A008288 -id:A008288 - cp's OEIS frontend

A211312 Square array of Delannoy numbers D(i,j) mod 3 (i >= 0, j >= 0) read by antidiagonals.

1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 0, 2, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 2, 2, 0, 0, 2, 2, 2, 1, 1, 1, 1, 2, 2, 0, 2, 2, 1, 1, 1, 1, 0, 1, 2, 0, 2, 2, 0, 2, 1, 0, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1

Offset: 0

N. J. A. Sloane, Apr 15 2012

			Written as a triangle:
1,
1, 1,
1, 0, 1,
1, 2, 2, 1,
1, 1, 1, 1, 1,
1, 0, 1, 1, 0, 1,
1, 2, 2, 0, 2, 2, 1,
1, 1, 1, 0, 0, 1, 1, 1,
1, 0, 1, 0, 0, 0, 1, 0, 1,
...

Marko Razpet, A self-similarity structure generated by king's walk, Algebraic and topological methods in graph theory (Lake Bled, 1999). Discrete Math. 244 (2002), no. 1-3, 423--433. MR1844050 (2002k:05022)
Rémy Sigrist, Colored representation of the first 1000 rows

Cf. A008288, A211312-A211315.

A211312 := proc(n,k): add(binomial(k, j) * binomial(n-j, k), j=0..n-k) mod 3 end: seq(seq(A211312(n,k), k=0..n), n=0..12); # Johannes W. Meijer, Jul 19 2013

a[n_, k_] := Mod[Binomial[n, k]*Hypergeometric2F1[-k, k-n, -n, -1], 3]; Table[a[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 14 2014, after Johannes W. Meijer *)

a(n) = sum(binomial(k, j) * binomial(n-j, k), j=0..n-k) mod 3. - Johannes W. Meijer, Jul 19 2013

A211313 Square array of Delannoy numbers D(i,j) mod 5 (i >= 0, j >= 0) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 0, 0, 1, 1, 2, 3, 2, 1, 1, 4, 0, 0, 4, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 4, 4, 1, 3, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 4, 0, 0, 4, 3, 4, 0, 0, 4, 1, 1, 1, 1, 3, 1, 3, 3, 1, 3, 1, 1, 1, 1, 3, 1, 4, 4, 3, 4, 3, 4, 4, 1, 3, 1

Offset: 0

N. J. A. Sloane, Apr 15 2012

			Written as a triangle:
1,
1, 1,
1, 3, 1,
1, 0, 0, 1,
1, 2, 3, 2, 1,
1, 4, 0, 0, 4, 1,
1, 1, 1, 3, 1, 1, 1,
1, 3, 1, 4, 4, 1, 3, 1,
1, 0, 0, 1, 1, 1, 0, 0, 1,
...

Marko Razpet, A self-similarity structure generated by king's walk, Algebraic and topological methods in graph theory (Lake Bled, 1999). Discrete Math. 244 (2002), no. 1-3, 423--433. MR1844050 (2002k:05022).
Rémy Sigrist, Colored representation of the first 1000 rows

Cf. A008288, A211312, A211314, A211315.

A008417 Crystal ball sequence for 8-dimensional cubic lattice.

Original entry on oeis.org

1, 17, 145, 833, 3649, 13073, 40081, 108545, 265729, 598417, 1256465, 2485825, 4673345, 8405905, 14546705, 24331777, 39490049, 62390545, 96220561, 145198913, 214828609, 312193553, 446304145, 628496897, 872893441, 1196924561, 1621925137, 2173806145, 2883810113, 3789356689, 4934985233

Offset: 0

N. J. A. Sloane

This is row/column 8 of the Delannoy numbers array, A008288, which is the main entry for these numbers, listing many more properties. - Shel Kaphan, Jan 06 2023

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).
Index entries for crystal ball sequences
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A001849, A099196.
Partial sums of A008416.
Cf. A240876.
Row/Column 8 of A008288.

CoefficientList[Series[-(z + 1)^8/(z - 1)^9, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,17,145,833,3649,13073,40081,108545,265729},40] (* Harvey P. Dale, May 26 2024 *)

G.f.: (1+x)^8/(1-x)^9.
First differences of A099196. - Alexander Adamchuk, May 23 2006
a(n) = (2*n^8 + 8*n^7 + 84*n^6 + 224*n^5 + 798*n^4 + 1232*n^3 + 1636*n^2 + 1056*n + 315)/315. - Alexander Adamchuk, May 23 2006
Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = log(2) - (1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - 1/8). - Peter Bala, Mar 23 2024

More terms from Alexander Adamchuk, May 23 2006

A008421 Crystal ball sequence for 10-dimensional cubic lattice.

