A113413 -id:A113413 - cp's OEIS frontend

A008416 Coordination sequence for 8-dimensional cubic lattice.

1, 16, 128, 688, 2816, 9424, 27008, 68464, 157184, 332688, 658048, 1229360, 2187520, 3732560, 6140800, 9785072, 15158272, 22900496, 33830016, 48978352, 69629696, 97364944, 134110592, 182192752, 244396544, 324031120, 425000576

Offset: 0

N. J. A. Sloane

Coordination sequence for 8-dimensional cyclotomic lattice Z[zeta_16].

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Ross McPhedran, Numerical Investigations of the Keiper-Li Criterion for the Riemann Hypothesis, arXiv:2311.06294 [math.NT], 2023. See p. 6.
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A008574, A005899, A008412, A008413, A008414, A008415, A008418, A008420.

Cf. A113413, A122542.

CoefficientList[Series[((1 + x)/(1 - x))^8, {x, 0, 26}], x] (* Michael De Vlieger, Dec 18 2017 *)

G.f.: ((1+x)/(1-x))^8.
a(n) = A008415(n) + A008415(n-1) + a(n-1). - Bruce J. Nicholson, Dec 17 2017
n*a(n) = 16*a(n-1) + (n-2)*a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018

A035599 Number of points of L1 norm 5 in cubic lattice Z^n.

Original entry on oeis.org

0, 2, 20, 102, 360, 1002, 2364, 4942, 9424, 16722, 28004, 44726, 68664, 101946, 147084, 207006, 285088, 385186, 511668, 669446, 864008, 1101450, 1388508, 1732590, 2141808, 2625010, 3191812, 3852630, 4618712, 5502170, 6516012

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A035596-A035606.

Column 5 of A035607, A266213. Row 5 of A113413, A119800, A122542.

[(4*n^4+20*n^2+6)*n/15: n in [0..30]]; // Vincenzo Librandi, Apr 23 2012

f := proc(d,m) local i; sum( 2^i*binomial(d,i)*binomial(m-1,i-1),i=1..min(d,m)); end; # n=dimension, m=norm

CoefficientList[Series[2*x*(1+x)^4/(1-x)^6,{x,0,33}],x] (* Vincenzo Librandi, Apr 23 2012 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,2,20,102,360,1002},40] (* Harvey P. Dale, Dec 30 2023 *)

a(n)=(4*n^5+20*n^3+6*n)/15 \\ Charles R Greathouse IV, Dec 07 2011

a(n) = (4*n^4+20*n^2+6)*n/15. - Frank Ellermann, Mar 16 2002
G.f.: 2*x*(1+x)^4/(1-x)^6. - Colin Barker, Mar 19 2012
a(n) = 2*A069038(n). - R. J. Mathar, Dec 10 2013
From Shel Kaphan, Mar 01 2023: (Start)
a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=5.
a(n) = A001847(n) - A001846(n).
a(n) = A008413(n)*n/5. (End)

A080246 Signed version of A035607.

Original entry on oeis.org

1, -2, 1, 2, -4, 1, -2, 8, -6, 1, 2, -12, 18, -8, 1, -2, 16, -38, 32, -10, 1, 2, -20, 66, -88, 50, -12, 1, -2, 24, -102, 192, -170, 72, -14, 1, 2, -28, 146, -360, 450, -292, 98, -16, 1, -2, 32, -198, 608, -1002, 912, -462, 128, -18, 1, 2, -36, 258, -952, 1970, -2364

Offset: 0

Paul Barry, Feb 15 2003

Written as lower triangular matrix this has inverse A080247. Row sums are (1,-1,-1,1,1,-1,-1,1,1,...) Diagonal sums are signed tribonacci numbers A078042

Riordan array((1-x)/(1+x), x*(1-x)/(1+x)). - Philippe Deléham, Jan 05 2014

			Rows are {1}, {-2,1}, {2,-4,1}, {-2,8,-6,1}, ...

