A035597 -id:A035597 - cp's OEIS frontend

A343599 T(n,k) is the coordination number of the (n+1)-dimensional cubic lattice for radius k; triangle read by rows, n>=0, 0<=k<=n.

1, 1, 4, 1, 6, 18, 1, 8, 32, 88, 1, 10, 50, 170, 450, 1, 12, 72, 292, 912, 2364, 1, 14, 98, 462, 1666, 4942, 12642, 1, 16, 128, 688, 2816, 9424, 27008, 68464, 1, 18, 162, 978, 4482, 16722, 53154, 148626, 374274, 1, 20, 200, 1340, 6800, 28004, 97880, 299660, 822560, 2060980, 1, 22, 242, 1782, 9922, 44726, 170610, 568150, 1690370, 4573910, 11414898

Offset: 0

R. J. Mathar, Apr 21 2021

			The full array starts
     1      2      2      2      2      2      2      2      2
     1      4      8     12     16     20     24     28     32
     1      6     18     38     66    102    146    198    258
     1      8     32     88    192    360    608    952   1408
     1     10     50    170    450   1002   1970   3530   5890
     1     12     72    292    912   2364   5336  10836  20256
     1     14     98    462   1666   4942  12642  28814  59906
     1     16    128    688   2816   9424  27008  68464 157184
     1     18    162    978   4482  16722  53154 148626 374274

J. Schroder, Generalized Schroder Numbers and the Rotation principle, J. Int. Seq. 10 (2007) # 07.7.7, Theorem 4.2.

Cf. A035607 (by antidiags), A008574 (n=1), A005899 (n=2), A008412 (n=3), A008413 (n=4), A008414 (n=5), A001105 (k=2), A035597 (k=3), A035598 (k=4).

Main diagonal gives A050146(n+1).

A343599 := proc(n,k)
    local g,x,y ;
    g := (1+y)/(1-x-y-x*y) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,k) ;
end proc:

T[n_, k_] := Module[{x, y}, SeriesCoefficient[(1 + y)/(1 - x - y - x*y), {x, 0, n}] // SeriesCoefficient[#, {y, 0, k}]&];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 16 2023 *)

G.f.: (1+y)/(1-x-y-x*y).

T(n,k) = A008288(n,k) + A008288(n,k-1).

A202241 Array F(n,m) read by antidiagonals: F(0,m)=1, F(n,0) = A130713(n), and column m+1 is recursively defined as the partial sums of column m.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 0, 4, 4, 1, 0, 4, 8, 5, 1, 0, 4, 12, 13, 6, 1, 0, 4, 16, 25, 19, 7, 1, 0, 4, 20, 41, 44, 26, 8, 1, 0, 4, 24, 61, 85, 70, 34, 9, 1, 0, 4, 28, 85, 146, 155, 104, 43, 10, 1, 0, 4, 32, 113, 231, 301, 259, 147, 53, 11, 1, 0, 4, 36, 145, 344, 532, 560, 406, 200, 64, 12, 1

Offset: 0

Paul Curtz, Dec 16 2011

The array F(n,m), beginning with row n=0, is:
  1, 1,  1,  1,   1,   1,    1,
  2, 3,  4,  5,   6,   7,    8,
  1, 4,  8, 13,  19,  26,   34,
  0, 4, 12, 25,  44,  70,  104,
  0, 4, 16, 41,  85, 155,  259,
  0, 4, 20, 61, 146, 301,  560,
  0, 4, 24, 85, 231, 532, 1092.
Columns after A130713, A113311, A008574 have signatures (3,-3,1), (4,-6,4,-1), (5,-10,10,-5,1), (6,-15,20,-15,6,-1) (from A135278(n+3)).
Inserting columns of zeros and pushing the columns down, plus alternating sign switches defines the following triangle T(n,2m) = (-1)^(m/2)*F(n-2m,m):
  1,
  2 0,
  1 0 -1,
  0 0 -3 0,
  0 0 -4 0 1,
  0 0 -4 0 4 0,
  0 0 -4 0 8 0 -1
The row sums in the triangle are (-1)^n*A099838(n).
The companion to A201863 is
  1
  1 0
  1 0   0
  1 0  -2 0
  1 0  -4 0  1
  1 0  -6 0  5 0
  1 0  -8 0 13 0   -2
  1 0 -10 0 25 0  -12 0
  1 0 -12 0 41 0  -38 0   4
  1 0 -14 0 61 0  -88 0  28 0
  1 0 -16 0 85 0 -170 0 104 0 -8
5th column: A001844; 7th column: -A035597=-2*A005900(n+1); 9th column: 4*A006325(n+2); 11th column: -8*(1,8,34,104) (from columns 4,5,6,7 of F(n,m)).
As a triangular array, this is the Riordan array ((1+x)^2, x/(1-x)). - Philippe Deléham, Feb 21 2012

