A188313
A (25,-29) Somos-4 sequence.
Original entry on oeis.org
1, 3, 23, 314, 8209, 620297, 126742987, 47301104551, 32606721084786, 61958046554226593, 334806306946199122193, 3416372868727801226636179, 62595441409640805744780870839, 2993270782370572835241882188306602, 424202695773047673359251734568172738737
Offset: 0
G.f.: 1 + 3*x + 23*x^2 + 314*x^3 + 8209*x^4 + ... - _Michael Somos_, Feb 28 2022
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I:=[8209, 620297, 126742987, 47301104551]; [1, 3, 23, 314] cat [n le 4 select I[n] else (25*Self(n-1)*Self(n-3) - 29*Self(n-2)^2)/Self(n-4): n in [1..30]]; // G. C. Greubel, Aug 14 2018
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Join[{1, 3, 23, 314}, RecurrenceTable[{a[n] == (25*a[n - 1]*a[n - 3] - 29*a[n - 2]^2)/a[n - 4], a[4] == 8209, a[5] == 620297, a[6] == 126742987, a[7] == 47301104551}, a, {n, 4, 25}]] (* G. C. Greubel, Aug 14 2018 *)
b[ n_] := If[OddQ[n], a[(n-3)/2], a[-n/2]]; a[ n_] := If[-2<=n<=2, {2, 1, 1, 3, 23}[[n+3]], 2*b[n+2]^3*b[n+3] + b[n+1]^2*(b[n+3]*b[n+4] - b[n+2]*b[n+5])]; (* Michael Somos, Feb 28 2022 *)
A215973
a(0) = 1, for n > 0: a(n) = Sum_{k=0..n-1} a(k) * (1 + a(n-1-k)).
Original entry on oeis.org
1, 2, 7, 28, 122, 565, 2735, 13682, 70188, 367248, 1952394, 10516141, 57265929, 314751625, 1743829163, 9728561418, 54604800126, 308137127382, 1747158309208, 9949001656704, 56872435967840, 326243091718978, 1877419829207578, 10835354636496321
Offset: 0
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a215973 n = a215973_list !! n
a215973_list = 1 : f [1] where
f xs = y : f (y:xs) where
y = sum $ zipWith (*) xs $ map (+ 1) $ reverse xs
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nmax = 30; CoefficientList[Series[(2*x - 1 + Sqrt[1-8*x+12*x^2-4*x^3]) / (2*x*(x-1)), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 31 2017 *)
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a(n):=sum(sum(binomial(j+1,i)*binomial(2*j-i,j-i)*binomial(n-j+i-1,n-j),i,0,j)/(j+1),j,0,n); /* Vladimir Kruchinin, May 04 2018 */
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a(n) = sum(j=0, n, sum(i=0,j, binomial(j+1,i)*binomial(2*j-i,j-i)*binomial(n-j+i-1,n-j)/(j+1))); \\ Altug Alkan, May 04 2018
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x='x+O('x^99); Vec((2*x-1+(1-8*x+12*x^2-4*x^3)^(1/2))/(2*x*(x-1))) \\ Altug Alkan, May 04 2018
A188314
Expansion of (1/(1-x))*c(x/((1-x)*(1-x^2))) where c(x) is the g.f. of A000108.
Original entry on oeis.org
1, 2, 5, 16, 57, 219, 883, 3687, 15803, 69128, 307363, 1385003, 6310869, 29028616, 134610771, 628612921, 2953640371, 13953726888, 66240021987, 315812059436, 1511569447859, 7260364084997, 34984937594741, 169073568381936, 819288294835939, 3979892232651125, 19377475499900015
Offset: 0
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m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x^2- Sqrt(1-4*x-6*x^2+x^4))/(2*x))); // G. C. Greubel, Aug 14 2018
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CoefficientList[Series[(1-x^2 - Sqrt[1-4*x-6*x^2+x^4])/(2*x), {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *)
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x='x+O('x^30); Vec((1-x^2- sqrt(1-4*x-6*x^2+x^4))/(2*x)) \\ G. C. Greubel, Aug 14 2018
Showing 1-3 of 3 results.
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