A290995 -id:A290995 - cp's OEIS frontend

A024495 a(n) = C(n,2) + C(n,5) + ... + C(n, 3*floor(n/3)+2).

0, 0, 1, 3, 6, 11, 21, 42, 85, 171, 342, 683, 1365, 2730, 5461, 10923, 21846, 43691, 87381, 174762, 349525, 699051, 1398102, 2796203, 5592405, 11184810, 22369621, 44739243, 89478486, 178956971, 357913941, 715827882, 1431655765, 2863311531, 5726623062

Offset: 0

Clark Kimberling

Trisections give A082365, A132804, A132805. - Paul Curtz, Nov 18 2007
If the offset is changed to 1, this is the maximal number of closed regions bounded by straight lines after n straight line cuts in a plane: a(n) = a(n-1) + n - 3, a(1)=0; a(2)=0; a(3)=1; and so on. - Srikanth K S, Jan 23 2008
M^n * [1,0,0] = [A024493(n), a(n), A024494(n)]; where M = a 3x3 matrix [1,1,0; 0,1,1; 1,0,1]. Sum of terms = 2^n. Example: M^5 * [1,0,0] = [11, 11, 10], sum = 2^5 = 32. - Gary W. Adamson, Mar 13 2009
For n>=1, a(n-1) is the number of generalized compositions of n when there are i^2/2 - 3*i/2 + 1 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010
Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1+M)^n = A024493(n) + A024494(n)*M + a(n)*M^2. - Stanislav Sykora, Jun 10 2012
{A024493, A131708, A024495} is the difference analog of the hyperbolic functions {h_1(x), h_2(x), h_3(x)} of order 3. For the definitions of {h_i(x)} and the difference analog {H_i(n)} see [Erdelyi] and the Shevelev link respectively. - Vladimir Shevelev, Aug 01 2017
This is the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S^3; see A291000.  - Clark Kimberling, Aug 24 2017

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, 2nd. ed., Problem 38, p. 70.

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Paul Barry, A note on Krawtchouk Polynomials and Riordan Arrays, JIS 11 (2008) 08.2.2, example 9.
Antoine-Augustin Cournot, Solution d'un problème d'analyse combinatoire, Bulletin des Sciences Mathématiques, Physiques et Chimiques, item 34, volume 11, 1829, pages 93-97. Also at Google Books. Page 97 case p=3 formula y^(2) = a(n).
Christian Ramus, Solution générale d'un problème d'analyse combinatoire, Journal für die Reine und Angewandte Mathematik (Crelle's journal), volume 11, 1834, pages 353-355. Page 353 case p=3 formula y^(2) = a(n).
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Plane division by lines
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A007283, A010892, A011655, A024493, A024494, A024495, A057079, A082365.
Cf. A083322, A139469, A111927, A113405, A131577, A131708, A132804, A132805.
Cf. A175805, A291000.
Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), this sequence (m=3), A000749 (m=4), A049016 (m=5), A192080 (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).

R:=PowerSeriesRing(Integers(), 30); [0,0] cat Coefficients(R!( x^2/((1-x)^3-x^3) )); // G. C. Greubel, Apr 11 2023

a:= proc(n) option remember; `if`(n=0, 0, 2*a(n-1)+
      [-1, 0, 1, 1, 0, -1, -1][1+(n mod 6)])
    end:
seq(a(n), n=0..33); # Paul Weisenhorn, May 17 2020

LinearRecurrence[{3,-3,2},{0,0,1},40] (* Harvey P. Dale, Sep 20 2016 *)

a(n) = sum(k=0,n\3,binomial(n,3*k+2)) /* Michael Somos, Feb 14 2006 */

a(n)=if(n<0, 0, ([1,0,1;1,1,0;0,1,1]^n)[3,1]) /* Michael Somos, Feb 14 2006 */

def A024495(n): return (2^n - chebyshev_U(n, 1/2) - chebyshev_U(n-1, 1/2))/3
[A024495(n) for n in range(41)] # G. C. Greubel, Apr 11 2023

