108689, 143982, 190736, 252672, 334719, 443408, 587391

Offset: 0

Michael Somos, Jan 01 2013

nonn

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 13*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 0, 0, -1).

Cf. A000931, A008621, A017818.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x-x^3+x^6)));  // G. C. Greubel, Aug 10 2018

CoefficientList[Series[1/(1-x-x^3+x^6),{x,0,50}],x] (* or *) LinearRecurrence[ {1,0,1,0,0,-1},{1,1,1,2,3,4},50] (* Harvey P. Dale, Jul 25 2017 *)

{a(n) = if( n<0, n = -n; polcoeff( - x^6 / (1 - x^3 - x^5 + x^6) + x * O(x^n), n), polcoeff( 1 / (1 - x - x^3 + x^6) + x * O(x^n), n))};

G.f.: 1 / (1 - x - x^3 + x^6) = 1 / (1 - x / (1 - x^2 / (1 + x^2 / (1 - x / (1 + x / (1 + x^2 / (1 - x^2))))))).
a(n) = a(n-1) + a(n-3) - a(n-6) for all n in Z.
Convolution of A008621 and A000931. PSUM transform of A017818.

A052614 E.g.f. 1/((1-x)(1-x^4)).

Original entry on oeis.org

1, 1, 2, 6, 48, 240, 1440, 10080, 120960, 1088640, 10886400, 119750400, 1916006400, 24908083200, 348713164800, 5230697472000, 104613949440000, 1778437140480000, 32011868528640000, 608225502044160000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 559

spec := [S,{S=Prod(Sequence(Z),Sequence(Prod(Z,Z,Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[1/((1-x)(1-x^4)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 30 2013 *)

E.g.f.: 1/(-1+x)/(-1+x^4)
Recurrence: {a(1)=1, a(0)=1, a(3)=6, a(2)=2, (-61*n-11*n^3-n^4-30-41*n^2)*a(n) +(-n^2-5*n-6)*a(n+1) +(-n-3)*a(n+2) +a(n+4) -a(n+3)=0}
(Sum(1/16*(2*_alpha+_alpha^2-1)*_alpha^(-1-n), _alpha=RootOf(1+_Z+_Z^2+_Z^3))+1/4*n+5/8)*n!
n!*[n/4+1].
a(n)=n!*A008621(n). - R. J. Mathar, Jun 03 2022

cp's OEIS Frontend

A348388 Irregular triangle read by rows: T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

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A110868 Molien series for real 32-dimensional Clifford group of genus 5 and order 96253116206284800.

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A193771 Expansion of 1 / (1 - x - x^3 + x^6) in powers of x.

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A052614 E.g.f. 1/((1-x)(1-x^4)).

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