A242091 -id:A242091 - cp's OEIS frontend

A010967 a(n) = binomial coefficient C(n,14).

1, 15, 120, 680, 3060, 11628, 38760, 116280, 319770, 817190, 1961256, 4457400, 9657700, 20058300, 40116600, 77558760, 145422675, 265182525, 471435600, 818809200, 1391975640, 2319959400, 3796297200, 6107086800, 9669554100, 15084504396, 23206929840, 35240152720

Offset: 14

N. J. A. Sloane

nonn

In this sequence there are no primes. - Artur Jasinski, Dec 02 2007

T. D. Noe, Table of n, a(n) for n = 14..1000
Milan Janjic, Two Enumerative Functions.
Index entries for linear recurrences with constant coefficients, signature (15, -105, 455, -1365, 3003, -5005, 6435, -6435, 5005, -3003, 1365, -455, 105, -15, 1).

Cf. A000581, A001787, A110555, A242091.

[ Binomial(n,14): n in [14..60]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,14),n=14..37); # Zerinvary Lajos, Aug 06 2008

Table[Binomial[n,14],{n,14,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

for(n=14, 50, print1(binomial(n,14), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n) = A110555(n+1,14). - Reinhard Zumkeller, Jul 27 2005
a(n+13) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)(n+11)(n+12)(n+13)/14!. - Artur Jasinski, Dec 02 2007, R. J. Mathar, Jul 07 2009
G.f.: x^14/(1-x)^15. - Zerinvary Lajos, Aug 06 2008, R. J. Mathar, Jul 07 2009
a(n) = n/(n-14) * a(n-1), n > 14. - Vincenzo Librandi, Mar 26 2011
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=14} 1/a(n) = 14/13.
Sum_{n>=14} (-1)^n/a(n) = A001787(14)*log(2) - A242091(14)/13! = 114688*log(2) - 102309709/1287 = 0.9404563356... (End)

Some formulas rewritten for the correct offset by R. J. Mathar, Jul 07 2009

A010972 a(n) = binomial(n,19).

Original entry on oeis.org

1, 20, 210, 1540, 8855, 42504, 177100, 657800, 2220075, 6906900, 20030010, 54627300, 141120525, 347373600, 818809200, 1855967520, 4059928950, 8597496600, 17672631900, 35345263800, 68923264410, 131282408400, 244662670200, 446775310800, 800472431850

Offset: 19

N. J. A. Sloane

nonn

In this sequence there are no primes. - Artur Jasinski, Dec 02 2007

T. D. Noe, Table of n, a(n) for n = 19..1000
Index entries for linear recurrences with constant coefficients, signature (20,-190,1140,-4845,15504,-38760,77520,-125970,167960,-184756,167960,-125970,77520,-38760,15504,-4845,1140,-190,20,-1).

Cf. A001787, A242091.

List([19..45], n-> Binomial(n,19) ); # G. C. Greubel, Aug 27 2019

[ Binomial(n,19): n in [19..45]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,19),n=19..39); # Zerinvary Lajos, Aug 04 2008

Table[Binomial[n,19],{n,19,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

vector(25, n, binomial(n+18,19)) \\ G. C. Greubel, Nov 23 2017

[binomial(n,19) for n in (19..45)] # G. C. Greubel, Aug 27 2019

a(n+18) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9)*(n+10)*(n+11)*(n+12)*(n+13)*(n+14)*(n+15)*(n+16)*(n+17)*(n+18)/19!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
G.f.: x^19/(1-x)^20. - Zerinvary Lajos, Aug 04 2008; R. J. Mathar, Jul 07 2009
a(n) = n/(n-19) * a(n-1), n > 19. - Vincenzo Librandi, Mar 26 2011
From Amiram Eldar, Dec 11 2020: (Start)
Sum_{n>=19} 1/a(n) = 19/18.
Sum_{n>=19} (-1)^(n+1)/a(n) = A001787(19)*log(2) - A242091(19)/18! = 4980736*log(2) - 10574853703013/3063060 = 0.9542064261... (End)

A010968 a(n) = binomial(n,15).

Original entry on oeis.org

1, 16, 136, 816, 3876, 15504, 54264, 170544, 490314, 1307504, 3268760, 7726160, 17383860, 37442160, 77558760, 155117520, 300540195, 565722720, 1037158320, 1855967520, 3247943160, 5567902560, 9364199760, 15471286560, 25140840660, 40225345056, 63432274896

Offset: 15

N. J. A. Sloane

nonn

There are no primes in this sequence. - Artur Jasinski, Dec 02 2007

T. D. Noe, Table of n, a(n) for n = 15..1000
Milan Janjic, Two Enumerative Functions University of Banja Luka (Bosnia and Herzegovina, 2017).
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Cf. A000581, A001787, A110555, A242091.

