A354734
a(0) = a(1) = 1; a(n) = 3 * Sum_{k=0..n-2} a(k) * a(n-k-2).
Original entry on oeis.org
1, 1, 3, 6, 21, 54, 189, 558, 1944, 6210, 21681, 72576, 254988, 878850, 3112101, 10935000, 39030660, 139001346, 499808232, 1797731496, 6506661798, 23583173328, 85847830965, 313063862436, 1145325387114, 4197826175634, 15424343762184, 56774049331356, 209400739623054
Offset: 0
-
a[0] = a[1] = 1; a[n_] := a[n] = 3 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 28}]
nmax = 28; CoefficientList[Series[(1 - Sqrt[1 - 12 x^2 (1 + x)])/(6 x^2), {x, 0, nmax}], x]
A354735
a(0) = a(1) = 1; a(n) = 4 * Sum_{k=0..n-2} a(k) * a(n-k-2).
Original entry on oeis.org
1, 1, 4, 8, 36, 96, 416, 1312, 5504, 19200, 79168, 293888, 1203712, 4648448, 19027968, 75411456, 309487616, 1248411648, 5144133632, 21011775488, 86971449344, 358540509184, 1490753372160, 6189315784704, 25843660619776, 107902536122368, 452308820819968, 1897178275512320
Offset: 0
-
a[0] = a[1] = 1; a[n_] := a[n] = 4 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 27}]
nmax = 27; CoefficientList[Series[(1 - Sqrt[1 - 16 x^2 (1 + x)])/(8 x^2), {x, 0, nmax}], x]
A354736
a(0) = a(1) = 1; a(n) = 5 * Sum_{k=0..n-2} a(k) * a(n-k-2).
Original entry on oeis.org
1, 1, 5, 10, 55, 150, 775, 2550, 12500, 46250, 219375, 875000, 4075000, 17071250, 78796875, 341100000, 1569350000, 6947531250, 31966000000, 143761750000, 662668906250, 3014440000000, 13932834296875, 63921914062500, 296358191406250, 1368603488281250, 6365085546875000
Offset: 0
-
a[0] = a[1] = 1; a[n_] := a[n] = 5 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 26}]
nmax = 26; CoefficientList[Series[(1 - Sqrt[1 - 20 x^2 (1 + x)])/(10 x^2), {x, 0, nmax}], x]
A367071
G.f. satisfies A(x) = 1 + 2*x + 2*x^2*A(x)^2.
Original entry on oeis.org
1, 2, 2, 8, 16, 48, 136, 384, 1184, 3520, 10944, 34048, 107008, 340480, 1087104, 3502080, 11333120, 36867072, 120491008, 395276288, 1301700608, 4300414976, 14250496000, 47353233408, 157747462144, 526740717568, 1762653863936, 5910312910848
Offset: 0
-
A367071 := proc(n)
add(2^(n-k) * binomial(k+1,n-2*k) * A000108(k),k=0..floor(n/2)) ;
end proc:
seq(A367071(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
-
a(n) = sum(k=0, n\2, 2^(n-k)*binomial(k+1, n-2*k)*binomial(2*k, k)/(k+1));
Showing 1-4 of 4 results.