Original entry on oeis.org

1, 21, 221, 1561, 8361, 36365, 134245, 433905, 1256465, 3317445, 8097453, 18474633, 39753273, 81270333, 158819253, 298199265, 540279585, 948062325, 1616336765, 2684641785, 4354393801, 6911195501, 10753517061, 16429137361, 24680949041, 36503969061

Offset: 0

N. J. A. Sloane

This is row/column 10 of the Delannoy numbers array, A008288, which is the main entry for these numbers, listing many more properties. - Peter Munn, Jan 05 2023

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).
Index entries for crystal ball sequences
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A001847, A001848, A001849, A008288, A240876.

CoefficientList[Series[-(z + 1)^10/(z - 1)^11, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)

G.f.: (1+x)^10/(1-x)^11.

a(n) = (4*n^10+20*n^9+330*n^8+1200*n^7+7392*n^6+18060*n^5+50270*n^4 +71800*n^3+83754*n^2+50670*n+14175)/14175 - Johannes W. Meijer, Jul 14 2013

A027618 c(i,j) is cost of evaluation of edit distance of two strings with lengths i and j, when you use recursion (every call has a unit cost, other computations are free); sequence gives c(n,n).

Original entry on oeis.org

1, 4, 19, 94, 481, 2524, 13483, 72958, 398593, 2193844, 12146179, 67570078, 377393953, 2114900428, 11885772379, 66963572734, 378082854913, 2138752086628, 12118975586803, 68774144872414, 390815720696161, 2223564321341884

Offset: 0

Bruno Petazzoni (Bruno.Petazzoni(AT)ac-idf.jussieu.fr)

Found by 7 students: Dufour, Hermon, Lesueur, Moynot, Schabanel, Sers and Wolf.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Delannoy numbers A008288, A001850 are given by c'(i, j)=(3c(i, j)-1)/2.

Table[SeriesCoefficient[(3/Sqrt[1-6*x+x^2]-1/(1-x))/2,{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)

x='x+O('x^66); Vec((3/sqrt(1-6*x+x^2)-1/(1-x))/2) \\ Joerg Arndt, May 04 2013

c(n, n) where c(i, 0)=c(0, j)=1 and c(i+1, j+1)=1+c(i+1, j)+c(i, j+1)+c(i, j) (c(i, j) is A047671).
G.f.: (3/sqrt(1-6*x+x^2)-1/(1-x))/2.
Recurrence: n*(2*n-3)*a(n) = (2*n-1)*(7*n-10)*a(n-1) - (2*n-3)*(7*n-4)*a(n-2) + (n-2)*(2*n-1)*a(n-3). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ 3*sqrt(8+6*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012

A143410 Form the difference table of the sequence {2^kk!}, then divide k-th column entries by 2^kk!. Array read by ascending antidiagonals, T(n, k) for n, k >= 0.

Original entry on oeis.org

1, 1, 1, 5, 3, 1, 29, 17, 5, 1, 233, 131, 37, 7, 1, 2329, 1281, 353, 65, 9, 1, 27949, 15139, 4105, 743, 101, 11, 1, 391285, 209617, 56189, 10049, 1349, 145, 13, 1, 6260561, 3325923, 883885, 156679, 20841, 2219, 197, 15, 1, 112690097, 59475329, 15700313

Offset: 0

Peter Bala, Aug 19 2008

This table is closely connected to the constant sqrt(e). The row, column and diagonal entries of this table occur in series acceleration formulas for sqrt(e). For a similar table based on the Euler-Seidel matrix of the sequence {2^k*k!} and related to the constant 1/sqrt(e), see A143411. For other arrays similarly related to constants see A086764 (for e), A143409 (for 1/e), A008288 (for log(2)), A108625 (for zeta(2)) and A143007 (for zeta(3)).

			Table of differences of {2^k*k!}
  =====================================================
  Column                0     1     2     3     4     5
  =====================================================
  Sequence 2^k*k!       1     2     8    48   384  3840
  First differences     1     6    40   336  3456
  Second differences    5    34   296  3120
  Third differences    29   262  2824
  Fourth differences  233  2562
  ...
Remove the common factor 2^k*k! from k-th column entries:
  ====================================
  n\k|   0      1      2      3      4
  ====================================
  0  |   1      1      1      1      1
  1  |   1      3      5      7      9
  2  |   5     17     37     65    101
  3  |  29    131    353    743   1349
  4  | 233   1281   4105  10049  20841

Eric Weisstein's World of Mathematics, Poisson-Charlier polynomial.

