A190315
Central coefficients of the Riordan matrix ((1-x-x^2)/(1-2x-x^2),(x-x^2-x^3)/(1-2x-x^2)) (A190215).
Original entry on oeis.org
1, 2, 9, 48, 265, 1500, 8638, 50360, 296325, 1756160, 10467556, 62683896, 376838098, 2272896626, 13747543035, 83354081728, 506467851061, 3083121435312, 18799746616104, 114804614071760, 702016963933404, 4297947201746176, 26342178216979384
Offset: 0
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Table[Sum[Binomial[n+i,n]Sum[Binomial[i+j-1,j]Binomial[j,n-i-j],{j,0,n-i}],{i,0,n}],{n,0,22}]
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makelist(sum(binomial(n+i,n)*sum(binomial(i+j-1,j)*binomial(j,n-i-j),j,0,n-i),i,0,n),n,0,22);
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for(n=0,30, print1(sum(k=0,n, binomial(n+k,n)*sum(j=0,n-k, binomial(k+j-1,j)*binomial(j,n-k-j))), ", ")) \\ G. C. Greubel, Mar 04 2018
A052960
Expansion of ( 1-x-x^2 ) / ( 1-2*x-2*x^2+x^3+x^4 ).
Original entry on oeis.org
1, 1, 3, 7, 18, 46, 118, 303, 778, 1998, 5131, 13177, 33840, 86905, 223182, 573157, 1471933, 3780093, 9707713, 24930522, 64024444, 164422126, 422254905, 1084399096, 2784861432, 7151844025, 18366756913, 47167941348, 121132691065
Offset: 0
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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a:=[1,1,3,7];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3] -a[n-4]; od; a; # G. C. Greubel, Oct 23 2019
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R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x-x^2)/(1-2*x-2*x^2+x^3+x^4) )); // G. C. Greubel, Oct 23 2019
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spec:= [S,{S=Sequence(Prod(Union(Sequence(Union(Prod(Z,Z),Z)),Z),Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(coeff(series((1-x-x^2)/(1-2*x-2*x^2+x^3+x^4), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 23 2019
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Table[Sum[Sum[Binomial[i+2k,2k]Binomial[i+k,n-i-2k],{k,0,n/2}],{i,0,n}],{n,0,12}] (* Emanuele Munarini, May 10 2011 *)
LinearRecurrence[{2,2,-1,-1}, {1,1,3,7}, 30] (* G. C. Greubel, Oct 23 2019 *)
CoefficientList[Series[(1-x-x^2)/(1-2x-2x^2+x^3+x^4),{x,0,50}],x] (* Harvey P. Dale, Jan 21 2021 *)
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makelist(sum(sum(binomial(i+2*k, 2*k)*binomial(i+k, n-i-2*k), k,0,n/2),i,0,n),n,0,24); /* Emanuele Munarini, May 10 2011 */
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my(x='x+O('x^30)); Vec((1-x-x^2)/(1-2*x-2*x^2+x^3+x^4)) \\ G. C. Greubel, Oct 23 2019
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def A052960_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P((1-x-x^2)/(1-2*x-2*x^2+x^3+x^4)).list()
A052960_list(30) # G. C. Greubel, Oct 23 2019
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