			Triangle T(n,k) begins:
  1
  2, 1
  1, 3,  1
  0, 4,  4,  1
  0, 4,  8,  5,   1
  0, 4, 12, 13,   6,   1
  0, 4, 16, 25,  19,   7,   1
  0, 4, 20, 41,  44,  26,   8,  1
  0, 4, 24, 61,  85,  70,  34,  9,  1
  0, 4, 28, 85, 146, 155, 104, 43, 10, 1
- _Philippe Deléham_, Feb 21 2012

Muniru A Asiru, Table of n, a(n) for n = 0..5151

Cf. A130713 (column 0), A113311 (column 1), A008574 (column 2), A001844 (column 3), A005900 (column 4), A006325 (column 5), A033455 (column 6).

Cf. A267633.

Flat(List([0..12],n->List([0..n],k->Binomial(n,n-k)+Binomial(n-1,n-k-1)-Binomial(n-2,n-k-2)-Binomial(n-3,n-k-3)))); # Muniru A Asiru, Mar 22 2018

A130713 := proc(n)
    if n <= 2 and n >=0 then
        op(n+1,[1,2,1]) ;
    else
        0;
    end if;
end proc:
A202241 := proc(n,m)
    option remember;
    if n < 0 then
        0 ;
    elif m = 0 then
        A130713(n);
    else
        procname(n,m-1)+procname(n-1,m) ;
    end if;
end proc:
for d from 0 to 12 do
    for m from 0 to d do
        printf("%d,",A202241(d-m,m)) ;
    end do:
end do: # R. J. Mathar, Dec 22 2011
C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if end proc:
for n from 0 to 10 do
     seq(C(n, n-k) + C(n-1, n-k-1) - C(n-2, n-k-2) - C(n-3, n-k-3), k = 0..n);
end do; # Peter Bala, Mar 20 2018

rows = 12;
T[0] = PadRight[{1, 2, 1}, rows];
T[n_ /; nJean-François Alcover, Jun 29 2019 *)

def Trow(n): return [binomial(n, n-k) + binomial(n-1, n-k-1) - binomial(n-2, n-k-2) - binomial(n-3, n-k-3) for k in (0..n)]
for n in (0..9): print(Trow(n)) # Peter Luschny, Mar 21 2018

F(1,m) = m+2.
F(2,m) = A034856(m+1).
F(3,m) = A000297(m-1).
Sum_{m=0..d} F(d-m,m) = A116453(d-3), d >= 3 (antidiagonal sums).
As a triangular array T(n,k), 0 <= k <= n, satisfies: T(n,k) = T(n-1,k) + T(n-1,k-1) with T(0,0) = 1, T(1,0) = 2, T(2,0) = 1, T(3,0) = 0. - Philippe Deléham, Feb 21 2012
Unsigned diagonals of A267633 (beginning with its main diagonal) appear to be the reverse rows of this entry's triangle beginning with the fourth row. - Tom Copeland, Jan 26 2016
T(n,k) = C(n, n-k) + C(n-1, n-k-1) - C(n-2, n-k-2) - C(n-3, n-k-3), where C(n, k) = n!/(k!*(n-k)!) if 0 <= k <= n, otherwise 0. - Peter Bala, Mar 20 2018

cp's OEIS Frontend

A343599 T(n,k) is the coordination number of the (n+1)-dimensional cubic lattice for radius k; triangle read by rows, n>=0, 0<=k<=n.

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