Manuel Kauers has authored 3508 sequences. Here are the ten most recent ones:
A181750
The number of paths of a chess rook in a 5D hypercube, from (0..0) to (n..n), where the rook may move in steps that are multiples of (1,0..0), (0,1,0..0), ..., (0..0,1).
Original entry on oeis.org
1, 120, 291720, 1085674320, 4927561419120, 25071989721176760, 137401053406474591320, 793279085081986319145120, 4760210822189950253433759120, 29426738284267047709626231969120, 186257720453050086737999575854359760, 1201788369927033696254110199515917069120
Offset: 0
a(1) = 120 because there are 120 rook paths from (0..0) to (1..1).
A181751
The number of paths of a chess rook in a 6D hypercube, from (0..0) to (n..n), where the rook may move in steps that are multiples of (1,0..0), (0,1,0..0), ..., (0..0,1).
Original entry on oeis.org
1, 720, 19445040, 906140159280, 54259623434853360, 3751591832963936408880, 284751719071722748492598160, 23074162542675887567516737575120, 1962510523766609850302634846653001840, 173256558851090756974435967247639154062480
Offset: 0
a(1) = 720 because there are 720 rook paths from (0..0) to (1..1).
A181752
The number of paths of a chess rook in a 7D hypercube, from (0..0) to (n..n), where the rook may move in steps that are multiples of (1,0..0), (0,1,0..0), ..., (0..0,1).
Original entry on oeis.org
1, 5040, 1781750880, 1224777388630320, 1139907455908263095760, 1261774625300967410997361200, 1562711749309729990152410055981840, 2093930745462011336917616846237237197680, 2973970378870521024883907518445574302750687520
Offset: 0
a(1) = 5040 because there are 5040 rook paths from (0..0) to (1..1).
A181749
The number of paths of a chess rook in a 4D hypercube, from (0..0) to (n..n), where the rook may move in steps that are multiples of (1,0..0), (0,1,0..0), ..., (0..0,1).
Original entry on oeis.org
1, 24, 6384, 2306904, 964948464, 439331916888, 211383647188320, 105734905550405400, 54434276297806242480, 28652982232251791825880, 15350736081585866511795024, 8343014042738696079671066904, 4588687856038215036178166258304
Offset: 0
a(1) = 24 because there are 24 rook paths from (0..0) to (1..1).
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a:= proc(n) option remember; `if`(n<4, [1, 24, 6384, 2306904][n+1],
((44148546*n^7-417566955*n^6+1582366209*n^5-3082719955*n^4
+3301523581*n^3-1923587242*n^2+559133416*n-61892160)*(n-1)^2*
a(n-1) -2*(n-2)*(131501097*n^8-1572004161*n^7+7935973542*n^6
-21971456652*n^5+36200366619*n^4-35926876063*n^3+20608609302*n^2
-6086148644*n+688049040)*a(n-2) +(393838614*n^7-4640973051*n^6
+22263043023*n^5-55659442951*n^4+77029268163*n^3
-57647348158*n^2+20864000120*n-2733950400)*(n-3)^2*a(n-3)
-5000*(34983*n^4-138138*n^3+184101*n^2-92498*n+14640)*(n-3)^2*
(n-4)^3*a(n-4))/ (2*n^3*(464360-1015046*n+808413*n^2
-278070*n^3+34983*n^4)*(n-1)^2))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Aug 31 2014
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b[l_List] := b[l] = If[Union[l]~Complement~{0} == {}, 1, Sum[Sum[b[Sort[ ReplacePart[l, i -> l[[i]] - j]]], {j, 1, l[[i]]}], {i, 1, Length[l]}]];
a[n_] := b[Array[n&, 4]];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz in A181731 *)
A181724
The number of paths of a chess rook in a 8D hypercube, from (0..0) to (n..n), where the rook may move in steps that are multiples of (1,0..0), (0,1,0..0), ..., (0..0,1).
Original entry on oeis.org
1, 40320, 214899027840, 2509137924026751360, 41795104403987233709518080, 852847938704373386478865686645120, 19846219244619878972245087341015659057280, 506348195597089273505079176351561351976609740160
Offset: 0
a(1) = 40320 because there are 40320 rook paths from (0..0) to (1..1).
