A104031
Column 1 of triangle A104030, which is the matrix inverse of the triangle of pairwise sums of trinomial coefficients.
Original entry on oeis.org
1, -5, 32, -299, 4015, -74080, 1801537, -55855829, 2150565968, -100668835739, 5630336915071, -370807060847872, 28403381397111649, -2503741873703550533, 251652633457563092096, -28608467296650576313259, 3652291148549164761101551, -520296573821474451201241696
Offset: 0
-
{a(n)=if(n<0,0,((matrix(n+2,n+2,m,j,if(m>=j, polcoeff((1+x+x^2)^(m-1)+O(x^(2*j)),2*j-2)+ polcoeff((1+x+x^2)^(m-1)+O(x^(2*j)),2*j-1))))^-1)[n+2,2])}
A104032
Unsigned row sums of triangle A104030, which is the matrix inverse of the triangle of pairwise sums of trinomial coefficients.
Original entry on oeis.org
1, 3, 13, 83, 781, 10515, 194125, 4721363, 146385805, 5636169363, 263831355085, 14755901813843, 971805538480141, 74439152531508243, 6561768848473177933, 659527417026466047443, 74976638559342869785741
Offset: 0
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{a(n)=if(n<0,0,(sum(k=0,n,abs((matrix(n+2,n+2,m,j,if(m>=j, polcoeff((1+x+x^2)^(m-1)+O(x^(2*j)),2*j-2)+ polcoeff((1+x+x^2)^(m-1)+O(x^(2*j)),2*j-1))))^-1)[n+1,k+1])))}
A006846
Hammersley's polynomial p_n(1).
Original entry on oeis.org
1, 1, 2, 7, 41, 376, 5033, 92821, 2257166, 69981919, 2694447797, 126128146156, 7054258103921, 464584757637001, 35586641825705882, 3136942184333040727, 315295985573234822561, 35843594275585750890976, 4575961401477587844760793, 651880406652100451820206941
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Vincenzo Librandi, Table of n, a(n) for n = 0..100
- J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.
- J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23. (Annotated scanned copy)
-
function A006846list(len::Int) # Algorithm of L. Seidel (1877)
R = Array{BigInt}(len)
A = fill(BigInt(0), len+1); A[1] = 1
for n in 1:len
for k in n:-1:2 A[k] += A[k+1] end
for k in 2:1:n A[k] += A[k-1] end
R[n] = A[n]
end
return R
end
println(A006846list(20)) # Peter Luschny, Jan 02 2018
-
A006846 := proc(n)
option remember ;
if n =0 then
return 1;
else
add(binomial(2*n,2*m)*procname(m)/(-4)^(n-m),m=0..n-1) ;
(3/4)^n-% ;
end if
end proc:
seq(A006846(n),n=0..20) ; # R. J. Mathar, Jan 10 2018
-
h[n_, x_] := Sum[c[k] x^k, {k, 0, n}]; eq[n_] := SolveAlways[h[n, x*(x-1)] == EulerE[2*n, x], x]; a[n_] := Sum[(-1)^(n+k)*c[k], {k, 0, n}] /. eq[n] // First; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Oct 02 2013, after Philippe Deléham *)
-
{a(n)=local(X=x+x*O(x^(2*n))); round((2*n)!*polcoeff(cosh(sqrt(3)*X/2)/cos(X/2),2*n))} \\ Paul D. Hanna
A104029
Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).
Original entry on oeis.org
1, 2, 1, 3, 5, 1, 4, 13, 9, 1, 5, 26, 35, 14, 1, 6, 45, 96, 75, 20, 1, 7, 71, 216, 267, 140, 27, 1, 8, 105, 427, 750, 623, 238, 35, 1, 9, 148, 770, 1800, 2123, 1288, 378, 44, 1, 10, 201, 1296, 3858, 6046, 5211, 2436, 570, 54, 1, 11, 265, 2067, 7590, 15115, 17303, 11505
Offset: 0
Row 3: {4,13,9,1} is formed from the pairwise sums
of row 3 of A027907: {1,3, 6,7, 6,3, 1}.
Rows begin:
1;
2, 1;
3, 5, 1;
4, 13, 9, 1;
5, 26, 35, 14, 1;
6, 45, 96, 75, 20, 1;
7, 71, 216, 267, 140, 27, 1;
8, 105, 427, 750, 623, 238, 35, 1;
9, 148, 770, 1800, 2123, 1288, 378, 44, 1;
10, 201, 1296, 3858, 6046, 5211, 2436, 570, 54, 1;
11, 265, 2067, 7590, 15115, 17303, 11505, 4302, 825, 65, 1;
12, 341, 3157, 13959, 34210, 49721, 43923, 23397, 7194, 1155, 77, 1; ...
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{T(n,k)=polcoeff((1+x+x^2)^n+x*O(x^(2*k)),2*k)+ polcoeff((1+x+x^2)^n+x*O(x^(2*k+1)),2*k+1)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
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{T(n,k)=polcoeff(polcoeff((1-x*y)/(1-2*x*(1+y)+x^2*(1+y+y^2)) +x*O(x^n),n,x)+y*O(y^k),k,y)}
Showing 1-4 of 4 results.
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