Cf. A008288, A076571, A086764, A108625, A143007, A143409, A143411, A143412.

T := (n, k) -> (-1)^n/k!*add((-2)^j*binomial(n,j)*(k+j)!, j = 0..n):
for n from 0 to 9 do seq(T(n, k), k = 0..9) end do;

T(n,k) = ((-1)^n/k!)*Sum {j = 0..n} (-2)^j*C(n,j)*(k+j)!.
Relation with Poisson-Charlier polynomials c_n(x,a): T(n,k) = c_n(-(k+1),-1/2).
Recurrence relations: T(n,k) = 2*n*T(n-1,k) + T(n,k-1); T(n,k) = 2*(n+k)*T(n-1,k) - T(n-1,k-1); T(n,k) = 2*(k+1)*T(n-1,k+1) - T(n-1,k);
Recurrence for row n entries: 2*k*T(n,k) = (2*n+2*k+1)*T(n,k-1) - T(n,k-2).
E.g.f. for column k: exp(-y)/(1-2*y)^(k+1).
E.g.f. for array: exp(-y)/(1-x-2*y) = (1 + x + x^2 + ...) + (1 + 3*x + 5*x^2 + ...)*y + (5 + 17*x + 37*x^2 + ...)*y^2/2! + ... .
Series acceleration formulas for sqrt(e):
Row n: sqrt(e) = 2^n*n!*(1/T(n,0) + (-1)^n*(1/(2*1!*T(n,0)*T(n,1)) + 1/(2^2*2!*T(n,1)*T(n,2)) + 1/(2^3*3!*T(n,2)*T(n,3)) + ...)). For example, row 3 gives sqrt(e) = 48*(1/29 - 1/(2*29*131) - 1/(8*131*353) - 1/(48*353*743) - ...).
Column k: sqrt(e) = (1 + (1/2)/1! + (1/2)^2 / 2! + ... + (1/2)^k/k!) + 1/(2^k*k!) * Sum_{n>= 0} ((-2)^n *n!/(T(n,k)*T(n+1,k))). For example, column 3 gives sqrt(e) = 79/48 + (1/48)*(1/(1*7) - 2/(7*65) + 8/(65*743) - 48/(743*10049) + ...).
Main diagonal: sqrt(e) = 1 + 2*(1/(1*3) - 1/(3*37) + 1/(37*743) - ...). See A143412.
T(n, k) = (-1)^n*(-1/2)^(k + 1)*KummerU(k + 1, k + n + 2, -1/2). - Peter Luschny, Jan 02 2020

A211315 Square array of Delannoy numbers D(i,j) mod 11 (i >= 0, j >= 0) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 2, 7, 1, 1, 9, 3, 3, 9, 1, 1, 0, 8, 8, 8, 0, 1, 1, 2, 6, 8, 8, 6, 2, 1, 1, 4, 8, 0, 2, 0, 8, 4, 1, 1, 6, 3, 3, 10, 10, 3, 3, 6, 1, 1, 8, 2, 3, 2, 0, 2, 3, 2, 8, 1, 1, 10, 5, 8, 8, 1, 1, 8, 8, 5, 10, 1, 1, 1, 1, 4, 8, 0, 2, 0, 8, 4, 1, 1, 1

Offset: 0

N. J. A. Sloane, Apr 15 2012

			Written as a triangle:
1,
1, 1,
1, 3, 1,
1, 5, 5, 1,
1, 7, 2, 7, 1,
1, 9, 3, 3, 9, 1,
1, 0, 8, 8, 8, 0, 1,
1, 2, 6, 8, 8, 6, 2, 1,
1, 4, 8, 0, 2, 0, 8, 4, 1,
...

Marko Razpet, A self-similarity structure generated by king's walk, Algebraic and topological methods in graph theory (Lake Bled, 1999). Discrete Math. 244 (2002), no. 1-3, 423--433. MR1844050 (2002k:05022).

Cf. A008288, A211312, A211313, A211314.

A341476 Coefficients related to the asymptotics of generalized Delannoy numbers.