a(n) = ( 2^n + 2*cos((n-4)*Pi/3) )/3 = (2^n - A057079(n))/3.
a(n) = 2*a(n-1) + A010892(n-2) = a(n-1) + A024494(n-1). With initial zero, binomial transform of A011655 which is effectively A010892 unsigned. - Henry Bottomley, Jun 04 2001
a(2) = 1, a(3) = 3, a(n+2) = a(n+1) - a(n) + 2^n. - Benoit Cloitre, Sep 04 2002
a(n) = Sum_{k=0..n} 2^k*2*sin(Pi*(n-k)/3 + Pi/3)/sqrt(3) (offset 0). - Paul Barry, May 18 2004
G.f.: x^2/((1-x)^3 - x^3) =  x^2 / ( (1-2*x)*(1-x+x^2) ).
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3). - Paul Curtz, Nov 18 2007
a(n) + A024493(n-1) = A131577(n). - Paul Curtz, Jan 24 2008
From Paul Curtz, May 29 2011: (Start)
a(n) + a(n+3) = 3*2^n = A007283(n).
a(n+6) - a(n) = 21*2^n = A175805(n).
a(n) + a(n+9) = 171*2^n.
a(n+12) - a(n) = 1365*2^n. (End)
a(n) = A113405(n) + A113405(n+1). - Paul Curtz, Jun 05 2011
Start with x(0)=1, y(0)=0, z(0)=0 and set x(n+1) = x(n) + z(n), y(n+1) = y(n) + x(n), z(n+1) = z(n) + y(n). Then a(n) = z(n). - Stanislav Sykora, Jun 10 2012
G.f.: -x^2/( x^3 - 1 + 3*x/Q(0) ) where Q(k) = 1 + k*(x+1) + 3*x - x*(k+1)*(k+4)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Mar 15 2013
a(n) = 1/18*(-4*(-1)^floor((n - 1)/3) - 6*(-1)^floor(n/3) - 3*(-1)^floor((n + 1)/3) + (-1)^(1 + floor((n + 2)/3)) + 3*2^(n + 1)). - John M. Campbell, Dec 23 2016
a(n) = (1/63)*(-40 + 21*2^n - 42*floor(n/6) + 32*floor((n+3)/6) + 16*floor((n+ 4)/6) - 24*floor((n+5)/6) - 22*floor((n+7)/6) + 21*floor((n+8)/6) + 10*floor((n+9)/6) + 5*floor((n+10)/6) + 3*floor((n+11)/6) + floor((n+ 13)/6)). - John M. Campbell, Dec 24 2016
a(n+m) = a(n)*A024493(m) + A131708(n)*A131708(m) + A024493(n)*a(m). - Vladimir Shevelev, Aug 01 2017
From Kevin Ryde, Sep 24 2020: (Start)
a(n) = (1/3)*2^n - (1/3)*cos((1/3)*Pi*n) - (1/sqrt(3))*sin((1/3)*Pi*n). [Cournot]
a(n) + A111927(n) + A131708(n) = 2^n - 1. [Cournot, page 96 last formula, but misprint should be 2^x - 1 rather than 2^p - 1] (End)
E.g.f.: (exp(2*x) - exp(x/2)*(cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/3. - Stefano Spezia, Feb 06 2025

A291000 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3.

Original entry on oeis.org

1, 3, 9, 26, 74, 210, 596, 1692, 4804, 13640, 38728, 109960, 312208, 886448, 2516880, 7146144, 20289952, 57608992, 163568448, 464417728, 1318615104, 3743926400, 10630080640, 30181847168, 85694918912, 243312448256, 690833811712, 1961475291648, 5569190816256

