A117662
a(n) = n*(n-1)*(n-2)*(n+3)/12.
Original entry on oeis.org
0, 0, 0, 3, 14, 40, 90, 175, 308, 504, 780, 1155, 1650, 2288, 3094, 4095, 5320, 6800, 8568, 10659, 13110, 15960, 19250, 23023, 27324, 32200, 37700, 43875, 50778, 58464, 66990, 76415, 86800, 98208, 110704, 124355, 139230, 155400, 172938, 191919
Offset: 0
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Aram Bingham, Lisa Johnston, Colin Lawson, Rosa Orellana, Jianping Pan, and Chelsea Sato, The Chromatic Symmetric Function for Unicyclic Graphs, arXiv:2505.06486 [math.CO], 2025. See p. 12.
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
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[n*(n-1)*(n-2)*(n+3)/12: n in [0..50]]; // Vincenzo Librandi, Oct 10 2013
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seq(n*(n-1)*(n-2)*(n+3)/12, n=0..40); # Wesley Ivan Hurt, Oct 10 2013
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Table[n(n-1)(n-2)(n+3)/12, {n,0,100}] (* Wesley Ivan Hurt, Sep 26 2013 *)
CoefficientList[Series[x^3 (3 - x)/(1 - x)^5, {x, 0, 80}], x] (* Vincenzo Librandi, Oct 10 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,0,0,3,14},80] (* Harvey P. Dale, Jan 01 2025 *)
A179898
Triangle V(l,p) (l>=0, p=0..l) read by rows: see Formula for definition, see Comments for motivation.
Original entry on oeis.org
1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 14, 0, 10, 0, 1, 14, 0, 40, 0, 15, 0, 1, 0, 84, 0, 90, 0, 21, 0, 1, 84, 0, 300, 0, 175, 0, 28, 0, 1, 0, 594, 0, 825, 0, 308, 0, 36, 0, 1, 594, 0, 2475, 0, 1925, 0, 504, 0, 45, 0, 1, 0, 4719, 0, 7865, 0, 4004, 0, 780, 0, 55, 0, 1, 4719, 0, 22022, 0, 21021, 0, 7644, 0, 1155, 0, 66, 0, 1, 0, 40898, 0, 78078, 0, 49686, 0, 13650, 0, 1650, 0, 78, 0, 1, 40898, 0, 208208, 0, 231868, 0, 107016, 0, 23100, 0, 2288, 0, 91, 0, 1, 0, 379236, 0, 804440, 0, 606424, 0, 214200, 0, 37400, 0, 3094, 0, 105, 0, 1
Offset: 0
Triangle begins:
1;
0, 1;
1, 0, 1;
0, 3, 0, 1;
3, 0, 6, 0, 1;
0, 14, 0, 10, 0, 1;
14, 0, 40, 0, 15, 0, 1;
0, 84, 0, 90, 0, 21, 0, 1;
84, 0, 300, 0, 175, 0, 28, 0, 1;
0, 594, 0, 825, 0, 308, 0, 36, 0, 1;
594, 0, 2475, 0, 1925, 0, 504, 0, 45, 0, 1;
0, 4719, 0, 7865, 0, 4004, 0, 780, 0, 55, 0, 1;
4719, 0, 22022, 0, 21021, 0, 7644, 0, 1155, 0, 66, 0, 1;
0, 40898, 0, 78078, 0, 49686, 0, 13650, 0, 1650, 0, 78, 0, 1;
40898, 0, 208208, 0, 231868, 0, 107016, 0, 23100, 0, 2288, 0, 91, 0, 1;
...
- D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableau de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986 (see |V_{l,p}| on page 114).
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V:=proc(l,p)
if ((l-p) mod 2) = 1 then 0 else l!*(l+2)!*(p+3)! / (((l-p)/2)!*((l-p)/2+1)!*p!*((l+p)/2+2)!*((l+p)/2+3)!); fi;
end;
r:=n->[seq( V(n,p),p=0..n)];
for n from 0 to 15 do lprint(r(n)); od:
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v[l_, p_] := If[Mod[l-p, 2] == 1, 0, l!*(l+2)!*(p+3)!/(((l-p)/2)!*((l-p)/2+1)!*p!*((l+p)/2+2)!*((l+p)/2+3)!)]; Table[v[l, p], {l, 0, 15}, {p, 0, l}] // Flatten (* Jean-François Alcover, Jan 09 2014, translated from Maple *)
A050297
Number of scalars which can be constructed from the Riemann tensor and metric tensor in n dimensions.
Original entry on oeis.org
0, 1, 3, 14, 40, 90, 175, 308, 504, 780, 1155, 1650, 2288, 3094, 4095, 5320, 6800, 8568, 10659, 13110, 15960, 19250, 23023, 27324, 32200, 37700, 43875, 50778, 58464, 66990, 76415, 86800, 98208, 110704, 124355, 139230, 155400, 172938
Offset: 1
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[0,1] cat [n*(n-1)*(n-2)*(n+3)/12: n in [3..60]]; // Vincenzo Librandi, May 13 2017
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CoefficientList[Series[x^2*(x^5 - 5*x^4 + 10*x^3 - 9*x^2 + 2*x - 1)/(x - 1)^5, {x, 0, 50}], x] (* G. C. Greubel, May 12 2017 *)
Join[{0, 1}, Table[n (n - 1) (n - 2) (n + 3) / 12, {n, 3, 40}]] (* Vincenzo Librandi, May 13 2017 *)
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my(x='x+O('x^50)); concat([0], Vec(x^2*(x^5-5*x^4+10*x^3-9*x^2+2*x-1)/(x-1)^5)) \\ G. C. Greubel, May 12 2017
A181571
Third column of triangle in A179898.
Original entry on oeis.org
1, 6, 40, 300, 2475, 22022, 208208, 2068560, 21414900, 229523800, 2533942752, 28698821320, 332357673375, 3925129083750, 47167131780000, 575637606165600, 7123515376299300, 89266155250239000, 1131410294476020000, 14489559984061890000, 187330691180608155180, 2443121585638964379864
Offset: 1
Showing 1-4 of 4 results.
Comments