[ Binomial(n,15): n in [15..70]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,15),n=15..37); # Zerinvary Lajos, Aug 06 2008

Table[Binomial[n,15],{n,15,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

for(n=15, 50, print1(binomial(n,15), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n) = -A110555(n+1,15). - Reinhard Zumkeller, Jul 27 2005
a(n+14) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)(n+11)(n+12)(n+13)(n+14)/15!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
G.f.: x^15/(1-x)^16. - Zerinvary Lajos, Aug 06 2008; R. J. Mathar, Jul 07 2009
a(n) = n/(n-15) * a(n-1), n > 15. - Vincenzo Librandi, Mar 26 2011
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=15} 1/a(n) = 15/14.
Sum_{n>=15} (-1)^(n+1)/a(n) = A001787(15)*log(2) - A242091(15)/14! = 245760*log(2) - 1023103525/6006 = 0.9438350048... (End)

Some formulas adjusted to the offset by R. J. Mathar, Jul 07 2009

A010970 a(n) = binomial(n,17).

Original entry on oeis.org

1, 18, 171, 1140, 5985, 26334, 100947, 346104, 1081575, 3124550, 8436285, 21474180, 51895935, 119759850, 265182525, 565722720, 1166803110, 2333606220, 4537567650, 8597496600, 15905368710, 28781143380, 51021117810, 88732378800, 151584480450, 254661927156

Offset: 17

N. J. A. Sloane

nonn

In this sequence there are no primes. - Artur Jasinski, Dec 02 2007

T. D. Noe, Table of n, a(n) for n = 17..1000
Milan Janjic, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (18,-153,816,-3060,8568,-18564,31824,-43758,48620,-43758,31824,-18564,8568,-3060,816,-153,18,-1).

Cf. A001787, A242091.

[ Binomial(n,17): n in [17..80]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,17),n=17..37); # Zerinvary Lajos, Aug 06 2008

Table[Binomial[n,17],{n,17,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

for(n=17,50, print1(binomial(n,17), ", ")) \\ G. C. Greubel, Nov 23 2017

a(n+16) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9)*(n+10)*(n+11)*(n+12)*(n+13)*(n+14)*(n+15)*(n+16)/17!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
G.f.: x^17/(1-x)^18. - Zerinvary Lajos, Aug 06 2008; R. J. Mathar, Jul 07 2009
a(n) = n/(n-17) * a(n-1), n > 17. - Vincenzo Librandi, Mar 26 2011
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=17} 1/a(n) = 17/16.
Sum_{n>=17} (-1)^(n+1)/a(n) = A001787(17)*log(2) - A242091(17)/16! = 1114112*log(2) - 556570716997/720720 = 0.9495520222... (End)

A010971 a(n) = binomial(n,18).

Original entry on oeis.org

1, 19, 190, 1330, 7315, 33649, 134596, 480700, 1562275, 4686825, 13123110, 34597290, 86493225, 206253075, 471435600, 1037158320, 2203961430, 4537567650, 9075135300, 17672631900, 33578000610, 62359143990, 113380261800, 202112640600, 353697121050, 608359048206

Offset: 18

N. J. A. Sloane

nonn

Coordination sequence for 18-dimensional cyclotomic lattice Z[zeta_19].
Product of 18 consecutive numbers divided by 18!. - Artur Jasinski, Dec 02 2007
In this sequence only 19 is prime. - Artur Jasinski, Dec 02 2007
With a different offset, number of n-permutations (n>=18) of 2 objects: u,v, with repetition allowed, containing exactly (18) u's. - Zerinvary Lajos, Aug 04 2008

T. D. Noe, Table of n, a(n) for n = 18..1000
Matthias Beck and Serkan Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
Milan Janjic, Two Enumerative Functions.
Index entries for linear recurrences with constant coefficients, signature (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).

Cf. A001787, A242091.

[Binomial(n, 18): n in [18..50]]; // Vincenzo Librandi, Aug 08 2017

seq(binomial(n,18),n=18..38); # Zerinvary Lajos, Aug 04 2008

Table[n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)(n+11)(n+12)(n+13)(n+14)(n+15)(n+16)(n+17)/18!,{n,1,100}] (* Artur Jasinski, Dec 02 2007 *)
Table[Binomial[n, 18], {n, 18, 50}] (* Vincenzo Librandi, Aug 08 2017 *)

for(n=18,50, print1(binomial(n,18), ", ")) \\ G. C. Greubel, Nov 23 2017

a(n+17) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9)*(n+10)*(n+11)*(n+12)*(n+13)*(n+14)*(n+15)*(n+16)*(n+17)/18!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
G.f.: x^18/(1-x)^19. - Zerinvary Lajos, Aug 04 2008; R. J. Mathar, Jul 07 2009
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=18} 1/a(n) = 18/17.
Sum_{n>=18} (-1)^n/a(n) = A001787(18)*log(2) - A242091(18)/17! = 2359296*log(2) - 556571077357/340340 = 0.9519925176... (End)

Some formulas adjusted to the offset by R. J. Mathar, Jul 07 2009

A010980 a(n) = binomial(n,27).

