A303488
a(n) = n! * [x^n] 1/(1 - 5*x)^(n/5).
Original entry on oeis.org
1, 1, 14, 312, 9576, 375000, 17873856, 1004306688, 65006637696, 4763494479744, 389812500000000, 35237024762075136, 3487065897634615296, 374960171943074285568, 43532820293400237735936, 5427359437500000000000000, 723181462895975365595529216, 102563963819340862347122245632
Offset: 0
a(1) = 1;
a(2) = 2*7 = 14;
a(3) = 3*8*13 = 312;
a(4) = 4*9*14*19 = 9576;
a(5) = 5*10*15*20*25 = 375000, etc.
Cf.
A008546,
A008548,
A034300,
A034301,
A034323,
A034325,
A047055,
A047056,
A051687,
A051688,
A051689,
A051690,
A051691,
A052562,
A113551,
A303486,
A303487.
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Table[n! SeriesCoefficient[1/(1 - 5 x)^(n/5), {x, 0, n}], {n, 0, 17}]
Table[Product[5 k + n, {k, 0, n - 1}], {n, 0, 17}]
Table[5^n Pochhammer[n/5, n], {n, 0, 17}]
A153271
Triangle T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 3, read by rows.
Original entry on oeis.org
5, 5, 30, 5, 35, 315, 5, 40, 440, 6160, 5, 45, 585, 9945, 208845, 5, 50, 750, 15000, 375000, 11250000, 5, 55, 935, 21505, 623645, 21827575, 894930575, 5, 60, 1140, 29640, 978120, 39124800, 1838865600, 99298742400, 5, 65, 1365, 39585, 1464645, 65909025, 3493178325, 213083877825, 14702787569925
Offset: 0
Triangle begins as:
5;
5, 30;
5, 35, 315;
5, 40, 440, 6160;
5, 45, 585, 9945, 208845;
5, 50, 750, 15000, 375000, 11250000;
5, 55, 935, 21505, 623645, 21827575, 894930575;
Sequences related to m values:
-
m:=3;
function T(n,k)
if k eq 0 then return NthPrime(m);
else return (&*[j*n + NthPrime(m): j in [0..k]]);
end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 03 2019
-
m:=3; seq(seq(`if`(k=0, ithprime(m), mul(j*n + ithprime(m), j=0..k)), k=0..n), n=0..10); # G. C. Greubel, Dec 03 2019
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T[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j,0,k}]];
Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten
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T(n,k) = my(m=3); if(k==0, prime(m), prod(j=0,k, j*n + prime(m)) ); \\ G. C. Greubel, Dec 03 2019
-
def T(n, k):
m=3
if (k==0): return nth_prime(m)
else: return product(j*n + nth_prime(m) for j in (0..k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 03 2019
A153270
Triangle T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 2, read by rows.
Original entry on oeis.org
3, 3, 12, 3, 15, 105, 3, 18, 162, 1944, 3, 21, 231, 3465, 65835, 3, 24, 312, 5616, 129168, 3616704, 3, 27, 405, 8505, 229635, 7577955, 295540245, 3, 30, 510, 12240, 379440, 14418720, 648842400, 33739804800, 3, 33, 627, 16929, 592515, 25478145, 1299385395, 76663738305, 5136470466435
Offset: 0
Triangle begins as:
3;
3, 12;
3, 15, 105;
3, 18, 162, 1944;
3, 21, 231, 3465, 65835;
3, 24, 312, 5616, 129168, 3616704;
3, 27, 405, 8505, 229635, 7577955, 295540245;
3, 30, 510, 12240, 379440, 14418720, 648842400, 33739804800;
-
m:=2;
function T(n,k)
if k eq 0 then return NthPrime(m);
else return (&*[j*n + NthPrime(m): j in [0..k]]);
end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 03 2019
-
m:=2; seq(seq(`if`(k=0, ithprime(m), mul(j*n + ithprime(m), j=0..k)), k=0..n), n=0..10); # G. C. Greubel, Dec 03 2019
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T[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j,0,k}]];
Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten
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T(n,k) = my(m=2); if(k==0, prime(m), prod(j=0,k, j*n + prime(m)) ); \\ G. C. Greubel, Dec 03 2019
-
def T(n, k):
m=2
if (k==0): return nth_prime(m)
else: return product(j*n + nth_prime(m) for j in (0..k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 03 2019
A020038
Nearest integer to Gamma(n + 3/5)/Gamma(3/5).
Original entry on oeis.org
1, 1, 1, 2, 9, 41, 231, 1528, 11610, 99850, 958562, 10160756, 117864768, 1485096081, 20197306708, 294880677942, 4600138575888, 76362300359740, 1343976486331426, 24997962645764522, 489960067856984638
Offset: 0
-
Digits := 64:f := proc(n,x) round(GAMMA(n+x)/GAMMA(x)); end;
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Table[Round[QGamma[n+3/5,1]/QGamma[3/5,1]],{n,0,20}] (* Harvey P. Dale, May 19 2019 *)
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