A083374 -id:A083374 - cp's OEIS frontend

A279437 Number of ways to place 3 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

0, 4, 78, 528, 2200, 6900, 17934, 40768, 83808, 159300, 284350, 482064, 782808, 1225588, 1859550, 2745600, 3958144, 5586948, 7739118, 10541200, 14141400, 18711924, 24451438, 31587648, 40380000, 51122500, 64146654, 79824528, 98571928, 120851700, 147177150, 178115584

Offset: 1

Heinrich Ludwig, Dec 12 2016

Column 4 of triangle A279445.
Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279447.
For condition "no more than 2 points on straight lines at any angle", see A045996.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A045996, A249447, A279444, A279445, A197458.

Same problem but 2, 4..9 points: A083374, A279438, A279439, A279440, A279441, A279442, A279443.

Table[(n^6 - 5 n^4 + 6 n^3 - 2 n^2)/6, {n, 32}] (* or *)
Rest@ CoefficientList[Series[2 x^2*(2 + 25 x + 33 x^2 + x^3 - x^4)/(1 - x)^7, {x, 0, 32}], x] (* Michael De Vlieger, Dec 12 2016 *)

concat(0, Vec(2*x^2*(2 + 25*x + 33*x^2 + x^3 - x^4) / (1 - x)^7 + O(x^50))) \\ Colin Barker, Dec 12 2016

a(n) = (n^6 - 5*n^4 + 6*n^3 - 2*n^2)/6.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 2*x^2*(2 + 25*x + 33*x^2 + x^3 - x^4) / (1 - x)^7. - Colin Barker, Dec 12 2016

A279438 Number of ways to place 4 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 1, 90, 1428, 10600, 51525, 190806, 584080, 1552608, 3701025, 8088850, 16470036, 31616520, 57743413, 101055150, 170433600, 278290816, 441610785, 683206218, 1033218100, 1530887400, 2226630021, 3184447750, 4484709648, 6227340000, 8535450625, 11559457026, 15481719540

Offset: 1

Heinrich Ludwig, Dec 12 2016

Column 5 of triangle A279445.
Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279448.
For condition "no more than 2 points on straight lines at any angle", see A175383.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A175383, A249448, A279444, A279445, A197458.

Same problem but 2,3,5..9 points: A083374, A279437, A279439, A279440, A279441, A279442, A279443.

Table[(n^8 - 14 n^6 + 30 n^5 - 17 n^4 - 6 n^3 + 6 n^2)/24, {n, 28}] (* Michael De Vlieger, Dec 12 2016 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,1,90,1428,10600,51525,190806,584080,1552608},30] (* Harvey P. Dale, Sep 05 2024 *)

concat(0, Vec(x^2*(1 + 81*x + 654*x^2 + 904*x^3 + 99*x^4 - 57*x^5 - 2*x^6) / (1 - x)^9 + O(x^30))) \\ Colin Barker, Dec 13 2016

a(n) = (n^6 - 14*n^4 + 30*n^3 - 17*n^2 - 6*n + 6)*n^2/24 \\ Charles R Greathouse IV, Dec 13 2016

a(n) = (n^8 - 14*n^6 + 30*n^5 - 17*n^4 - 6*n^3 + 6*n^2)/24.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).
G.f.: x^2*(1 + 81*x + 654*x^2 + 904*x^3 + 99*x^4 - 57*x^5 - 2*x^6) / (1 - x)^9. - Colin Barker, Dec 13 2016

A279439 Number of ways to place 5 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 45, 2304, 34020, 270720, 1475145, 6209280, 21654864, 65422080, 176467005, 434206080, 990140580, 2117816064, 4288771305, 8284308480, 15355471680, 27446584320, 47501098029, 79872376320, 130866406020, 209448328320, 328150139625, 504222960384, 761083938000

Offset: 1

Heinrich Ludwig, Dec 21 2016

Column 6 of triangle A279445.
Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279449.
For condition "no more than 2 points on straight lines at any angle", see A194190.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A194190, A279449, A279444, A279445, A197458.

Same problem but 2,3,4,6..9 points: A083374, A279437, A279438, A279440, A279441, A279442, A279443.

