A307047 -id:A307047 - cp's OEIS frontend

A000748 Expansion of bracket function.

1, -3, 6, -9, 9, 0, -27, 81, -162, 243, -243, 0, 729, -2187, 4374, -6561, 6561, 0, -19683, 59049, -118098, 177147, -177147, 0, 531441, -1594323, 3188646, -4782969, 4782969, 0, -14348907, 43046721, -86093442, 129140163, -129140163, 0, 387420489, -1162261467

Offset: 0

N. J. A. Sloane

It appears that the sequence coincides with its third-order absolute difference. - John W. Layman, Sep 05 2003

It appears that, for n > 0, the (unsigned) a(n) = 3*|A057682(n)| = 3*|Sum_{j=0..floor(n/3)} (-1)^j*binomial(n,3*j+1)|. - John W. Layman, Sep 05 2003

			G.f. = 1 - 3*x + 6*x^2 - 9*x^3 + 9*x^4 - 27*x^6 + 81*x^7 - 162*x^8 + ...

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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
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
Index entries for linear recurrences with constant coefficients, signature (-3,-3).

Column 3 of A307047.
Cf. A000749, A000750, A001659.
Cf. A057682.

I:=[1,-3]; [n le 2 select I[n] else -3*Self(n-1)-3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Feb 11 2016

A000748:=(-1-2*z-3*z**2-3*z**3+18*z**5)/(-1+z+9*z**5); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence apart from signs
a:= n-> (Matrix([[ -3,1], [ -3,0]])^n)[1,1]: seq(a(n), n=0..40); # Alois P. Heinz, Sep 06 2008

a[n_] := 2*3^(n/2)*Sin[(1-5*n)*Pi/6]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 12 2014 *)
LinearRecurrence[{-3, -3}, {1, -3}, 40] (* Jean-François Alcover, Feb 11 2016 *)

{a(n) = if( n<0, 0, polcoeff(1 / (1 + 3*x + 3*x^2) + x * O(x^n), n))}; /* Michael Somos, Jun 07 2005 */

{a(n) = if( n<0, 0, 3^((n+1)\2) * (-1)^(n\6) * ((-1)^n + (n%3==2)))}; /* Michael Somos, Sep 29 2007 */

G.f.: 1/((1+x)^3-x^3).
a(n) = A007653(3^n).
a(n) = -3*a(n-1) - 3*a(n-2). - Paul Curtz, May 12 2008
a(n) = Sum_{k=1..n} binomial(k,n-k)*(-3)^(k) for n > 0; a(0)=1. - Vladimir Kruchinin, Feb 07 2011
G.f.: 1/(1 + 3*x /(1 - x /(1+x))). - Michael Somos, May 12 2012
G.f.: G(0)/2, where G(k) = 1 + 1/( 1 - 3*x*(2*k+1 + x)/(3*x*(2*k+2 + x) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Feb 09 2014
a(n) = 2*3^(n/2)*sin((1-5*n)*Pi/6). - Jean-François Alcover, Mar 12 2014
a(n) = (-1)^n * Sum_{k=0..floor(n/3)} (-1)^k * binomial(n+2,3*k+2). - Seiichi Manyama, Aug 05 2024
a(n) = (i*sqrt(3)/3)*((-3/2 - i*sqrt(3)/2)^(n+1) - (-3/2 + i*sqrt(3)/2)^(n+1)), where i = sqrt(-1). - Taras Goy, Jan 20 2025
a(n) = -2*a(n-1) + 3*a(n-3). - Taras Goy, Jan 26 2025

A000750 Expansion of bracket function.

Original entry on oeis.org

1, -5, 15, -35, 70, -125, 200, -275, 275, 0, -1000, 3625, -9500, 21250, -42500, 76875, -124375, 171875, -171875, 0, 621875, -2250000, 5890625, -13171875, 26343750, -47656250, 77109375, -106562500, 106562500, 0

Offset: 0

N. J. A. Sloane

It appears that the (unsigned) sequence is identical to its 5th-order absolute difference. - John W. Layman, Sep 23 2003

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).

Seiichi Manyama, Table of n, a(n) for n = 0..3000
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
Index entries for linear recurrences with constant coefficients, signature (-5, -10, -10, -5).

Column 5 of A307047.

Cf. A000748, A000749, A001659, A006090, A049016.

LinearRecurrence[{-5, -10, -10, -5}, {1, -5, 15, -35}, 30] (* Jean-François Alcover, Feb 11 2016 *)

Vec(1/((1+x)^5-x^5) + O(x^40)) \\ Michel Marcus, Feb 11 2016

{a(n) = (-1)^n*sum(k=0, n\5, (-1)^k*binomial(n+4, 5*k+4))} \\ Seiichi Manyama, Mar 21 2019

G.f.: 1/((1+x)^5-x^5).

a(n) = (-1)^n * Sum_{k=0..floor(n/5)} (-1)^k * binomial(n+4,5*k+4). - Seiichi Manyama, Mar 21 2019

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

A006090 Expansion of bracket function.

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

Offset: 0

David Callan

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2, issue 4, (1964), 241-260.
Problems Drive, Eureka, 37 (1974), 8-11, 32-33, 24-27. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (-6,-15,-20,-15,-6).

Column 6 of A307047.

Cf. A000748, A000749, A000750, A001659.

CoefficientList[Series[1/((1+x)^6-x^6),{x,0,30}],x] (* or *) LinearRecurrence[ {-6,-15,-20,-15,-6},{1,-6,21,-56,126},31] (* Harvey P. Dale, Oct 14 2016 *)

x='x+O('x^50); Vec(1/((1+x)^6-x^6)) \\ G. C. Greubel, Jul 02 2017

G.f.: 1/((1+x)^6-x^6).

a(n) = (-1)^n * Sum_{k=0..floor(n/6)} binomial(n+5,6*k+5). - Seiichi Manyama, Aug 05 2024

A049018 Expansion of 1/((1+x)^7 - x^7).

Original entry on oeis.org

1, -7, 28, -84, 210, -462, 924, -1715, 2989, -4900, 7448, -9996, 9996, 0, -38759, 149205, -422576, 1041348, -2350922, 4970070, -9940140, 18874261, -33957343, 57374296, -89125120, 120875944, -120875944, 0, 459957169, -1749692735, 4904887652

Offset: 0

N. J. A. Sloane

It appears that the (unsigned) sequence is identical to its 7th-order absolute difference. - John W. Layman, Oct 02 2003

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).

Column 7 of A307047.

Cf. A049017.

R:=PowerSeriesRing(Integers(), 35); Coefficients(R!( 1/((1+x)^7 - x^7) )); // G. C. Greubel, Mar 17 2019

LinearRecurrence[{-7,-21,-35,-35,-21,-7},{1,-7,28,-84,210,-462}, 35] (* Ray Chandler, Sep 23 2015 *)

Vec(1/((1+x)^7-x^7)+O(x^35)) \\ Charles R Greathouse IV, Sep 27 2012

{a(n) = (-1)^n*sum(k=0, n\7, (-1)^k*binomial(n+6, 7*k+6))} \\ Seiichi Manyama, Mar 21 2019

(1/((1+x)^7 - x^7)).series(x, 35).coefficients(x, sparse=False) # G. C. Greubel, Mar 17 2019

a(n) = (-1)^n * Sum_{k=0..floor(n/7)} (-1)^k * binomial(n+6,7*k+6). - Seiichi Manyama, Mar 21 2019

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A000748 Expansion of bracket function.

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A000750 Expansion of bracket function.

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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).

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A006090 Expansion of bracket function.

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A049018 Expansion of 1/((1+x)^7 - x^7).

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