Offset: 0

Clark Kimberling, Aug 22 2017

Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).  Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
In the following guide to p-INVERT sequences using s = (1,1,1,1,1,...) = A000012, in some cases t(1,1,1,1,1,...) is a shifted version of the cited sequence:
p(S)                             t(1,1,1,1,1,...)
1 - S                                A000079
1 - S^2                              A000079
1 - S^3                              A024495
1 - S^4                              A000749
1 - S^5                              A139761
1 - S^6                              A290993
1 - S^7                              A290994
1 - S^8                              A290995
1 - S - S^2                          A001906
1 - S - S^3                          A116703
1 - S - S^4                          A290996
1 - S^3 - S^6                        A290997
1 - S^2 - S^3                        A095263
1 - S^3 - S^4                        A290998
1 - 2 S^2                            A052542
1 - 3 S^2                            A002605
1 - 4 S^2                            A015518
1 - 5 S^2                            A163305
1 - 6 S^2                            A290999
1 - 7 S^2                            A291008
1 - 8 S^2                            A291001
(1 - S)^2                            A045623
(1 - S)^3                            A058396
(1 - S)^4                            A062109
(1 - S)^5                            A169792
(1 - S)^6                            A169793
(1 - S^2)^2                          A024007
1 - 2 S - 2 S^2                      A052530
1 - 3 S - 2 S^2                      A060801
(1 - S)(1 - 2 S)                     A053581
(1 - 2 S)(1 - 3 S)                   A291002
(1 - S)(1 - 2 S)(1 - 3 S)(1 - 4 S)   A291003
(1 - 2 S)^2                          A120926
(1 - 3 S)^2                          A291004
1 + S - S^2                          A000045  (Fibonacci numbers starting with -1)
1 - S - S^2 - S^3                    A291000
1 - S - S^2 - S^3 - S^4              A291006
1 - S - S^2 - S^3 - S^4 - S^5        A291007
1 - S^2 - S^4                        A290990
(1 - S)(1 - 3 S)                     A291009
(1 - S)(1 - 2 S)(1 - 3 S)            A291010
(1 - S)^2 (1 - 2 S)                  A291011
(1 - S^2)(1 - 2 S)                   A291012
(1 - S^2)^3                          A291013
(1 - S^3)^2                          A291014
1 - S - S^2 + S^3                    A045891
1 - 2 S - S^2 + S^3                  A291015
1 - 3 S + S^2                        A136775
1 - 4 S + S^2                        A291016
1 - 5 S + S^2                        A291017
1 - 6 S + S^2                        A291018
1 - S - S^2 - S^3 + S^4              A291019
1 - S - S^2 - S^3 - S^4 + S^5        A291020
1 - S - S^2 - S^3 + S^4 + S^5        A291021
1 - S - 2 S^2 + 2 S^3                A175658
1 - 3 S^2 + 2 S^3                    A291023
(1 - 2 S^2)^2                        A291024
(1 - S^3)^3                          A291143
(1 - S - S^2)^2                      A209917

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, -4, 2)

Cf. A000012, A289780.

z = 60; s = x/(1 - x); p = 1 - s - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291000 *)

G.f.: (-1 + x - x^2)/(-1 + 4 x - 4 x^2 + 2 x^3).

a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-3) for n >= 4.

A000749 a(n) = 4a(n-1) - 6a(n-2) + 4*a(n-3), n > 3, with a(0)=a(1)=a(2)=0, a(3)=1.

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 20, 36, 64, 120, 240, 496, 1024, 2080, 4160, 8256, 16384, 32640, 65280, 130816, 262144, 524800, 1049600, 2098176, 4194304, 8386560, 16773120, 33550336, 67108864, 134225920, 268451840, 536887296, 1073741824, 2147450880