Table[(n^10 - 30 n^8 + 90 n^7 - 27 n^6 - 270 n^5 + 500 n^4 - 360 n^3 + 96 n^2)/120, {n, 25}] (* or *)
Rest@ CoefficientList[Series[9 x^3*(5 + 201 x + 1239 x^2 + 1755 x^3 + 335 x^4 - 165 x^5 - 11 x^6 + x^7)/(1 - x)^11, {x, 0, 25}], x] (* Michael De Vlieger, Dec 22 2016 *)

concat(vector(2), Vec(9*x^3*(5 +201*x +1239*x^2 +1755*x^3 +335*x^4 -165*x^5 -11*x^6 +x^7) / (1 -x)^11 + O(x^30))) \\ Colin Barker, Dec 22 2016

a(n) = (n^10 -30*n^8 +90*n^7 -27*n^6 -270*n^5 +500*n^4 -360*n^3 +96*n^2)/120.
a(n) = 11*a(n-1) -55*a(n-2) +165*a(n-3) -330*a(n-4) +462*a(n-5) -462*a(n-6) +330*a(n-7) -165*a(n-8) +55*a(n-9) -11*a(n-10) +*a(n-11).
G.f.: 9*x^3*(5 +201*x +1239*x^2 +1755*x^3 +335*x^4 -165*x^5 -11*x^6 +x^7) / (1 -x)^11. - Colin Barker, Dec 22 2016

A279440 Number of ways to place 6 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 6, 2040, 71400, 1005720, 8421630, 50092896, 233483040, 905925600, 3045791430, 9125544120, 24868110696, 62593429080, 147255640350, 326843422080, 689604309120, 1391614736256, 2699616160710, 5055848825400, 9173923662120, 16177675640280, 27798546316926, 46651469520480

Offset: 1

Heinrich Ludwig, Dec 22 2016

Column 7 of triangle A279445.
Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279450.
For condition "no more than 2 points on straight lines at any angle", see A194191.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A194191, A279450, A279444, A279445, A197458.

Same problem but 2..5,7,8,9 points: A083374, A279437, A279438, A279439, A279441, A279442, A279443.

Table[(n^12 - 55 n^10 + 210 n^9 + 93 n^8 - 2220 n^7 + 5855 n^6 - 7350 n^5 + 4786 n^4 - 1440 n^3 + 120 n^2)/720, {n, 24}] (* or *)
Rest@ CoefficientList[Series[6 x^3 (1 + 327 x + 7558 x^2 + 39154 x^3 + 56220 x^4 + 14724 x^5 - 6262 x^6 - 978 x^7 + 131 x^8 + 5 x^9)/(1 - x)^13, {x, 0, 24}], x] (* Michael De Vlieger, Dec 22 2016 *)

concat(vector(2), Vec(6*x^3*(1 +327*x +7558*x^2 +39154*x^3 +56220*x^4 +14724*x^5 -6262*x^6 -978*x^7 +131*x^8 +5*x^9) / (1 -x)^13 + O(x^30))) \\ Colin Barker, Dec 22 2016

a(n) = (n^12 -55*n^10 +210*n^9 +93*n^8 -2220*n^7 +5855*n^6 -7350*n^5 +4786*n^4 -1440*n^3 +120*n^2)/720.
a(n) = 13*a(n-1) -78*a(n-2) +286*a(n-3) -715*a(n-4) +1287*a(n-5) -1716*a(n-6) +1716*a(n-7) -1287*a(n-8) +715*a(n-9) -286*a(n-10) +78*a(n-11) -13*a(n-12) +*a(n-13).
G.f.: 6*x^3*(1 +327*x +7558*x^2 +39154*x^3 +56220*x^4 +14724*x^5 -6262*x^6 -978*x^7 +131*x^8 +5*x^9) / (1 -x)^13. - Colin Barker, Dec 22 2016

A279441 Number of ways to place 7 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 0, 816, 93000, 2602800, 35526120, 309328320, 1972234656, 9989784000, 42369069600, 155993500080, 511660972680, 1524225598896, 4185197289000, 10715254368000, 25817751281280, 58981960615680, 128554066935936, 268691201838000, 540886175310600, 1052558059827120

Offset: 1

Heinrich Ludwig, Dec 22 2016

Column 8 of triangle A279445.
Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279451.
For condition "no more than 2 points on straight lines at any angle", see A194192.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Cf. A194192, A279451, A279444, A279445, A197458.

Same problem but 2..6,8,9 points: A083374, A279437, A279438, A279439, A279440, A279442, A279443.