A181725
The number of paths of a chess rook in a 9D hypercube, from (0..0) to (n..n), where the rook may move in steps that are multiples of (1,0..0), (0,1,0..0), ..., (0..0,1).
Original entry on oeis.org
1, 362880, 33007837322880, 7408611474125625953280, 2499611266020127048565292881280, 1064141699563485513180737844317706666240, 526577363627345975232160422620146408876598167680, 289514065258843883748159731480148589989905149052842682880
Offset: 0
a(1) = 362880 because there are 362880 rook paths from (0..0) to (1..1).
A181726
The number of paths of a chess rook in a 10D hypercube, from (0..0) to (n..n), where the rook may move in steps that are multiples of (1,0..0), (0,1,0..0), ..., (0..0,1).
Original entry on oeis.org
1, 3628800, 6290830003852800, 30306073546461323055916800, 231242270452155338291905203314956800, 2293197130058463838438742129627609575368940800, 26941822036577030394903099245279465611395585827577676800
Offset: 0
a(1) = 3628800 because there are 3628800 rook paths from (0..0) to (1..1).
A181727
The number of paths of a chess rook in a 11D hypercube, from (0..0) to (n..n), where the rook may move in steps that are multiples of (1,0..0), (0,1,0..0), ..., (0..0,1).
Original entry on oeis.org
1, 39916800, 1456812592995513600, 166369951631853645510591187200, 31707078596527364069316526441204831526400, 8089435115221815003427192379950659547969112311680000, 2492107900477900258313589438717998843635090670139189341868499200
Offset: 0
a(1) = 39916800 because there are 39916800 rook paths from (0..0) to (1..1).
A181728
The number of paths of a chess rook in a 12D hypercube, from (0..0) to (n..n), where the rook may move in steps that are multiples of (1,0..0), (0,1,0..0), ..., (0..0,1).
Original entry on oeis.org
1, 479001600, 402910665227270323200, 1193980357099103775859825737292800, 6221349234739584150822122029143772173312614400, 44698730304001991182769831137859339764690493418024756096000, 395245742455869432937361185087176756463979731526578123254618890928614400
Offset: 0
a(1) = 479001600 because there are 479001600 rook paths from (0..0) to (1..1).
A181731
Table A(d,n) of the number of paths of a chess rook in a d-dimensional hypercube from (0...0) to (n...n) where the rook may move in steps that are multiples of (1,0..0), (0,1,0..0), ..., (0..0,1).
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 1, 6, 14, 4, 1, 24, 222, 106, 8, 1, 120, 6384, 9918, 838, 16, 1, 720, 291720, 2306904, 486924, 6802, 32, 1, 5040, 19445040, 1085674320, 964948464, 25267236, 56190, 64, 1, 40320, 1781750880, 906140159280, 4927561419120, 439331916888, 1359631776, 470010, 128, 1, 362880, 214899027840, 1224777388630320, 54259623434853360
Offset: 1
A(3,1) = 6 because there are 6 rook paths on 3D chessboards from (0,0,0) to (1,1,1).
Square table A(d,n) begins:
1, 1, 2, 4, 8, ...
1, 2, 14, 106, 838, ...
1, 6, 222, 9918, 486924, ...
1, 24, 6384, 2306904, 964948464, ...
1, 120, 291720, 1085674320, 4927561419120, ...
Rows d=1-12 give:
A011782,
A051708 (from [1,1]),
A144045 (from [1,1,1]),
A181749,
A181750,
A181751,
A181752,
A181724,
A181725,
A181726,
A181727,
A181728.
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b:= proc(l) option remember; `if`({l[]} minus {0}={}, 1, add(add
(b(sort(subsop(i=l[i]-j, l))), j=1..l[i]), i=1..nops(l)))
end:
A:= (d, n)-> b([n$d]):
seq(seq(A(h-n, n), n=0..h-1), h=1..10); # Alois P. Heinz, Jul 21 2012
-
b[l_List] := b[l] = If[Union[l] ~Complement~ {0} == {}, 1, Sum[ Sum[ b[ Sort[ ReplacePart[l, i -> l[[i]] - j]]], {j, 1, l[[i]]}], {i, 1, Length[l]}]]; A[d_, n_] := b[Array[n&, d]]; Table[Table[A[h-n, n], {n, 0, h-1}], {h, 1, 10}] // Flatten (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)
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