Paul Barry, Laurent Biorthogonal Polynomials and Riordan Arrays, arXiv preprint arXiv:1311.2292, 2013
Heinrich Niederhausen, Inverses of Motzkin and Schroeder Paths, arXiv preprint arXiv:1105.3713, 2011.

Cf. A035607, A080247.

Columns are generated by (1-x)^k/(1+x)^k
T(n,k)=(-1)^(n+k)*A113413(n,k). - Philippe Deléham, Jan 05 2014
T(n,k)=T(n-1,k-1)-T(n-1,k)-T(n-2,k-1), T(0,0)=1, T(1,0)=-2, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 05 2014

A270715 a(n) = ((n+2)/2)Sum_{k=0..n/2}(Sum_{i=0..n-2k} binomial(k+1,n-2k-i)binomial(k+i,k))/(k+1).

Original entry on oeis.org

1, 3, 5, 10, 19, 35, 65, 120, 221, 407, 749, 1378, 2535, 4663, 8577, 15776, 29017, 53371, 98165, 180554, 332091, 610811, 1123457, 2066360, 3800629, 6990447, 12857437, 23648514, 43496399, 80002351, 147147265

Offset: 0

Vladimir Kruchinin, Mar 22 2016

nonn

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

Cf. A113413, A270709.

LinearRecurrence[{2,0,0,-1},{1,3,5,10},40] (* Harvey P. Dale, May 23 2017 *)

a(n):=(n+2)/2*(sum(sum(binomial(k+1,n-2*k-i)*binomial(k+i,k),i,0,n-2*k)/(k+1),k,0,n/2));

x='x+O('x^200); Vec((-x^2+x+1)/((1-x)*(-x^3-x^2-x+1))) \\ Altug Alkan, Mar 22 2016

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

A270724 a(n) = ((n+2)/2)Sum_{k=0..n/2} (Sum_{i=0..n-2k} (binomial(k+1,n-2k-i)binomial(k+i,k))/(k+1)*C(k)), where C(k) is Catalan numbers.

Original entry on oeis.org

1, 3, 5, 10, 20, 42, 93, 213, 504, 1221, 3014, 7553, 19158, 49087, 126845, 330174, 864884, 2278138, 6030218, 16031950, 42790362, 114616360, 307996874, 830084080, 2243193198, 6076953906, 16500486744, 44897830740, 122406923038, 334333367602

Offset: 0

Vladimir Kruchinin, Mar 22 2016

nonn

Cf. A000108, A113413, A270715.

A270724 := proc(n)
    a := 0 ;
    for k from 0 to n/2 do
        for i from 0 to n-2*k do
            a := a+binomial(k+1,n-2*k-i)*binomial(k+i,k)/(k+1)*A000108(k) ;
        end do:
    end do:
    %*(n+2)/2 ;
end proc: # R. J. Mathar, Oct 07 2016

Table[((n + 2)/2) Sum[Sum[(Binomial[k + 1, n - 2 k - i] Binomial[k + i, k]) Binomial[2 k, k]/(k + 1)^2, {i, 0, n - 2 k}], {k, 0, n/2}], {n, 0, 29}] (* or *)
CoefficientList[Series[((-x^2 + x + 1) (1 - Sqrt[1 - (4 x^2 (x + 1))/(1 - x)]))/(2 x^2*(1 - x^2)), {x, 0, 29}], x] (* Michael De Vlieger, Mar 25 2016 *)

a(n):=((n+2)/2)*(sum(sum(binomial(k+1,n-2*k-i)*binomial(k+i,k),i,0,n-2*k)/(k+1)^2*binomial(2*k,k),k,0,n/2));

x='x+O('x^200); Vec(((-x^2+x+1)*(1-sqrt(1-(4*x^2*(x+1))/(1-x))))/(2*x^2*(1-x^2))) \\ Altug Alkan, Mar 22 2016