Table[(n^14 - 91 n^12 + 420 n^11 + 693 n^10 - 10500 n^9 + 33647 n^8 - 45780 n^7 + 5866 n^6 + 65940 n^5 - 89796 n^4 + 50400 n^3 - 10800 n^2)/5040, {n, 23}] (* or *)
Rest@ CoefficientList[Series[24 x^4*(34 + 3365 x + 53895 x^2 + 244910 x^3 + 355390 x^4 + 115542 x^5 - 42490 x^6 - 11570 x^7 + 1500 x^8 + 145 x^9 - x^10)/(1 - x)^15, {x, 0, 23}], x] (* Michael De Vlieger, Dec 22 2016 *)

concat(vector(3), Vec(24*x^4*(34 +3365*x +53895*x^2 +244910*x^3 +355390*x^4 +115542*x^5 -42490*x^6 -11570*x^7 +1500*x^8 +145*x^9 -x^10) / (1 -x)^15 + O(x^30))) \\ Colin Barker, Dec 22 2016

a(n) = (n^12 -91*n^10 +420*n^9 +693*n^8 -10500*n^7 +33647*n^6 -45780*n^5 +5866*n^4 +65940*n^3 -89796*n^2 +50400*n -10800)*n^2/5040 \\ Charles R Greathouse IV, Dec 22 2016

a(n) = (n^14 -91*n^12 +420*n^11 +693*n^10 -10500*n^9 +33647*n^8 -45780*n^7 +5866*n^6 +65940*n^5 -89796*n^4 +50400*n^3 -10800*n^2)/5040.
a(n) = 15*a(n-1) -105*a(n-2) +455*a(n-3) -1365*a(n-4) +3003*a(n-5) -5005*a(n-6) +6435*a(n-7) -6435*a(n-8) +5005*a(n-9) -3003*a(n-10) +1365*a(n-11) -455*a(n-12) +105*a(n-13) -15*a(n-14) +a(n-15).
G.f.: 24*x^4*(34 +3365*x +53895*x^2 +244910*x^3 +355390*x^4 +115542*x^5 -42490*x^6 -11570*x^7 +1500*x^8 +145*x^9 -x^10) / (1 -x)^15. - Colin Barker, Dec 22 2016

A279442 Number of ways to place 8 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 0, 90, 67950, 4531950, 109425330, 1460297160, 13112872920, 88456195800, 480149029800, 2196080372970, 8743233946590, 31033043111070, 99992483914050, 296626638016800, 819218054279520, 2125440234303840, 5218743585428640, 12201529135725450, 27304286810701950

Offset: 1

Heinrich Ludwig, Dec 22 2016

Column 9 of triangle A279445.

Rotations and reflections of placements are counted.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Cf. A279444, A279445, A197458.

Same problem but 2..7,9 points: A083374, A279437, A279438, A279439, A279440, A279441, A279443.

Table[n^2*(n - 1)^2*(n - 2)^2*(n - 3)^2*(n^8 + 12 n^7 - 54 n^6 - 444 n^5 + 1845 n^4 + 1392 n^3 - 11332 n^2 + 9660 n + 1260)/8!, {n, 21}] (* Michael De Vlieger, Dec 22 2016 *)

concat(vector(3), Vec(90*x^4*(1 +738*x +37656*x^2 +461802*x^3 +1893555*x^4 +2767824*x^5 +1058064*x^6 -331920*x^7 -140913*x^8 +15950*x^9 +3096*x^10 -90*x^11 -3*x^12) / (1 -x)^17 + O(x^30))) \\ Colin Barker, Dec 23 2016

a(n) = (n^16 -140*n^14 +756*n^13 +2506*n^12 -36540*n^11 +130940*n^10 -117432*n^9 -559615*n^8 +2186100*n^7 -3622360*n^6 +3228876*n^5 -1439892*n^4 +181440*n^3 +45360*n^2)/40320; factorized: a(n) = n^2*(n-1)^2*(n-2)^2*(n-3)^2*(n^8 +12*n^7 -54*n^6 -444*n^5 +1845*n^4 +1392*n^3 -11332*n^2 +9660*n +1260)/8!.
a(n) = SUM(1<=j<=17, C(17,j)*(-1)^(j-1)*a(n-j)).
G.f.: 90*x^4*(1 +738*x +37656*x^2 +461802*x^3 +1893555*x^4 +2767824*x^5 +1058064*x^6 -331920*x^7 -140913*x^8 +15950*x^9 +3096*x^10 -90*x^11 -3*x^12) / (1 -x)^17. - Colin Barker, Dec 23 2016

A279443 Number of ways to place 9 points on an n X n board so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 0, 0, 22650, 4987800, 240023070, 5219088000, 68483325960, 630486309600, 4456523194200, 25647802519680, 125166919041450, 533442526857240, 2029603476250350, 7011735609715200, 22291042191643680, 65914292362262400, 182880685655641440, 479548000781222400

Offset: 1

Heinrich Ludwig, Dec 24 2016

Column 9 of triangle A279445.

