Search: a005994 -id:a005994
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A282011
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Number T(n,k) of k-element subsets of [n] having an even sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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+10
20
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1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 4, 6, 3, 0, 1, 3, 6, 10, 9, 3, 0, 1, 3, 9, 19, 19, 9, 3, 1, 1, 4, 12, 28, 38, 28, 12, 4, 1, 1, 4, 16, 44, 66, 60, 40, 20, 5, 0, 1, 5, 20, 60, 110, 126, 100, 60, 25, 5, 0, 1, 5, 25, 85, 170, 226, 226, 170, 85, 25, 5, 1, 1, 6, 30, 110, 255, 396, 452, 396, 255, 110, 30, 6, 1
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OFFSET
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0,12
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COMMENTS
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Row n is symmetric if and only if n mod 4 in {0,3} (or if T(n,n) = 1).
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LINKS
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FORMULA
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T(n,k) = Sum_{j=0..floor((n+1)/4)} C(ceiling(n/2),2*j) * C(floor(n/2),k-2*j).
Sum_{k=0..n} k * T(n,k) = A057711(n-1) for n>0.
Sum_{k=0..n} (k+1) * T(n,k) = A087447(n) + [n=2].
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EXAMPLE
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T(5,0) = 1: {}.
T(5,1) = 2: {2}, {4}.
T(5,2) = 4: {1,3}, {1,5}, {2,4}, {3,5}.
T(5,3) = 6: {1,2,3}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,5}, {3,4,5}.
T(5,4) = 3: {1,2,3,4}, {1,2,4,5}, {2,3,4,5}.
T(5,5) = 0.
T(7,7) = 1: {1,2,3,4,5,6,7}.
Triangle T(n,k) begins:
1;
1, 0;
1, 1, 0;
1, 1, 1, 1;
1, 2, 2, 2, 1;
1, 2, 4, 6, 3, 0;
1, 3, 6, 10, 9, 3, 0;
1, 3, 9, 19, 19, 9, 3, 1;
1, 4, 12, 28, 38, 28, 12, 4, 1;
1, 4, 16, 44, 66, 60, 40, 20, 5, 0;
1, 5, 20, 60, 110, 126, 100, 60, 25, 5, 0;
1, 5, 25, 85, 170, 226, 226, 170, 85, 25, 5, 1;
1, 6, 30, 110, 255, 396, 452, 396, 255, 110, 30, 6, 1;
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MAPLE
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b:= proc(n, s) option remember; expand(
`if`(n=0, s, b(n-1, s)+x*b(n-1, irem(s+n, 2))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
seq(T(n), n=0..16);
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MATHEMATICA
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Flatten[Table[Sum[Binomial[Ceiling[n/2], 2j]Binomial[Floor[n/2], k-2j], {j, 0, Floor[(n+1)/4]}], {n, 0, 10}, {k, 0, n}]] (* Indranil Ghosh, Feb 26 2017 *)
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PROG
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(PARI) a(n, k)=sum(j=0, floor((n+1)/4), binomial(ceil(n/2), 2*j)*binomial(floor(n/2), k-2*j));
tabl(nn)={for(n=0, nn, for(k=0, n, print1(a(n, k), ", "); ); print(); ); } \\ Indranil Ghosh, Feb 26 2017
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CROSSREFS
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Columns k=0..10 give (offsets may differ): A000012, A004526, A002620, A005993, A005994, A032092, A032093, A018211, A018212, A282077, A282078.
Lower diagonals T(n+j,n) for j=1..10 give: A004525(n+1), A282079, A228705, A282080, A282081, A282082, A282083, A282084, A282085, A282086.
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KEYWORD
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AUTHOR
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STATUS
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approved
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1, 4, 7, 1, 2, 3, 3, 4, 6, 1, 1, 2, 2, 2, 3, 3, 3, 4, 6, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2
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OFFSET
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1,2
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COMMENTS
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all terms are of the form u*v mod 10, where u <= v and belonging to {1,3,4,6,7}, the distinct elements of A254397:
length of k-th run of consecutive 1s = A005993(k-2), k > 1;
length of k-th run of consecutive 2s = k*(k+1)/2 = A000217(k), k >= 1;
length of k-th run of consecutive 3s = k+1, k >= 1;
length of k-th run of consecutive 4s = A065033(k-1);
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LINKS
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PROG
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(Haskell)
a254338 = a000030 . a254143
(PARI) listA237424(lim)=my(v=List(), a, t); while(1, for(b=0, a, t=(10^a+10^b+1)/3; if(t>lim, return(Set(v))); listput(v, t)); a++)
do(lim)=my(v=List(), u=listA237424(lim), t); for(i=1, #u, for(j=1, i, t=u[i]*u[j]; if(t>lim, break); listput(v, t))); apply(n->digits(n)[1], Set(v)) \\ Charles R Greathouse IV, May 13 2015
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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A283113
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Triangle read by rows: T(n,k) is the number of nonequivalent ways (mod D_3) to place k points on an n X n X n triangular grid so that no two of them are on the same row, column or diagonal (n >= 1).