G.f.: ((-x^2+x+1)*(1-sqrt(1-(4*x^2*(x+1))/(1-x))))/(2*x^2*(1-x^2)).
a(n) = ((n+2)/2)*Sum_{k=0..n/2} (Sum_{i=0..n-2*k} (binomial(k+1,n-2*k-i)*binomial(k+i,k))*binomial(2*k,k)/(k+1)^2).
Conjecture: (n+2)*a(n) +(-n-2)*a(n-1) +(-7*n+6)*a(n-2) +10*a(n-3) +(13*n-32)*a(n-4) +(5*n-32)*a(n-5) +(-11*n+52)*a(n-6) +4*(-n+6)*a(n-7) +4*(n-7)*a(n-8)=0. - R. J. Mathar, Oct 07 2016

A270737 a(n) = ((n+2)/2)Sum_{k=0..n/2} (Sum_{i=0..n-2k} (binomial(k+1,n-2k-i)binomial(k+i,k))*F(k+1)/(k+1)), where F(k) is Fibonacci numbers.

Original entry on oeis.org

1, 3, 5, 10, 20, 42, 91, 195, 415, 880, 1864, 3952, 8385, 17795, 37765, 80138, 170044, 360810, 765595, 1624515, 3447071, 7314368, 15520400, 32932800, 69880225, 148279107, 314634021, 667623210, 1416632420, 3005958090, 6378354619

Offset: 0

Vladimir Kruchinin, Mar 22 2016

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

Cf. A000045, A113413.

Table[((n + 2)/2) Sum[Sum[(Binomial[k + 1, n - 2 k - i] Binomial[k + i, k])/(k + 1) Fibonacci[k + 1], {i, 0, n - 2 k}], {k, 0, n/2}], {n, 0, 30}] (* or *)
CoefficientList[Series[(-x^2 + x + 1)/(-x^6 - 2 x^5 - 2 x + 1), {x, 0, 30}], x] (* Michael De Vlieger, Mar 25 2016 *)
LinearRecurrence[{2, 0, 0, 0, 2, 1}, {1, 3, 5, 10, 20, 42}, 100] (* G. C. Greubel, Mar 25 2016 *)

a(n):=(n+2)/2*(sum(sum(binomial(k+1,n-2*k-i)*binomial(k+i,k),i,0,n-2*k)*fib(k+1)/(k+1),k,0,n/2));

my(x='x+O('x^40)); Vec((-x^2+x+1)/(-x^6-2*x^5-2*x+1)) \\ Altug Alkan, Mar 22 2016

G.f.: (-x^2+x+1)/(-x^6-2*x^5-2*x+1).

a(n) = 2*a(n-1) + 2*a(n-5) + a(n-6). - G. C. Greubel, Mar 25 2016

cp's OEIS Frontend

A008416 Coordination sequence for 8-dimensional cubic lattice.

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A035599 Number of points of L1 norm 5 in cubic lattice Z^n.

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A080246 Signed version of A035607.

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A270715 a(n) = ((n+2)/2)*Sum_{k=0..n/2}(Sum_{i=0..n-2*k} binomial(k+1,n-2*k-i)*binomial(k+i,k))/(k+1).

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A270724 a(n) = ((n+2)/2)*Sum_{k=0..n/2} (Sum_{i=0..n-2*k} (binomial(k+1,n-2*k-i)*binomial(k+i,k))/(k+1)*C(k)), where C(k) is Catalan numbers.

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A270737 a(n) = ((n+2)/2)*Sum_{k=0..n/2} (Sum_{i=0..n-2*k} (binomial(k+1,n-2*k-i)*binomial(k+i,k))*F(k+1)/(k+1)), where F(k) is Fibonacci numbers.

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A270715 a(n) = ((n+2)/2)Sum_{k=0..n/2}(Sum_{i=0..n-2k} binomial(k+1,n-2k-i)binomial(k+i,k))/(k+1).

A270724 a(n) = ((n+2)/2)Sum_{k=0..n/2} (Sum_{i=0..n-2k} (binomial(k+1,n-2k-i)binomial(k+i,k))/(k+1)*C(k)), where C(k) is Catalan numbers.

A270737 a(n) = ((n+2)/2)Sum_{k=0..n/2} (Sum_{i=0..n-2k} (binomial(k+1,n-2k-i)binomial(k+i,k))*F(k+1)/(k+1)), where F(k) is Fibonacci numbers.