Rotations and reflections of placements are counted.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (19,-171,969,-3876,11628, -27132,50388,-75582,92378,-92378,75582,-50388,27132,-11628,3876, -969,171,-19,1).

Cf. A279444, A279445, A197458.

Same problem but 2..8 points: A083374, A279437, A279438, A279439, A279440, A279441, A279442.

concat(vector(4), Vec(30*x^5*(755 +151915*x +4970934*x^2 +49653854*x^3 +187307071*x^4 +275138271*x^5 +119386656*x^6 -31251744*x^7 -19595619*x^8 +1706821*x^9 +667466*x^10 -26334*x^11 -2543*x^12 +17*x^13) / (1 -x)^19 + O(x^30))) \\ Colin Barker, Dec 24 2016

a(n) = (n^18 -204*n^16 +1260*n^15 +6846*n^14 -104076*n^13 +394504*n^12 +128520*n^11 -6237075*n^10 +24018372*n^9 -43820196*n^8 +30400020*n^7 +34251148*n^6 -99199296*n^5 +98504496*n^4 -47779200*n^3 +9434880*n^2)/362880; factorized: a(n) = n^2*(n-1)^2*(n-2)^2*(n-3)^2*(n-4)^2*(n^8 +20*n^7 +26*n^6 -820*n^5 -247*n^4 +9704*n^3 -9104*n^2 -14700*n +16380)/9!.
a(n) = SUM(1<=j<=19, C(19,j)*(-1)^(j-1)*a(n-j)).
G.f.: 30*x^5*(755 +151915*x +4970934*x^2 +49653854*x^3 +187307071*x^4 +275138271*x^5 +119386656*x^6 -31251744*x^7 -19595619*x^8 +1706821*x^9 +667466*x^10 -26334*x^11 -2543*x^12 +17*x^13) / (1 -x)^19. - Colin Barker, Dec 24 2016

A194193 Square array read by antidiagonals downwards: T(n,k) = number of ways to arrange k indistinguishable points on an n X n square grid so that no three points are collinear at any angle.

Original entry on oeis.org

1, 0, 4, 0, 6, 9, 0, 4, 36, 16, 0, 1, 76, 120, 25, 0, 0, 78, 516, 300, 36, 0, 0, 28, 1278, 2148, 630, 49, 0, 0, 2, 1668, 9498, 6768, 1176, 64, 0, 0, 0, 998, 25052, 47331, 17600, 2016, 81, 0, 0, 0, 204, 36698, 215448, 175952, 40120, 3240, 100, 0, 0, 0, 11, 26700, 620210

Offset: 1

R. H. Hardin, Aug 18 2011

Columns 4..7 are A175383, A194190, A194191, A194192 respectively. - Heinrich Ludwig, Nov 16 2016

			Table starts:
...1.....0.......0........0..........0...........0............0............0
...4.....6.......4........1..........0...........0............0............0
...9....36......76.......78.........28...........2............0............0
..16...120.....516.....1278.......1668.........998..........204...........11
..25...300....2148.....9498......25052.......36698........26700.........8242
..36...630....6768....47331.....215448......620210......1073076......1035097
..49..1176...17600...175952....1189868.....5367308.....15657764.....28228158
..64..2016...40120...545764....5199888....34678364....159413700....491910848
..81..3240...82608..1461672...18520572...169259212...1108580092...5122725512
.100..4950..157252..3507553...56978440...682686652...6030207624..38914424892
.121..7260..280988..7701638..155627304..2356999994..26852315940.229093733030
.144.10296..477012.15773526..388897892..7294368210.104865006648
.169.14196..775172.30375194..894254904.20227526910
.196.19110.1214768.55695587.1932504496
.225.25200.1844512.97777392
.256.32640.2725000
...
Some solutions for n=4, k=4:
..0..0..1..0....0..0..0..0....0..0..0..0....0..0..1..0....1..0..0..0
..1..0..0..0....1..0..0..0....0..0..1..0....1..0..0..0....0..0..0..1
..0..0..0..0....0..1..0..1....1..0..1..0....1..0..0..0....0..0..0..1
..0..0..1..1....0..1..0..0....0..1..0..0....0..0..0..1....1..0..0..0

R. H. Hardin and Heinrich Ludwig, Table of n, a(n) for n = 1..199, (first 181 terms from R. H. Hardin)

Column 1 is A000290.
Column 2 is A083374.
Column 3 is A045996.
Column 4 is A175383.
Column 5 is A194190.
Column 6 is A194191.
Column 7 is A194192.

