A020042
a(n) = round( Gamma(n+1/4)/Gamma(1/4) ).
Original entry on oeis.org
1, 0, 0, 1, 2, 10, 51, 319, 2310, 19061, 176310, 1807181, 20330789, 249052162, 3299941145, 47024161317, 717118460090, 11653174976468, 201017268344078, 3668565147279424, 70619879085128903, 1430052551473860293
Offset: 0
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[Round(Gamma(n+1/4)/Gamma(1/4)): n in [0..30]]; // G. C. Greubel, Dec 06 2019
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Digits := 64:f := proc(n,x) round(GAMMA(n+x)/GAMMA(x)); end;
seq( round(pochhammer(1/4,n)), n=0..30); # G. C. Greubel, Dec 06 2019
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Table[Round[Pochhammer[1/4,n]], {n,0,30}] (* G. C. Greubel, Dec 06 2019 *)
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x=1/4; vector(30, n, round(gamma(n-1+x)/gamma(x)) ) \\ G. C. Greubel, Dec 06 2019
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[round(rising_factorial(1/4,n)) for n in (0..30)] # G. C. Greubel, Dec 06 2019
A370397
a(n) = floor(g(n + 1/4)), where g(x) is the function defined for all real numbers except the negative integers by the recurrence formula g(x+1) = g(x)*(x+1), with g(x) = 1 for 0 <= x < 1.
Original entry on oeis.org
1, 1, 2, 9, 38, 203, 1274, 9241, 76242, 705241, 7228724, 81323154, 996208647, 13199764580, 188096645269, 2868473840361, 46612699905873, 804069073376312, 14674260589117694, 282479516340515613, 5720210205895441171
Offset: 0
a(0) = 1 because: floor(g(0 + 1/4)) = floor(g(1/4)) = floor(1) = 1.
a(1) = 1 because: floor(g(1 + 1/4)) = floor(g(5/4)) = floor(g(1/4)*(5/4)) = floor((1)*(5/4)) = 1.
a(2) = 2 because: floor(g(2 + 1/4)) = floor(g(9/4)) = floor(g(5/4)*(9/4)) = floor((5/4)*(9/4)) = 2.
a(3) = 9 because: floor(g(3 + 1/4)) = floor(g(13/4)) = floor(g(9/4)*(13/4)) = floor((5/4)*(9/4)*(13/4)) = 9.
a(4) = 38 because: floor(g(4 + 1/4)) = floor(g(17/4)) = floor(g(13/4)*(17/4)) = floor((5/4)*(9/4)*(13/4)*(17/4)) = 38.
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makelist(floor(product(4*k+1, k, 1, n)/(4^n)), n, 0, 50);
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from math import prod
def A370397(n): return prod(range(5,(n<<2)+2,4))>>(n<<1) # Chai Wah Wu, Apr 28 2024
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