A004988
a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k + 2).
Original entry on oeis.org
1, 6, 45, 360, 2970, 24948, 212058, 1817640, 15677145, 135868590, 1182056733, 10316131488, 90266150520, 791564704560, 6954461332920, 61199259729696, 539318476367946, 4758692438540700, 42035116540442850, 371678925199705200, 3289358488017391020
Offset: 0
Joe Keane (jgk(AT)jgk.org)
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List([0..20], n-> 3^n*Product([0..n-1], k-> 3*k+2)/Factorial(n) ); # G. C. Greubel, Aug 22 2019
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[1] cat [3^n*&*[3*k+2: k in [0..n-1]]/Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 22 2019
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A004988 := proc(n)
binomial(-2/3,n)*(-9)^n ;
end proc: # R. J. Mathar, Sep 16 2012
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Table[FullSimplify[9^n*Gamma[n+2/3]/(Gamma[2/3]*Gamma[n+1])],{n,0,20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1-9x)^(-2/3), {x, 0, 20}], x] (* Vincenzo Librandi, Feb 10 2014 *)
Table[9^n*Pochhammer[2/3, n]/n!, {n,0,20}] (* G. C. Greubel, Aug 22 2019 *)
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a(n)=if(n<0,0,prod(k=0,n-1,3*k+2)*3^n/n!)
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[9^n*rising_factorial(2/3, n)/factorial(n) for n in (0..20)] # G. C. Greubel, Aug 22 2019
A004990
a(n) = (3^n/n!)*Product_{k=0..n-1} (3*k - 1).
Original entry on oeis.org
1, -3, -9, -45, -270, -1782, -12474, -90882, -681615, -5225715, -40760577, -322379109, -2579032872, -20830650120, -169621008120, -1390892266584, -11474861199318, -95173848770814, -793115406423450, -6637123664280450, -55751838779955780
Offset: 0
Joe Keane (jgk(AT)jgk.org)
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List([0..20], n-> 3^n*Product([0..n-1], k-> 3*k-1)/Factorial(n) ); # G. C. Greubel, Aug 22 2019
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[1] cat [3^n*(&*[3*k-1: k in [0..n-1]])/Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 22 2019
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a:= n-> (3^n/n!)*mul(3*k-1, k=0..n-1): seq(a(n), n=0..20); # G. C. Greubel, Aug 22 2019
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FullSimplify[Table[3^(2*n) * Gamma[n-1/3] / (n! * Gamma[-1/3]),{n,0,20}]] (* Vaclav Kotesovec, Dec 03 2014 *)
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for(n=0,30,print1( (3^n/n!)*prod(k=0,n-1,(3*k-1) ),","))
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[9^n*rising_factorial(-1/3, n)/factorial(n) for n in (0..20)] # G. C. Greubel, Aug 22 2019
A004989
a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k - 2).
Original entry on oeis.org
1, -6, -9, -36, -189, -1134, -7371, -50544, -360126, -2640924, -19806930, -151252920, -1172210130, -9197341020, -72921775230, -583374201840, -4703454502335, -38180983607190, -311811366125385, -2560135427134740, -21121117273861605, -175003543126281870, -1455711290550435555
Offset: 0
Joe Keane (jgk(AT)jgk.org)
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List([0..25], n-> 3^n*Product([0..n-1], k-> 3*k-2)/Factorial(n) ); # G. C. Greubel, Aug 22 2019
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[1] cat [3^n*(&*[3*k-2: k in [0..n-1]])/Factorial(n): n in [1..25]]; // G. C. Greubel, Aug 22 2019
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a:= n-> (3^n/n!)*product(3*k-2, k=0..n-1); seq(a(n), n=0..25); # G. C. Greubel, Aug 22 2019
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Table[9^n*Pochhammer[-2/3, n]/n!, {n,0,25}] (* G. C. Greubel, Aug 22 2019 *)
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a(n)=if(n<0,0,prod(k=0,n-1,3*k-2)*3^n/n!)
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[9^n*rising_factorial(-2/3, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019
A004991
a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k + 4).
Original entry on oeis.org
1, 12, 126, 1260, 12285, 117936, 1120392, 10563696, 99034650, 924323400, 8596207620, 79710288840, 737320171770, 6806032354800, 62712726697800, 576957085619760, 5300793224131545, 48642573115560060, 445890253559300550, 4083416006279910300, 37363256457461179245, 341606916182502210240
Offset: 0
Joe Keane (jgk(AT)jgk.org)
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List([0..25], n-> 3^n*Product([0..n-1], k-> 3*k+4)/Factorial(n) ); # G. C. Greubel, Aug 22 2019
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[1] cat [3^n*(&*[3*k+4: k in [0..n-1]])/Factorial(n): n in [1..25]]; // G. C. Greubel, Aug 22 2019
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a:= n-> (3^n/n!)*product(3*k+4, k=0..n-1); seq(a(n), n=0..25); # G. C. Greubel, Aug 22 2019
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Table[9^n*Pochhammer[4/3, n]/n!, {n,0,25}] (* G. C. Greubel, Aug 22 2019 *)
Table[3^n/n! Product[3k+4,{k,0,n-1}],{n,0,30}] (* or *) CoefficientList[ Series[ 1/Surd[(1-9x)^4,3],{x,0,30}],x] (* Harvey P. Dale, Aug 02 2021 *)
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a(n) = 3^n*prod(k=0,n-1, 3*k+4)/n!;
vector(25, n, n--; a(n)) \\ G. C. Greubel, Aug 22 2019
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[9^n*rising_factorial(4/3, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019
A382517
Expansion of 1/(1 - x/(1 - 9*x)^(5/3)).
Original entry on oeis.org
1, 1, 16, 211, 2611, 31426, 373099, 4397527, 51623530, 604629688, 7072089076, 82652922457, 965513250832, 11275328397061, 131649767277064, 1536953772789256, 17941954844917198, 209439428952580837, 2444747948094707815, 28536537876362681194, 333091044353156790346
Offset: 0
Showing 1-5 of 5 results.