A000750 -id:A000750 - cp's OEIS frontend

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.

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

A000748 Expansion of bracket function.

Original entry on oeis.org

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

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)).

A001659 Expansion of bracket function.

Original entry on oeis.org

1, 1, -1, 2, -5, 13, -33, 80, -184, 402, -840, 1699, -3382, 6750, -13716, 28550, -60587, 129579, -275915, 579828, -1197649, 2431775, -4870105, 9672634, -19173013, 38151533, -76521331, 154941608, -316399235, 649807589, -1337598675, 2751021907, -5640238583, 11513062785, -23389948481, 47310801199, -95345789479, 191616365385

Offset: 1

N. J. A. Sloane

sign

Inverse binomial transform of A006218.

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).

G. C. Greubel, Table of n, a(n) for n = 1..1000
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2, issue 4, (1964), 241-260.

Cf. A000005, A000748, A000749, A000750, A006090, A006218.

Equals A038200(n-1) + A038200(n), n>1.

Table[Sum[(-1)^(n - k)*Binomial[n, k]*Sum[Floor[k/j], {j, 1, k}], {k, 0, n}], {n, 1, 50}] (* G. C. Greubel, Jul 02 2017 *)

a(n)=sum(j=0,n,(-1)^(n-j)*binomial(n,j)*sum(k=1,j,j\k))

a(n)=polcoeff(sum(k=1,n,x^k/((1+x)^k-x^k),x*O(x^n)),n)

a(n) = Sum_{j=0..n} ((-1)^(n-j)*binomial(n,j)*Sum_{k=1..j} floor(j/k)).
G.f.: Sum_{k>0} x^k/((1+x)^k-x^k).
G.f.: Sum_{k>0} tau(k)*x^k/(1+x)^k. - Vladeta Jovovic, Jun 24 2003
G.f.: Sum_{n>=1} z^n/(1-z^n) (Lambert series) where z=x/(1+x). - Joerg Arndt, Jan 30 2011
a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n-1,k-1)*tau(k). - Ridouane Oudra, Aug 21 2021

Edited by Michael Somos, Jun 14 2003

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

A307047 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, 0, 1, -2, 0, 1, -3, 4, 0, 1, -4, 6, -8, 0, 1, -5, 10, -9, 16, 0, 1, -6, 15, -20, 9, -32, 0, 1, -7, 21, -35, 36, 0, 64, 0, 1, -8, 28, -56, 70, -64, -27, -128, 0, 1, -9, 36, -84, 126, -125, 120, 81, 256, 0, 1, -10, 45, -120, 210, -252, 200, -240, -162, -512, 0

Offset: 0

Seiichi Manyama, Mar 21 2019

			Square array begins:
   1,    1,    1,    1,    1,    1,     1,     1, ...
   0,   -2,   -3,   -4,   -5,   -6,    -7,    -8, ...
   0,    4,    6,   10,   15,   21,    28,    36, ...
   0,   -8,   -9,  -20,  -35,  -56,   -84,  -120, ...
   0,   16,    9,   36,   70,  126,   210,   330, ...
   0,  -32,    0,  -64, -125, -252,  -462,  -792, ...
   0,   64,  -27,  120,  200,  463,   924,  1716, ...
   0, -128,   81, -240, -275, -804, -1715, -3432, ...
   0,  256, -162,  496,  275, 1365,  2989,  6436, ...

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

Columns 1-7 give A000007, A122803, A000748, (-1)^n * A000749(n+3), A000750, A006090, A049018.

Cf. A039912 (square array A(n,k), n >= 0, k >= 2), A306913, A306914, A306915.

T[n_, k_] := (-1)^n * Sum[(-1)^(j * Mod[k, 2]) * Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

A(n,k) = (-1)^n * Sum_{j=0..floor(n/k)} (-1)^((k mod 2) * j) * binomial(n+k-1,k*j+k-1).

cp's OEIS Frontend

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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A000748 Expansion of bracket function.

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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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A001659 Expansion of bracket function.

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A006090 Expansion of bracket function.

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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.