A008383 -id:A008383 - cp's OEIS frontend

A103881 Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n.

1, 1, 2, 1, 6, 2, 1, 12, 12, 2, 1, 20, 42, 18, 2, 1, 30, 110, 92, 24, 2, 1, 42, 240, 340, 162, 30, 2, 1, 56, 462, 1010, 780, 252, 36, 2, 1, 72, 812, 2562, 2970, 1500, 362, 42, 2, 1, 90, 1332, 5768, 9492, 7002, 2570, 492, 48, 2, 1, 110, 2070, 11832, 26474, 27174, 14240, 4060, 642, 54, 2, 1, 132, 3080, 22530, 66222, 91112, 65226, 26070, 6040, 812, 60, 2

Offset: 1

Ralf Stephan, Feb 20 2005

T(n,k) is the number of integer sequences of length n+1 with sum zero and sum of absolute values 2k. - R. H. Hardin, Feb 23 2009

			Array begins:
  1,   2,     2,      2,       2,        2,         2,          2, ... A040000;
  1,   6,    12,     18,      24,       30,        36,         42, ... A008458;
  1,  12,    42,     92,     162,      252,       362,        492, ... A005901;
  1,  20,   110,    340,     780,     1500,      2570,       4060, ... A008383;
  1,  30,   240,   1010,    2970,     7002,     14240,      26070, ... A008385;
  1,  42,   462,   2562,    9492,    27174,     65226,     137886, ... A008387;
  1,  56,   812,   5768,   26474,    91112,    256508,     623576, ... A008389;
  1,  72,  1332,  11832,   66222,   271224,    889716,    2476296, ... A008391;
  1,  90,  2070,  22530,  151560,   731502,   2777370,    8809110, ... A008393;
  1, 110,  3080,  40370,  322190,  1815506,   7925720,   28512110, ... A008395;
  1, 132,  4422,  68772,  643632,  4197468,  20934474,   85014204, ... A035837;
  1, 156,  6162, 112268, 1219374,  9129276,  51697802,  235895244, ... A035838;
  1, 182,  8372, 176722, 2206932, 18827718, 120353324,  614266354, ... A035839;
  1, 210, 11130, 269570, 3838590, 37060506, 265953170, 1511679210, ... A035840;
  ...
Antidiagonals:
  1;
  1,  2;
  1,  6,    2;
  1, 12,   12,    2;
  1, 20,   42,   18,    2;
  1, 30,  110,   92,   24,    2;
  1, 42,  240,  340,  162,   30,    2;
  1, 56,  462, 1010,  780,  252,   36,   2;
  1, 72,  812, 2562, 2970, 1500,  362,  42,  2;
  1, 90, 1332, 5768, 9492, 7002, 2570, 492, 48,  2;

Muniru A Asiru, Table of n, a(n) for n = 1..5050 (antidiagonals 1 to 100, flattened)
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Arun Padakandla, P. R. Kumar, and Wojciech Szpankowski, On the Discrete Geometry of Differential Privacy via Ehrhart Theory, November 2017.
Arun Padakandla, P. R. Kumar, and Wojciech Szpankowski, Preserving Privacy and Fidelity via Ehrhart Theory, July 2017.
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Rows include A040000, A008458, A005901, A008383, A008385, A008387, A008389, A008391, A008393, A008395, A035837, A035838, A035839, A035840, A035841 - A035876.
Columns include A002376, A001621.
Main diagonal is in A103882.
Cf. A047085, A103884, A103903, A103998, A143007.

T:=Flat(List([1..12],n->Concatenation([1],List([1..n-1],k->Sum([1..n],i->Binomial(n-k+1,i)*Binomial(k-1,i-1)*Binomial(n-i,k)))))); # Muniru A Asiru, Oct 14 2018

A103881:= func< n,k | k le 0 select 1 else (&+[Binomial(n-k+1, j)*Binomial(k-1, j-1)*Binomial(n-j, k): j in [1..n-k]]) >;
[A103881(n,k): k in [0..n-1], n in [1..15]]; // G. C. Greubel, Oct 16 2018; May 24 2023

T:=proc(n,k) option remember; local i;
if k=0 then 1 else
add( binomial(n+1,i)*binomial(k-1,i-1)*binomial(n-i+k,k),i=1..n); fi;
end:
g:=n->[seq(T(n-i,i),i=0..n-1)]:
for n from 1 to 14 do lprint(op(g(n))); od:

T[n_, k_]:= (n+1)*(n+k-1)!*HypergeometricPFQ[{1-k,1-n,-n}, {2,-n-k+1}, 1]/(k!*(n-1)!); T[, 0]=1; Flatten[Table[T[n-k, k], {n,12}, {k,0,n-1}]] (* _Jean-François Alcover, Dec 27 2012 *)

A103881(n,k) = if(k==0, 1, sum(j=1, n-k, binomial(n-k+1, j)*binomial(k-1, j-1)*binomial(n-j, k)));
for(n=1, 15, for(k=0, n-1, print1(A103881(n,k), ", "))) \\ G. C. Greubel, Oct 16 2018; May 24 2023

def A103881(n,k): return 1 if k==0 else (n-k+1)*binomial(n-1,k)*hypergeometric([k-n,1+k-n,1-k], [2,1-n], 1).simplify()
flatten([[A103881(n,k) for k in range(n)] for n in range(1,16)]) # G. C. Greubel, May 24 2023