Offset: 0

N. J. A. Sloane

Number of strings over Z_2 of length n with trace 1 and subtrace 1.
Same as number of strings over GF(2) of length n with trace 1 and subtrace 1.
Also expansion of bracket function.
a(n) is also the number of induced subgraphs with odd number of edges in the complete graph K(n-1). - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 02 2009
From Gary W. Adamson, Mar 13 2009: (Start)
M^n * [1,0,0,0] = [A038503(n), a(n), A038505(n), A038504(n)];
where M = the 4 X 4 matrix [1,1,0,0; 0,1,1,0; 0,0,1,1; 1,0,0,1].
Sum of the 4 terms = 2^n.
Example; M^6 * [1,0,0,0] = [16, 20, 16, 12] sum = 64 = 2^6. (End)
Binomial transform of the period 4 repeat: [0,0,0,1], which is the same as A011765 with offset 0. - Wesley Ivan Hurt, Dec 30 2015
{A038503, A038504, A038505, A000749} is the difference analog of the hyperbolic functions of order 4, {h_1(x), h_2(x), h_3(x), h_4(x)}. For a definition see the reference "Higher Transcendental Functions" and the Shevelev link. - Vladimir Shevelev, Jun 14 2017
This is the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S^4; see A291000.  - Clark Kimberling, Aug 24 2017

			a(4;1,1)=4 since the four binary strings of trace 1, subtrace 1 and length 4 are { 0111, 1011, 1101, 1110 }.

Higher Transcendental Functions, Bateman Manuscript Project, Vol. 3, ed. A. Erdelyi, 1983 (chapter XVIII).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
Maran van Heesch, The multiplicative complexity of symmetric functions over a field with characteristic p, Thesis, 2014.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
F. Ruskey, Strings over Z_2 with given trace and subtrace
F. Ruskey, Strings over GF(2) with given trace and subtrace
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4).

Cf. A000748, A000750, A001659, A006090, A007877, A009545, A011765.
Cf. A021913, A038503, A038504, A038505, A133209, A133212, A291000.
Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), this sequence (m=4), A049016 (m=5), A192080 (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).

a000749 n = a000749_list !! n
a000749_list = 0 : 0 : 0 : 1 : zipWith3 (\u v w -> 4 * u - 6 * v + 4 * w)
   (drop 3 a000749_list) (drop 2 a000749_list) (drop 1 a000749_list)
-- Reinhard Zumkeller, Jul 15 2013

I:=[0,0,0,1]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Dec 31 2015

A000749 := proc(n) local k; add(binomial(n,4*k+3),k=0..floor(n/4)); end;
A000749:=-1/((2*z-1)*(2*z**2-2*z+1)); # Simon Plouffe in his 1992 dissertation
a:= n-> if n=0 then 0 else (Matrix(3, (i,j)-> if (i=j-1) then 1 elif j=1 then [4,-6,4][i] else 0 fi)^(n-1))[1,3] fi: seq(a(n), n=0..33); # Alois P. Heinz, Aug 26 2008
# Alternatively:
s := sqrt(2): h := n -> [0,-s,-2,-s,0,s,2,s][1+(n mod 8)]:
a := n -> `if`(n=0,0,(2^n+2^(n/2)*h(n))/4):
seq(a(n),n=0..33); # Peter Luschny, Jun 14 2017

Join[{0},LinearRecurrence[{4,-6,4},{0,0,1},40]] (* Harvey P. Dale, Mar 31 2012 *)
CoefficientList[Series[x^3/(1 -4x +6x^2 -4x^3), {x,0,80}], x] (* Vincenzo Librandi, Dec 31 2015 *)

PARI
```
a(n)=sum(k=0,n\4,binomial(n,4*k+3))
```

@CachedFunction
def a(n): # a = A000749
    if (n<4): return (n//3)
    else: return 4*a(n-1) -6*a(n-2) +4*a(n-3)
[a(n) for n in range(41)] # G. C. Greubel, Apr 11 2023

G.f.: x^3/((1-x)^4 - x^4).
a(n) = Sum_{k=0..n} binomial(n, 4*k+3).
a(n) = a(n-1) + A038505(n-2) = 2*a(n-1) + A009545(n-2) for n>=2.
Without the two initial zeros, binomial transform of A007877. - Henry Bottomley, Jun 04 2001
From Paul Barry, Aug 30 2004: (Start)
a(n) = (2^n - 2^(n/2+1)*sin(Pi*n/4) - 0^n)/4.
a(n+1) is the binomial transform of A021913. (End)
a(n; t, s) = a(n-1; t, s) + a(n-1; t+1, s+t+1) where t is the trace and s is the subtrace.
Without the initial three zeros, = binomial transform of [1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 3, ...]. - Gary W. Adamson, Jun 19 2008
From Vladimir Shevelev, Jun 14 2017: (Start)
1) For n>=1, a(n) = (1/4)*(2^n + i*(1+i)^n - i*(1-i)^n), where i=sqrt(-1);
2) a(n+m) = a(n)*H_1(m) + H_3(n)*H_2(m) + H_2(n)*H_3(m) + H_1(n)*a(m),
where H_1 = A038503, H_2 = A038504, H_3 = A038505. (End)
a(n) = (2^n - 2*A009545(n) - [n=0])/4. - G. C. Greubel, Apr 11 2023