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+10
6
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1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 9, 5, 1, 5, 19, 23, 3, 1, 7, 38, 82, 40, 1, 1, 8, 66, 230, 242, 45, 1, 10, 110, 560, 1038, 533, 29, 1, 12, 170, 1208, 3504, 3546, 821, 6, 1, 14, 255, 2392, 9998, 16917, 9137, 807, 1, 16, 365, 4405, 25158, 64345, 63755, 17408, 422
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OFFSET
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1,6
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COMMENTS
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Length of n-th row is A004396(n) + 1, for 1 <= n <= 21, where A004396(n) is the maximal number of points that can be placed under the condition mentioned above.
Rotations and reflections of placements are not counted. If they are to be counted, see A193986.
In terms or triangular chess: Number of nonequivalent ways (mod D_3) to arrange k nonattacking rooks on an n X n X n board, k>=0, n>=1.
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LINKS
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EXAMPLE
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The table begins with T(1,0), T(1,1);
1, 1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 9, 5;
1, 5, 19, 23, 3;
1, 7, 38, 82, 40, 1;
1, 8, 66, 230, 242, 45;
1, 10, 110, 560, 1038, 533, 29;
...
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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A006010
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Number of paraffins (see Losanitsch reference for precise definition).
(Formerly M3897)
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+10
4
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1, 5, 20, 52, 117, 225, 400, 656, 1025, 1525, 2196, 3060, 4165, 5537, 7232, 9280, 11745, 14661, 18100, 22100, 26741, 32065, 38160, 45072, 52897, 61685, 71540, 82516, 94725, 108225, 123136, 139520, 157505, 177157, 198612, 221940, 247285, 274721, 304400
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OFFSET
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1,2
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COMMENTS
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This is also the square of the sum of the odd numbers plus the square of the sum of the even numbers, up to n. E.g., a(4) = (1+3)^2 + (2+4)^2 = 52. - Scott R. Shannon, Feb 20 2019
The area of a square whose side is a segment connecting the ends of a broken line (snake), the adjacent links of which are perpendicular and equal to the numbers 1, 2, 3, 4, ..., n. For example, a(5) = 9^2 + 6^2 = 117. - Nicolay Avilov, Aug 02 2022
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Sum of [ 1, 3, 9, ... ](A005994) + [ 0, 0, 1, 3, 9, ... ] + 2*[ 0, 1, 5, 15, 35, ... ](binomial(n, 4)).
If n is odd then a(n) = (1/8) * (n^4 + 2*n^3 + 2*n^2 + 2*n + 1) = Det(Transpose[M]*M) where M is the 2 X 3 matrix whose rows are [(n-1)/2, (n-1)/2], [(n-1)/2 + 1, 0] and [(n-1)/2 + 1, (n-1)/2 + 1]. If n is even then a(n) = (1/8) * (n^4 + 2*n^3 + 2*n^2) = Det(Transpose[M]*M) where M is the 2 X 3 matrix whose rows are [n/2, 0], [n/2, n/2] and [n/2 + 1, 0]. - Gerald McGarvey, Oct 30 2007
G.f.: -x*(x^4+2*x^3+6*x^2+2*x+1) / ((x-1)^5*(x+1)^2). - Colin Barker, Mar 20 2013
E.g.f.: (x*(7 + 15*x + 8*x^2 + x^3)*cosh(x) + (1 + 5*x + 15*x^2 + 8*x^3 + x^4)*sinh(x))/8. - Stefano Spezia, Jul 08 2020
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MATHEMATICA
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CoefficientList[Series[-(x^4 + 2 x^3 + 6 x^2 + 2 x + 1)/((x - 1)^5 (x + 1)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 14 2013 *)
LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {1, 5, 20, 52, 117, 225, 400}, 40] (* Harvey P. Dale, Dec 13 2018 *)
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PROG
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(PARI) Vec(-x*(x^4+2*x^3+6*x^2+2*x+1)/((x-1)^5*(x+1)^2) + O(x^100)) \\ Colin Barker, Oct 05 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A006009
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Number of paraffins.