T(n,k) = Sum_{i=1..n} C(n+1, i)*C(k-1, i-1)*C(n-i+k, k), T(n,0)=1.
G.f. of n-th row: (Sum_{i=0..n} C(n, i)^2*x^i)/(1-x)^n.
From G. C. Greubel, May 24 2023: (Start)
T(n, k) = Sum_{j=0..n} binomial(n,j)^2 * binomial(n+k-j-1, n-1) (array).
T(n, k) = (n+1)*binomial(n+k-1,k)*hypergeometric([-n,1-n,1-k], [2,1-n-k], 1), with T(n, k) = 1 (array).
t(n, k) = (n-k+1)*binomial(n-1,k)*hypergeometric([k-n,1+k-n,1-k], [2,1-n], 1), with t(n, 0) = 1 (antidiagonals).
Sum_{k=0..n-1} t(n, k) = A047085(n). (End)
From Peter Bala, Jul 09 2023: (Start)
T(n,k) = [x^k] Legendre_P(n, (1 + x)/(1 - x)).
(n+1)*T(n+1,k) = (n+1)*T(n+1,k-1) + (2*n+1)*(T(n,k) + T(n,k-1)) - n*(T(n-1,k) - T(n-1,k-1)). (End)

Corrected by N. J. A. Sloane, Dec 15 2012, at the suggestion of Manuel Blum

A008384 Crystal ball sequence for A_4 lattice.

Original entry on oeis.org

1, 21, 131, 471, 1251, 2751, 5321, 9381, 15421, 24001, 35751, 51371, 71631, 97371, 129501, 169001, 216921, 274381, 342571, 422751, 516251, 624471, 748881, 891021, 1052501, 1235001, 1440271

Offset: 0

N. J. A. Sloane and J. H. Conway

Partial sums of A008383.

T. D. Noe, 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).
H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013; Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
Index entries for crystal ball sequences
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A008383.

Table[1/12 (12-50 n+85 n^2-70 n^3+35 n^4),{n,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,21,131,471,1251},30] (* Harvey P. Dale, Aug 22 2011 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,-5,10,-10,5]^n*[1;21;131;471;1251])[1,1] \\ Charles R Greathouse IV, Jun 15 2015

a(n) = 1 +5*n*(n+1)*(7*n^2+7*n+10)/12. - T. D. Noe, Apr 29 2007
G.f.: (-1-x^4-16*x^3-36*x^2-16*x)/(x-1)^5. [Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009]
a(n) = 5*a(n-1)-10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5), n> 4. - Harvey P. Dale, Aug 22 2011

A358632 Coordination sequence for the faces of the uniform infinite surface that is formed from congruent regular pentagons and from which there is a continuous function that maps the faces 1:1 to regular pentagons in the plane.

Original entry on oeis.org

1, 5, 20, 50, 110, 200, 340, 525, 780, 1095, 1500, 1980, 2570, 3250, 4060, 4975, 6040, 7225, 8580, 10070, 11750

Offset: 0

Peter Munn and Allan C. Wechsler, Nov 24 2022

Each pentagon shares each of its edges with a congruent pentagon. Obviously, the images of many pentagonal faces overlap.

Equivalently, at step 0, embed the edges of a regular pentagon in the plane. At step n >= 1, embed the edges to form all regular pentagons that include an edge that was present after step n-1. a(n) is the number of pentagons whose set of embedded edges is completed in step n.

			Position the seed pentagon in the complex plane centered at 0, scaled so that the centers of adjacent pentagons are at unit distance. Let the centers of the pentagons completed in step 1 be at 1, exp(2 Pi i / 5), exp(4 Pi i / 5), exp(6 Pi i / 5) and exp(8 Pi i / 5). Call the seed pentagon an "even" pentagon, for reasons that become obvious.
The step-1 pentagons have orientation opposite to that of the seed pentagon, so have neighbors whose centers are displaced by exp(Pi i / 5), exp(3 Pi i / 5), -1, exp (7 Pi i / 5) and exp(9 Pi i / 5). So call these pentagons "odd".
This arrangement is clearly like a checkerboard in that even pentagons are adjacent to odd pentagons and vice versa.
We give an "address" to every pentagon in the embedding by starting from the seed pentagon and giving the coefficient of Pi i / 5 in the argument of each displacement exponential. These coefficients range from 0 to 9. They alternate in parity, starting with an even coefficient. To illustrate, the seed pentagon has address [], an empty string of coefficients. Its neighbors are [0], [2], [4], [6], and [8]. [0]'s neighbors are [01], [03], [05], [07], and [09] -- of these, all but [05] are completed in step 2, as [05] is another name for the seed pentagon. In general, an occurrence of (n) cancels with an occurrence of (n+5) anywhere in the string, since the corresponding exponentials negate each other.
There are no other "coincidences" at step 2. So there are 20 new pentagons:
[01], [03], [07], [09], [21], [23], [25], [29], [41], [43], [45], [47], [63], [65], [67], [69], [81], [85], [87], [89]
-- after deleting the five "revisits" to the seed pentagon: [05], [27], [49], [61], and [83].
At step 3, the addresses beginning [01] are [010], [012], [014], [018] (with (1) and (6) canceling from [016] to give [0]). But every pentagon that is addressed as [abc] can be seen to be addressed as [cba] just by commutativity of addition. So we may allocate to [01] a half share of each of [012], [014] and [018], plus all of [010], counting 1 + 3/2 = 2.5 pentagons. Each of the 20 step-2 addresses generates step-3 addresses in a similar way, so giving 20 * 2.5 = 50 new pentagons.

Bisections (conjectured): A008383 (even), A063490 (odd, divided by 5).

Partial sums: A175898 (apparently).

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A103881 Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n.

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A008384 Crystal ball sequence for A_4 lattice.

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A358632 Coordination sequence for the faces of the uniform infinite surface that is formed from congruent regular pentagons and from which there is a continuous function that maps the faces 1:1 to regular pentagons in the plane.

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