Additional comments from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 22 2002
New definition from Paul Curtz, Oct 29 2007
Edited by N. J. A. Sloane, Jun 13 2008

A049016 Expansion of 1/((1-x)^5 - x^5).

Original entry on oeis.org

1, 5, 15, 35, 70, 127, 220, 385, 715, 1430, 3004, 6385, 13380, 27370, 54740, 107883, 211585, 416405, 826045, 1652090, 3321891, 6690150, 13455325, 26985675, 53971350, 107746282, 214978335, 429124630, 857417220, 1714834440, 3431847189

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,2).

Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), A000749 (m=4), this sequence (m=5), A192080 (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).

Cf. A000750, A078789.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/((1-x)^5-x^5) )); // G. C. Greubel, Apr 11 2023

CoefficientList[Series[1/((1-x)^5-x^5),{x,0,30}],x] (* or *) LinearRecurrence[ {5,-10,10,-5,2},{1,5,15,35,70},40] (* Harvey P. Dale, Jan 20 2014 *)

def A049016_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)^5-x^5) ).list()
A049016_list(30) # G. C. Greubel, Apr 11 2023

G.f.: 1/((1-x)^5-x^5) = 1/( (1-2*x)*(1-3*x+4*x^2-2*x^3+x^4) ).
a(10*n+3) = A078789(5*n+3).
a(10*n+5) = A078789(5*n+4).
a(n) = (-1)^n * A000750(n).
Binomial transform of expansion of (1+x)^4/(1-x^5), or (1, 4, 6, 4, 1, 1, 4, 6, 4, 1, ...). - Paul Barry, Mar 19 2004
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 2*a(n-5). - Paul Curtz, May 24 2008
G.f.: -1/( x^5 - 1 + 5*x/Q(0) ) where Q(k) = 1 + k*(x+1) + 5*x - x*(k+1)*(k+6)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 15 2013

A049017 Expansion of 1/((1-x)^7 - x^7).

Original entry on oeis.org

1, 7, 28, 84, 210, 462, 924, 1717, 3017, 5110, 8568, 14756, 27132, 54264, 116281, 257775, 572264, 1246784, 2641366, 5430530, 10861060, 21242341, 40927033, 78354346, 150402700, 291693136, 574274008, 1148548016, 2326683921, 4749439975, 9714753412, 19818498700, 40199107690

Offset: 0

N. J. A. Sloane

Differs for n >= 7 (1717 vs. 1716) from A000579(n+6) = binomial(n+6,6); see also row 6 of A027555, A059481 and A213808. - M. F. Hasler, Mar 05 2017

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,2).

Cf. A000579, A000750, A027555, A049018, A059481, A213808.

Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), A000749 (m=4), A049016 (m=5), A192080 (m=6), this sequence (m=7), A290995 (m=8), A306939 (m=9).

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-x)^7 - x^7) )); // G. C. Greubel, Apr 11 2023

CoefficientList[Series[1/((1-x)^7-x^7),{x,0,30}],x]  (* Harvey P. Dale, Feb 18 2011 *)

Vec(1/((1-x)^7-x^7)+O(x^99)) \\ M. F. Hasler, Mar 05 2017

def A049017_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)^7 - x^7) ).list()
A049017_list(40) # G. C. Greubel, Apr 11 2023

G.f.: 1/((1-x)^7 - x^7) = 1/((1-2*x)*(1-5*x+11*x^2-13*x^3+9*x^4-3*x^5+x^6)).

