A213221
Riordan array (f(x), x*g(x)) where f(x) is the g.f. of A157004 and g(x) is the g.f. of A157003.
Original entry on oeis.org
1, 2, 1, 6, 3, 1, 18, 10, 4, 1, 58, 32, 15, 5, 1, 192, 106, 52, 21, 6, 1, 650, 357, 180, 79, 28, 7, 1, 2232, 1222, 624, 288, 114, 36, 8, 1, 7746, 4230, 2178, 1035, 439, 158, 45, 9, 1, 27096, 14770, 7648, 3706, 1642, 643, 212, 55, 10, 1
Offset: 0
Triangle begins
1
2, 1
6, 3, 1
18, 10, 4, 1
58, 32, 15, 5, 1
192, 106, 52, 21, 6, 1
650, 357, 180, 79, 28, 7, 1
2232, 1222, 624, 288, 114, 36, 8, 1
7746, 4230, 2178, 1035, 439, 158, 45, 9, 1
27096, 14770, 7648, 3706, 1642, 643, 212, 55, 10, 1
95376, 51918, 27000, 13265, 6056, 2508, 911, 277, 66, 11, 1
337404, 183472, 95744, 47532, 22174, 9552, 3708, 1255, 354, 78, 12, 1
- Baccherini, D.; Merlini, D.; Sprugnoli, R. Binary words excluding a pattern and proper Riordan arrays. Discrete Math. 307 (2007), no. 9-10, 1021--1037. MR2292531 (2008a:05003). See page 1032. - N. J. A. Sloane, Mar 25 2014
A157005
A Somos-4 variant.
Original entry on oeis.org
1, 2, 8, 24, 112, 736, 3776, 40192, 391424, 4203008, 85312512, 1270368256, 32235102208, 1038278549504, 27640704385024, 1549962593927168, 73624753456480256, 4273828146025070592, 435765959975516766208
Offset: 0
-
a:=[1,2,8,24];; for n in [5..20] do a[n]:=(a[n-1]*a[n-3] + a[n-2]^2)/a[n-4]; od; a; # G. C. Greubel, Feb 23 2019
-
I:=[1,2,8,24]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4): n in [1..20]]; // G. C. Greubel, Feb 23 2019
-
RecurrenceTable[{a[0]==1,a[1]==2,a[2]==8,a[3]==24,a[n]==(a[n-1] a[n-3]+a[n-2]^2)/a[n-4]},a,{n,20}] (* Harvey P. Dale, Apr 30 2011 *)
-
m=20; v=concat([1,2,8,24], vector(m-4)); for(n=5, m, v[n] = (v[n-1]*v[n-3] +v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Feb 23 2019
-
def a(n):
if (n==0): return 1
elif (n==1): return 2
elif (n==2): return 8
elif (n==3): return 24
else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4)
[a(n) for n in (0..20)] # G. C. Greubel, Feb 23 2019
A360266
a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k,k) * binomial(2*(n-2*k),n-2*k).
Original entry on oeis.org
1, 2, 6, 22, 82, 312, 1210, 4752, 18834, 75184, 301856, 1217604, 4930626, 20032052, 81615072, 333328532, 1364264250, 5594210292, 22977466864, 94517423444, 389316529512, 1605533230256, 6628467569292, 27393187077144, 113310732332274, 469101108803052
Offset: 0
-
a(n) = sum(k=0, n\3, binomial(n-2*k, k)*binomial(2*(n-2*k), n-2*k));
-
my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x^2)))
A360219
a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * binomial(2*(n-3*k),n-3*k).
Original entry on oeis.org
1, 2, 6, 20, 68, 240, 864, 3152, 11616, 43136, 161152, 604992, 2280416, 8624832, 32714688, 124399488, 474066560, 1810053120, 6922776576, 26517173760, 101710338048, 390603984896, 1501732753408, 5779500226560, 22263437981184, 85835254221824, 331193445626880
Offset: 0
-
A360219 := proc(n)
add((-1)^k*binomial(n-3*k,k)*binomial(2*(n-3*k),n-3*k),k=0..n/3) ;
end proc:
seq(A360219(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
-
a(n) = sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)*binomial(2*(n-3*k), n-3*k));
-
my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1-x^3)))
A374598
Expansion of 1/sqrt(1 - 4*x - 8*x^3).
Original entry on oeis.org
1, 2, 6, 24, 94, 372, 1508, 6192, 25638, 106908, 448356, 1889040, 7989676, 33902504, 144259944, 615330784, 2630199942, 11263613484, 48315367076, 207556060816, 892819376964, 3845161246424, 16578320962104, 71548426931616, 309070048163676, 1336223562436632
Offset: 0
-
my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x-8*x^3))
-
a(n) = sum(k=0, n\3, 2^k*binomial(n-2*k, k)*binomial(2*(n-2*k), n-2*k));
A168151
Riordan array (1/u,(1-u)/2), u=sqrt(1-4x+4*x^3).
Original entry on oeis.org
1, 2, 1, 6, 3, 1, 18, 9, 4, 1, 58, 29, 13, 5, 1, 192, 96, 44, 18, 6, 1, 650, 325, 151, 64, 24, 7, 1, 2232, 1116, 524, 228, 90, 31, 8, 1, 7746, 3873, 1833, 813, 333, 123, 39, 9, 1, 27096, 13548, 6452, 2904, 1222, 473, 164, 48, 10, 1
Offset: 0
Triangle begins:
1
2 1
6 3 1
18 9 4 1
58 29 13 5 1
192 96 44 18 6 1
650 325 151 64 24 7 1
...
- Baccherini, D.; Merlini, D.; Sprugnoli, R. Binary words excluding a pattern and proper Riordan arrays. Discrete Math. 307 (2007), no. 9-10, 1021--1037. MR2292531 (2008a:05003). See Example 5.6. - N. J. A. Sloane, Mar 25 2014
Showing 1-6 of 6 results.
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