A250853 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction

100, 400, 543, 1225, 2457, 2670, 3136, 8037, 13097, 12311, 7056, 21436, 44797, 63631, 54410, 14400, 49599, 123016, 223933, 291165, 233683, 27225, 103293, 290646, 626416, 1043885, 1280447, 983950, 48400, 198297, 614965, 1499679, 2955136

Offset

1,1

Comments

Table starts

......100.......400.......1225.......3136........7056.......14400.......27225

......543......2457.......8037......21436.......49599......103293......198297

.....2670.....13097......44797.....123016......290646......614965.....1195457

....12311.....63631.....223933.....626416.....1499679.....3204951.....6279401

....54410....291165....1043885....2955136.....7134786....15344785....30214465

...233683...1280447....4648157...13263136....32201019....69543783...137379337

...983950...5480917...20067117...57570016...140301126...303858745...601566177

..4085631..23024631...84805533..244213216...596722599..1294875471..2567402601

.16796370..95448605..353060845.1019415136..2495502666..5422612945.10763029505

.68555723.391939087.1454214877.4206874336.10311967539.22429374423.44552408777

Links

R. H. Hardin, Table of n, a(n) for n = 1..160

Formula

Empirical T(n,k) = (((31/36)*k^6+(25/2)*k^5+(1229/18)*k^4+(620/3)*k^3+(10759/36)*k^2+(1181/6)*k+48)*4^n -((5/3)*k^6+(133/6)*k^5+(320/3)*k^4+(1717/6)*k^3+(944/3)*k^2+(344/3)*k)*3^n +(k^6+12*k^5+47*k^4+103*k^3+54*k^2-13*k)*2^n -((1/9)*k^6+(3/2)*k^5+(25/9)*k^4+(13/6)*k^3-(89/9)*k^2+(4/3)*k))/12

Empirical for column k:

k=1: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (832*4^n-846*3^n+204*2^n+2)/12

k=2: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (4838*4^n-6300*3^n+2214*2^n-80)/12

k=3: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (18104*4^n-26144*3^n+10680*2^n-644)/12

k=4: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (52650*4^n-80640*3^n+35820*2^n-2688)/12

k=5: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (129528*4^n-206190*3^n+96660*2^n-8190)/12

k=6: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (282492*4^n-462196*3^n+224994*2^n-20568)/12

k=7: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (562288*4^n-939120*3^n+470064*2^n-45220)/12

Empirical for row n:

n=1: a(n) = (1/36)*n^6 + (1/2)*n^5 + (133/36)*n^4 + (43/3)*n^3 + (277/9)*n^2 + (104/3)*n + 16

n=2: a(n) = (2/9)*n^6 + (47/12)*n^5 + (953/36)*n^4 + (1141/12)*n^3 + (6527/36)*n^2 + 172*n + 64

n=3: a(n) = (3/2)*n^6 + (74/3)*n^5 + (621/4)*n^4 + (3161/6)*n^3 + (3691/4)*n^2 + 783*n + 256

n=4: a(n) = (76/9)*n^6 + (1595/12)*n^5 + (28765/36)*n^4 + (31373/12)*n^3 + (155683/36)*n^2 + (10223/3)*n + 1024

n=5: a(n) = (763/18)*n^6 + (1949/3)*n^5 + (136493/36)*n^4 + (72691/6)*n^3 + (693923/36)*n^2 + (43319/3)*n + 4096

n=6: a(n) = 198*n^6 + (35807/12)*n^5 + (204911/12)*n^4 + (214827/4)*n^3 + (998209/12)*n^2 + (180451/3)*n + 16384

n=7: a(n) = (15887/18)*n^6 + (39464/3)*n^5 + (2674189/36)*n^4 + (462227/2)*n^3 + (12645859/36)*n^2 + (743119/3)*n + 65536

Example

Some solutions for n=3 k=4
..2..2..0..0..0....1..2..3..2..2....2..2..1..0..0....3..2..1..1..1
..0..0..0..0..0....0..1..2..2..3....0..0..0..0..0....0..0..0..0..0
..1..1..1..1..1....0..1..2..2..3....1..1..1..1..3....0..0..1..1..2
..0..1..1..1..3....0..1..2..2..3....0..0..0..1..3....0..0..1..1..2

Crossrefs

Row 1 is A001249(n+1)

Sequence in context: A017174 A202334 A250806 * A017270 A105089 A334707

Adjacent sequences: A250850 A250851 A250852 * A250854 A250855 A250856

Keyword

nonn,tabl

Author

R. H. Hardin, Nov 28 2014

Status

approved