A192080 Expansion of 1/((1-x)^6 - x^6).

Original entry on oeis.org

1, 6, 21, 56, 126, 252, 463, 804, 1365, 2366, 4368, 8736, 18565, 40410, 87381, 184604, 379050, 758100, 1486675, 2884776, 5592405, 10919090, 21572460, 43144920, 87087001, 176565486, 357913941, 723002336, 1453179126, 2906358252

Offset: 0

Bruno Berselli, Jun 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6).

Cf. A006090, A049016, A049017, A057079, A057083, A057681.

Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), A000749 (m=4), A049016 (m=5), this sequence (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).

m:=30; R:=PowerSeriesRing(Integers(),m); Coefficients(R!(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2))));

CoefficientList[Series[1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)), {x,0,50}], x] (* Vincenzo Librandi, Oct 15 2012 *)
LinearRecurrence[{6,-15,20,-15,6},{1,6,21,56,126},30] (* Harvey P. Dale, Feb 22 2017 *)

makelist(coeff(taylor(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)), x, 0, n), x, n), n, 0, 29);

Vec(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2))+O(x^99)) \\ Charles R Greathouse IV, Jun 23 2011

def A192080_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)^6-x^6) ).list()
A192080_list(51) # G. C. Greubel, Apr 11 2023

a(n) = abs(A006090(n)) = (-1)^n * A006090(n).
G.f.: 1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)).
From G. C. Greubel, Apr 11 2023: (Start)
a(n) = (2^(n+5) + A010892(n) - 2*A010892(n-1) - 27*(A057083(n) - 2*A057083(n-1)))/6.
a(n) = (2^(n+5) + A057079(n+2) - 27*A057681(n+1))/6. (End)

A306915 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1-x)^k-x^k).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 3, 4, 8, 1, 4, 6, 8, 16, 1, 5, 10, 11, 16, 32, 1, 6, 15, 20, 21, 32, 64, 1, 7, 21, 35, 36, 42, 64, 128, 1, 8, 28, 56, 70, 64, 85, 128, 256, 1, 9, 36, 84, 126, 127, 120, 171, 256, 512, 1, 10, 45, 120, 210, 252, 220, 240, 342, 512, 1024

Offset: 0

Seiichi Manyama, Mar 16 2019

			Square array begins:
     1,   1,   1,   1,   1,    1,    1,    1, ...
     2,   2,   3,   4,   5,    6,    7,    8, ...
     4,   4,   6,  10,  15,   21,   28,   36, ...
     8,   8,  11,  20,  35,   56,   84,  120, ...
    16,  16,  21,  36,  70,  126,  210,  330, ...
    32,  32,  42,  64, 127,  252,  462,  792, ...
    64,  64,  85, 120, 220,  463,  924, 1716, ...
   128, 128, 171, 240, 385,  804, 1717, 3432, ...
   256, 256, 342, 496, 715, 1365, 3017, 6436, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns (1+2),3-9 give A000079, A024495(n+2), A000749(n+3), A049016, A192080, A049017, A290995(n+7), A306939.

Cf. A039912, A101508, A306846, A306913, A306914, A307047, A307078, A307393.

A[n_, k_] := Sum[Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[A[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 25 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+k-1,k*j+k-1).

A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i+k-1,k*j+k-1) * binomial(n-i+k-1,k*j+k-1). - Seiichi Manyama, Apr 07 2019

A290993 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 6, 21, 56, 126, 252, 463, 804, 1365, 2366, 4368, 8736, 18565, 40410, 87381, 184604, 379050, 758100, 1486675, 2884776, 5592405, 10919090, 21572460, 43144920, 87087001, 176565486, 357913941, 723002336, 1453179126, 2906358252, 5791193143

Offset: 0

Clark Kimberling, Aug 21 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

See A291000 for a guide to related sequences.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6).

Cf. A000012, A033453, A192080, A289780, A290991, A290992, A291000.