(Formerly M3513)
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+10
3
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4, 16, 48, 108, 216, 384, 640, 1000, 1500, 2160, 3024, 4116, 5488, 7168, 9216, 11664, 14580, 18000, 22000, 26620, 31944, 38016, 44928, 52728, 61516, 71344, 82320, 94500, 108000, 122880, 139264, 157216, 176868, 198288, 221616, 246924, 274360, 304000, 336000
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OFFSET
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1,1
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REFERENCES
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S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = 2*(A005994(n) + binomial(n, 4)).
G.f.: 4*x*(1-x^3) / ((1-x)^4*(1-x^2)^2). - Alois P. Heinz, Aug 13 2008
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MAPLE
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a:= n-> (Matrix([[0$4, 4, 16, 48, 108]]). Matrix(8, (i, j)-> if (i=j-1) then 1 elif j=1 then [4, -4, -4, 10, -4, -4, 4, -1][i] else 0 fi)^n)[1, 1]: seq(a(n), n=1..40); # Alois P. Heinz, Aug 13 2008
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MATHEMATICA
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a[n_] := 1/16*(2*n^4+12*n^3+24*n^2+2*(9-(-1)^n)*n-3*(-1)^n+3); Array[a, 40] (* Jean-François Alcover, Mar 17 2014 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A141783
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Number of bracelets (turn over necklaces) with n beads: 1 blue, 12 green, and r = n - 13 red.
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+10
3
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1, 7, 49, 231, 924, 3108, 9324, 25236, 63090, 147070, 323554, 676270, 1352540, 2600612, 4829708, 8692788, 15212379, 25949469, 43249115, 70562765, 112900424, 177412664, 274183208, 417232088, 625848132, 926250780, 1353751140
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OFFSET
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13,2
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LINKS
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FORMULA
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a(n) = 1/2*(binomial(n-1,12) + binomial((n-2+n mod 2)/2, 6)).
a(n) = (1/(2*12!))*(n+2)*(n+4)*(n+6)*(n+8)*(n+10)*(n+12)*((n+1)*(n+3)*(n+5)*(n+7)*(n+9)*(n+11) + 1*3*5*7*9*11) - (1/15)*(1/2^10)*(n^5+(65/2)*n^4+400*n^3+(4615/2)*n^2+6154*n+(11895/2))*(1/2)*(1-(-1)^n) [Yosu Yurramendi, Jun 24 2013]
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MAPLE
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MATHEMATICA
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Table[(1/2) (Binomial[n - 1, 12] + Binomial[(n - 2 + Mod[n, 2])/2, 6]), {n, 13, 50}] (* Wesley Ivan Hurt, Jan 30 2014 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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Revised by Washington Bomfim, Jul 24 2012
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STATUS
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approved
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A301740
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The number of trees with 5 nodes labeled by positive integers, where each tree's label sum is n.
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+10
2
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3, 9, 24, 50, 96, 164, 267, 408, 603, 856, 1186, 1598, 2115, 2742, 3505, 4411, 5489, 6746, 8215, 9904, 11849, 14059, 16573, 19401, 22586, 26138, 30103, 34493, 39357, 44707, 50596, 57037, 64086, 71757, 80109, 89157, 98964, 109545, 120966, 133244, 146448, 160595, 175758, 191955
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OFFSET
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5,1
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COMMENTS
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Computed by the sum over the A000055(5)=3 shapes of the trees: the linear graph of the n-Pentane, the branched 2-Methyl-Butane, and the star graph of (1,1)-Bimethyl-Propane.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-2,2,0,1,0,-2,1).
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FORMULA
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EXAMPLE
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a(5)=3 because there is a linear tree with all labels equal 1, the branched tree with all labels equal to 1, and the star tree with all labels equal 1.
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MAPLE
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-x^5*(3+3*x+6*x^2+5*x^3+5*x^4+2*x^5+x^6)/(1+x^2)/(1+x+x^2)/(1+x)^2/(x-1)^5 ;
taylor(%, x=0, 80) ;
gfun[seriestolist](%) ;
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CROSSREFS
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Cf. A002620 (labeled trees with 3 nodes), A301739 (labeled trees with 4 nodes).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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