A349011
G.f. A(x) satisfies: A(x) = (1 - x * A(-x)) / (1 - 2 * x * A(x)).
Original entry on oeis.org
1, 1, 5, 17, 105, 433, 2925, 13185, 93425, 443009, 3233205, 15840209, 117950745, 591187953, 4466545245, 22766535297, 173906505825, 897941153665, 6918379345125, 36089242700049, 279988660639305, 1472715584804529, 11490841104036045, 60857608450349313, 477104721264920145
Offset: 0
-
A349011 := proc(n)
option remember ;
if n = 0 then
1;
else
(-1)^n*procname(n-1)+2*add(procname(k)*procname(n-k-1),k=0..n-1) ;
end if;
end proc:
seq(A349011(n),n=0..40) ; # R. J. Mathar, Aug 19 2022
-
nmax = 24; A[] = 0; Do[A[x] = (1 - x A[-x])/(1 - 2 x A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = (-1)^n a[n - 1] + 2 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
A235608
Triangle read by rows: a non-Riordan array serving as a counterexample to a conjecture about Riordan arrays.
Original entry on oeis.org
1, 2, 1, 10, 5, 1, 62, 31, 7, 1, 430, 215, 51, 10, 1, 3194, 1597, 389, 87, 12, 1, 24850, 12425, 3077, 740, 117, 15, 1, 199910, 99955, 25035, 6305, 1076, 168, 17, 1, 1649350, 824675, 208255, 54150, 9705, 1700, 208, 20, 1, 13879538, 6939769, 1763473, 469399, 87048
Offset: 0
Triangle begins:
1;
2, 1;
10, 5, 1;
62, 31, 7, 1;
430, 215, 51, 10, 1;
3194, 1597, 389, 87, 12, 1;
24850, 12425, 3077, 740, 117, 15, 1;
199910, 99955, 25035, 6305, 1076, 168, 17, 1;
1649350, 824675, 208255, 54150, 9705, 1700, 208, 20, 1;
13879538, 6939769, 1763473, 469399, 87048, 16449, 2248, 274, 22, 1;
... - Extended by _Philippe Deléham_, Jan 31 2014
-
f[x_] := (1+x-Sqrt[1-10*x+x^2])/(6*x); g[x_] := (1-x-Sqrt[1-10*x+x^2])/(4*x); t[n_, k_] := SeriesCoefficient[f[x]^Floor[(k+2)/2]*g[x]^Floor[(k+1)/2], {x, 0, n}]; Table[t[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 31 2014, after Philippe Deléham *)
A239488
Expansion of 1/x-4/(-sqrt(x^2-10*x+1)-x+1)-3.
Original entry on oeis.org
6, 30, 186, 1290, 9582, 74550, 599730, 4948050, 41638614, 356007630, 3083837802, 27006251610, 238704231102, 2126733078630, 19079571337314, 172209370246050, 1562686251141030, 14248144422407550, 130467052593799962
Offset: 1
-
ogf := 1/x-4/(-sqrt(x^2-10*x+1)-x+1)-3;
series(ogf, x=0, 20): seq(coeff(%,x,n), n=0..19); # Peter Luschny, Mar 21 2014
-
a(n):=sum(2^i*binomial(n,n-i+1)*binomial(n+i-1,n-1),i,0,n+1)/n;
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