Sequences of the form x^(m-1)/((1-x)^m - x^m): A000079 (m=1), A131577 (m=2), A024495 (m=3), A000749 (m=4), A139761 (m=5), this sequence (m=6), A290994 (m=7), A290995 (m=8).

a:=[0,0,0,0,1];;  for n in [6..35] do a[n]:=6*a[n-1]-15*a[n-2]+20*a[n-3]-15*a[n-4]+6*a[n-5]; od; Concatenation([0],a); # Muniru A Asiru, Oct 23 2018

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0,0] cat Coefficients(R!( x^5/((1-x)^6 - x^6) )); // G. C. Greubel, Apr 11 2023

seq(coeff(series(x^5/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 23 2018

z = 60; s = x/(1 - x); p = 1 - s^6;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A290993 *)

concat(vector(5), Vec(x^5 / ((1 - 2*x)*(1 - x + x^2)*(1 - 3*x + 3*x^2)) + O(x^50))) \\ Colin Barker, Aug 24 2017

def A290993_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^5/((1-x)^6 - x^6) ).list()
A290993_list(60) # G. C. Greubel, Apr 11 2023

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) for n>5. Corrected by Colin Barker, Aug 24 2017
G.f.: x^5 / ((1 - 2*x)*(1 - x + x^2)*(1 - 3*x + 3*x^2)). - Colin Barker, Aug 24 2017
a(n) = A192080(n-5) for n > 5. - Georg Fischer, Oct 23 2018
G.f.: x^5/((1-x)^6 - x^6). - G. C. Greubel, Apr 11 2023

A290994 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 7, 28, 84, 210, 462, 924, 1717, 3017, 5110, 8568, 14756, 27132, 54264, 116281, 257775, 572264, 1246784, 2641366, 5430530, 10861060, 21242341, 40927033, 78354346, 150402700, 291693136, 574274008, 1148548016, 2326683921, 4749439975

Offset: 0

Clark Kimberling, Aug 22 2017

See A291000 for a guide to related sequences.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,2).

Cf. A000012, A033453, A049017, A289780, A291000.

Sequences of the form x^(m-1)/((1-x)^m - x^m): A000079 (m=1), A131577 (m=2), A024495 (m=3), A000749 (m=4), A139761 (m=5), A290993 (m=6), this sequence (m=7), A290995 (m=8).

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0,0,0] cat Coefficients(R!( x^6/((1-x)^7 - x^7) )); // G. C. Greubel, Apr 11 2023

z = 60; s = x/(1 - x); p = 1 - s^7;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A290994 *)

concat(vector(6), Vec(x^6 / ((1 - 2*x)*(1 - 5*x + 11*x^2 - 13*x^3 + 9*x^4 - 3*x^5 + x^6)) + O(x^50))) \\ Colin Barker, Aug 22 2017

def A290994_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^6/((1-x)^7 - x^7) ).list()
A290994_list(60) # G. C. Greubel, Apr 11 2023

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + 2*a(n-7) for n >= 8.
G.f.: x^6 / ((1 - 2*x)*(1 - 5*x + 11*x^2 - 13*x^3 + 9*x^4 - 3*x^5 + x^6)). - Colin Barker, Aug 22 2017
a(n) = A049017(n-6) for n > 5. - Georg Fischer, Oct 23 2018
G.f.: x^6/((1-x)^7 - x^7). - G. C. Greubel, Apr 11 2023

cp's OEIS Frontend

A024495 a(n) = C(n,2) + C(n,5) + ... + C(n, 3*floor(n/3)+2).

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A291000 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3.

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A000749 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3), n > 3, with a(0)=a(1)=a(2)=0, a(3)=1.

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A049016 Expansion of 1/((1-x)^5 - x^5).

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A049017 Expansion of 1/((1-x)^7 - x^7).

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A192080 Expansion of 1/((1-x)^6 - x^6).

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A000749 a(n) = 4a(n-1) - 6a(n-2) + 4*a(n-3), n > 3, with a(0)=a(1)=a(2)=0, a(3)=1.