A379378
The n-th term of the n-th forward differences of partition numbers A000041.
Original entry on oeis.org
1, 1, 1, 2, 5, 10, 22, 64, 159, 328, 747, 1914, 4608, 10252, 23339, 55405, 128034, 287855, 660549, 1541383, 3528645, 7921187, 17870633, 40689873, 92248847, 207911243, 469331387, 1059603243, 2377923972, 5313383490, 11889346697, 26641635997, 59560543885
Offset: 0
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b:= proc(n, k) option remember; `if`(k=0,
combinat[numbpart](n), b(n+1, k-1)-b(n, k-1))
end:
a:= n-> b(n$2):
seq(a(n), n=0..32);
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A379378list[nmax_] := Module[{p = PartitionsP[Range[0, nmax*2]]}, Join[{First[p]}, Table[First[p = Differences[Rest[p]]], nmax]]];
A379378list[50] (* Paolo Xausa, Jul 26 2025 *)
A338152
a(n) is the number of acyclic orientations of the edges of an n-dimensional demihypercube.
Original entry on oeis.org
1, 2, 24, 24024, 193270310, 767795414400
Offset: 1
-
Table[Abs[ChromaticPolynomial[GraphData[{"HalvedCube",n}]][-1]],{n,1,6}]
A116854
First differences of the rows in the triangle of A116853, starting with 0.
Original entry on oeis.org
1, 1, 1, 3, 1, 2, 11, 3, 4, 6, 53, 11, 14, 18, 24, 309, 53, 64, 78, 96, 120, 2119, 309, 362, 426, 504, 600, 720, 16687, 2119, 2428, 2790, 3216, 3720, 4320, 5040, 148329, 16687, 18806, 21234, 24024, 27240, 30960, 35280, 40320, 1468457, 148329, 165016, 183822, 205056, 229080, 256320, 287280, 322560, 362880
Offset: 1
First few rows of the triangle are:
[1] 1;
[2] 1, 1;
[3] 3, 1, 2;
[4] 11, 3, 4, 6;
[5] 53, 11, 14, 18, 24;
[6] 309, 53, 64, 78, 96, 120;
[7] 2119, 309, 362, 426, 504, 600, 720;
...
For example, row 4 (11, 3, 4, 6) are first differences along row 4 of A116853: ((0), 11, 14, 18, 24).
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a116854 n k = a116854_tabl !! (n-1) !! (k-1)
a116854_row n = a116854_tabl !! (n-1)
a116854_tabl = [1] : zipWith (:) (tail $ map head tss) tss
where tss = a116853_tabl
-- Reinhard Zumkeller, Aug 31 2014
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A116853 := proc(n,k) option remember ; if n = k then n! ; else procname(n,k+1)-procname(n-1,k) ; end if; end proc:
A116854 := proc(n,k) if k = 1 then A116853(n,1) ; else A116853(n,k) -A116853(n,k-1) ; end if; end proc:
seq(seq(A116854(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Mar 27 2010
-
rows = 10;
rr = Range[rows]!;
dd = Table[Differences[rr, n], {n, 0, rows - 1}];
T = Array[t, {rows, rows}];
Do[Thread[Evaluate[Diagonal[T, -k+1]] = dd[[k, ;; rows-k+1]]], {k, rows}];
Table[({0}~Join~Table[t[n, k], {k, 1, n}]) // Differences, {n, 1, rows}] // Flatten (* Jean-François Alcover, Dec 21 2019 *)
Definition made concrete and sequence extended by
R. J. Mathar, Mar 27 2010
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