Original entry on oeis.org

1, 3, 22, 223, 2792, 42671, 761984, 15707707, 365122688, 9491746747, 271962399232, 8539383210711, 290937486190592, 10710312199270503, 422984587596455936, 17864076455770831219, 802450164859200372736, 38242916911507537149427, 1925477163696152909447168, 102213291475268656299164879

Offset: 1

Vaclav Kotesovec, Feb 13 2021

nonn

			Lim_{n->infinity} A001850(n)^(1/n) = (    3 +    2 * sqrt(1^2 + 1)) / 1^1.
Lim_{n->infinity} A026000(n)^(1/n) = (   22 +   10 * sqrt(2^2 + 1)) / 2^2.
Lim_{n->infinity} A026001(n)^(1/n) = (  223 +   70 * sqrt(3^2 + 1)) / 3^3.
Lim_{n->infinity} A331329(n)^(1/n) = ( 2792 +  680 * sqrt(4^2 + 1)) / 4^4.
Lim_{n->infinity} A341491(n)^(1/n) = (42671 + 8346 * sqrt(5^2 + 1)) / 5^5.

Vaclav Kotesovec, Table of n, a(n) for n = 1..100

Cf. A008288, A001850, A026000, A026001, A331329, A341491, A341477.

Lim_{n->infinity} (binomial(k*n, n) * hypergeom([(1-k)*n, -n], [-k*n], -1))^(1/n) = (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k-1)^(k-1), for k>1.
Lim_{n->infinity} hypergeom([(1-k)*n, -n], [-k*n], -1)^(1/n) = (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / k^k.
For k > 1, A341476(k)^2 - ((k-1)^2 + 1) * A341477(k)^2 = (-1)^k * (k-1)^(2*k-2).
Lim_{k->infinity} (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k * (k-1)^(k-1)) = 2*exp(1).
a(n) ~ n^n.

A341477 Coefficients related to the asymptotics of generalized Delannoy numbers.

Original entry on oeis.org

0, 2, 10, 70, 680, 8346, 125504, 2218350, 45335680, 1047314578, 27079557632, 772687787510, 24172386314240, 821114930966890, 30146801401143296, 1187943632192716894, 50068690149298438144, 2245175953053786221730, 106828553482726336102400, 5371204894269759411503910

Offset: 1

Vaclav Kotesovec, Feb 13 2021

nonn

			Lim_{n->infinity} A001850(n)^(1/n) = (    3 +    2 * sqrt(1^2 + 1)) / 1^1.
Lim_{n->infinity} A026000(n)^(1/n) = (   22 +   10 * sqrt(2^2 + 1)) / 2^2.
Lim_{n->infinity} A026001(n)^(1/n) = (  223 +   70 * sqrt(3^2 + 1)) / 3^3.
Lim_{n->infinity} A331329(n)^(1/n) = ( 2792 +  680 * sqrt(4^2 + 1)) / 4^4.
Lim_{n->infinity} A341491(n)^(1/n) = (42671 + 8346 * sqrt(5^2 + 1)) / 5^5.

Vaclav Kotesovec, Table of n, a(n) for n = 1..100

Cf. A008288, A001850, A026000, A026001, A331329, A341491, A341476.

Lim_{n->infinity} (binomial(k*n, n) * hypergeom([(1-k)*n, -n], [-k*n], -1))^(1/n) = (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k-1)^(k-1), for k>1.
Lim_{n->infinity} hypergeom([(1-k)*n, -n], [-k*n], -1)^(1/n) = (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / k^k.
For k > 1, A341476(k)^2 - ((k-1)^2 + 1) * A341477(k)^2 = (-1)^k * (k-1)^(2*k-2).
Lim_{k->infinity} (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k * (k-1)^(k-1)) = 2*exp(1).
a(n) ~ n^(n-1).

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.

Original entry on oeis.org

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).

cp's OEIS Frontend

A211312 Square array of Delannoy numbers D(i,j) mod 3 (i >= 0, j >= 0) read by antidiagonals.

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A211313 Square array of Delannoy numbers D(i,j) mod 5 (i >= 0, j >= 0) read by antidiagonals.

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A008417 Crystal ball sequence for 8-dimensional cubic lattice.

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A008421 Crystal ball sequence for 10-dimensional cubic lattice.

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A027618 c(i,j) is cost of evaluation of edit distance of two strings with lengths i and j, when you use recursion (every call has a unit cost, other computations are free); sequence gives c(n,n).

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A143410 Form the difference table of the sequence {2^k*k!}, then divide k-th column entries by 2^k*k!. Array read by ascending antidiagonals, T(n, k) for n, k >= 0.

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A211315 Square array of Delannoy numbers D(i,j) mod 11 (i >= 0, j >= 0) read by antidiagonals.

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A341476 Coefficients related to the asymptotics of generalized Delannoy numbers.

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A341477 Coefficients related to the asymptotics of generalized Delannoy numbers.

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A143410 Form the difference table of the sequence {2^kk!}, then divide k-th column entries by 2^kk!. Array read by ascending antidiagonals, T(n, k) for n, k >= 0.