A212084 Triangle T(n,k), n>=0, 0<=k<=2n, read by rows: row n gives the coefficients of the chromatic polynomial of the complete bipartite graph K_(n,n), highest powers first.

Original entry on oeis.org

1, 1, -1, 0, 1, -4, 6, -3, 0, 1, -9, 36, -75, 78, -31, 0, 1, -16, 120, -524, 1400, -2236, 1930, -675, 0, 1, -25, 300, -2200, 10650, -34730, 75170, -102545, 78610, -25231, 0, 1, -36, 630, -6915, 52080, -279142, 1074822, -2942445, 5552680, -6796926, 4787174

Offset: 0

Alois P. Heinz, Apr 30 2012

The complete bipartite graph K_(n,n) has 2n vertices and n^2 = A000290(n) edges. The chromatic polynomial of K_(n,n) has 2n+1 = A005408(n) coefficients.

			3 example graphs:                     +-----------+
.                 o        o   o      o   o   o   |
.                 |        |\ /|      |\ /|\ /|\ /
.                 |        | X |      | X | X | X
.                 |        |/ \|      |/ \|/ \|/ \
.                 o        o   o      o   o   o   |
.                                     +-----------+
Graph:         K_(1,1)    K_(2,2)      K_(3,3)
Vertices:         2          4            6
Edges:            1          4            9
The complete bipartite graph K_(2,2) is the cycle graph C_4 with chromatic polynomial q^4 -4*q^3 +6*q^2 -3*q => row 2 = [1, -4, 6, -3, 0].
Triangle T(n,k) begins:
  1;
  1,  -1,   0;
  1,  -4,   6,    -3,     0;
  1,  -9,  36,   -75,    78,     -31,       0;
  1, -16, 120,  -524,  1400,   -2236,    1930,     -675, ...
  1, -25, 300, -2200, 10650,  -34730,   75170,  -102545, ...
  1, -36, 630, -6915, 52080, -279142, 1074822, -2942445, ...
  ...

Alois P. Heinz, Rows n = 0..90, flattened
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Wikipedia, Chromatic Polynomial

Columns k=0-2 give: A000012, (-1)*A000290, A083374.
Row sums and last elements of rows give: A000007.
Row lengths give: A005408.
Sums of absolute values of row elements give: A048163(n+1).
T(n,2n-1) = (-1)*A092552(n).
Cf. A008277, A212085, A182368, A185442, A193233, A193277, A193283, A266695, A266972.

P:= n-> add(Stirling2(n, k) *mul(q-i, i=0..k-1) *(q-k)^n, k=0..n):
T:= n-> seq(coeff(P(n), q, 2*n-k), k=0..2*n):
seq(T(n), n=1..8);

T(n,k) = [q^(2n-k)] Sum_{j=0..n} (q-j)^n * S2(n,j) * Product_{i=0..j-1} (q-i).

T(0,0)=1 prepended by Alois P. Heinz, May 03 2024

A052296 Triangle read by rows: T(n,k) = number of labeled digraphs with n nodes and k arcs and without directed paths of length >=2, with 0 <= k <= floor(n^2/4).

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 6, 1, 12, 36, 32, 6, 1, 20, 120, 280, 280, 120, 20, 1, 30, 300, 1320, 2910, 3492, 2400, 960, 210, 20, 1, 42, 630, 4480, 17220, 39144, 56294, 53760, 35070, 15680, 4662, 840, 70, 1, 56, 1176, 12320, 73220, 269136, 654304, 1108928, 1362900

Offset: 0

Vladeta Jovovic, Feb 08 2000

			  1;
  1;
  1,  2;
  1,  6,   6;
  1, 12,  36,   32,    6;
  1, 20, 120,  280,  280,  120,   20;
  1, 30, 300, 1320, 2910, 3492, 2400, 960, 210, 20;
  ...

Row sums give A001831.

Cf. A002378 (k=1), A083374 (k=2).

A052296 := proc(n,k)
    local x,l ;
    add(binomial(n,l)*((1+x)^l-1)^(n-l),l=0..n) ;
    expand(%) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Mar 16 2021

G.f. for n-th row: Sum_{k=0..n} binomial(n, k)*((1+x)^k-1)^(n-k). - Vladeta Jovovic, Apr 04 2003
E.g.f.: Sum_{n>=0} exp(y*((1+x)^n-1))*y^n/n!. - Vladeta Jovovic, May 28 2004
T(n,3) = n*(n-1)*(n-2)*(n-3)*(n^2-3*n+4)/6, n>=4. - R. J. Mathar, Mar 16 2021

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