Original entry on oeis.org

1, 28, 406, 4060, 31465, 201376, 1107568, 5379616, 23535820, 94143280, 348330136, 1203322288, 3910797436, 12033222880, 35240152720, 98672427616, 265182149218, 686353797976, 1715884494940, 4154246671960, 9762479679106, 22314239266528, 49699896548176

Offset: 27

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 27..1000
Index entries for linear recurrences with constant coefficients, signature (28,-378,3276,-20475,98280,-376740,1184040,-3108105,6906900,-13123110,21474180,-30421755,37442160,-40116600,37442160,-30421755,21474180,-13123110,6906900,-3108105,1184040,-376740,98280,-20475,3276,-378,28,-1).

Cf. A010976, A010977, A010978, A001787, A242091.

[Binomial(n, 27): n in [27..60]]; // Vincenzo Librandi, Jun 12 2013

seq(binomial(n,27),n=27..51); # Zerinvary Lajos, Aug 18 2008

Table[Binomial[n,27],{n,27,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2011 *)

x='x+O('x^50); Vec(x^27/(1-x)^28) \\ G. C. Greubel, Nov 23 2017

From Zerinvary Lajos, Aug 18 2008: (Start)
a(n) = C(n,27), n >= 27.
G.f.: x^27/(1-x)^28. (End)
From Amiram Eldar, Dec 11 2020: (Start)
Sum_{n>=27} 1/a(n) = 27/26.
Sum_{n>=27} (-1)^(n+1)/a(n) = A001787(27)*log(2) - A242091(27)/26! = 1811939328*log(2) - 233492834118075846/185910725 = 0.9665300296... (End)

A010984 Binomial coefficient C(n,31).

Original entry on oeis.org

1, 32, 528, 5984, 52360, 376992, 2324784, 12620256, 61523748, 273438880, 1121099408, 4280561376, 15338678264, 51915526432, 166871334960, 511738760544, 1503232609098, 4244421484512, 11554258485616, 30405943383200, 77535155627160, 191991813933920

Offset: 31

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 31..1000
Index entries for linear recurrences with constant coefficients, signature (32, -496, 4960, -35960, 201376, -906192, 3365856, -10518300, 28048800, -64512240, 129024480, -225792840, 347373600, -471435600, 565722720, -601080390, 565722720, -471435600, 347373600, -225792840, 129024480, -64512240, 28048800, -10518300, 3365856, -906192, 201376, -35960, 4960, -496, 32, -1).

Cf. A010980, A010981, A010982, A001787, A242091

[Binomial(n, 31): n in [31..70]]; // Vincenzo Librandi, Jun 12 2013

seq(binomial(n,31),n=31..53); # Zerinvary Lajos, Dec 19 2008

Table[Binomial[n,31],{n,31,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2011 *)

x='x+O('x^50); Vec(x^31/(1-x)^32) \\ G. C. Greubel, Nov 23 2017

G.f.: x^31/(1-x)^32. - Zerinvary Lajos, Dec 19 2008; adapted to offset by Enxhell Luzhnica, Jan 21 2017
From Amiram Eldar, Dec 12 2020: (Start)
Sum_{n>=31} 1/a(n) = 31/30.
Sum_{n>=31} (-1)^(n+1)/a(n) = A001787(31)*log(2) - A242091(31)/30! = 33285996544*log(2) - 6717121856795533085173/291136195350 = 0.9704936372... (End)

A010985 Binomial coefficient C(n,32).

Original entry on oeis.org

1, 33, 561, 6545, 58905, 435897, 2760681, 15380937, 76904685, 350343565, 1471442973, 5752004349, 21090682613, 73006209045, 239877544005, 751616304549, 2254848913647, 6499270398159, 18053528883775, 48459472266975, 125994627894135, 317986441828055

Offset: 32

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 32..1000
Index entries for linear recurrences with constant coefficients, signature (33, -528, 5456, -40920, 237336, -1107568, 4272048, -13884156, 38567100, -92561040, 193536720, -354817320, 573166440, -818809200, 1037158320, -1166803110, 1166803110, -1037158320, 818809200, -573166440, 354817320, -193536720, 92561040, -38567100, 13884156, -4272048, 1107568, -237336, 40920, -5456, 528, -33, 1).

Cf. A010982, A010984, A001787, A242091.

[Binomial(n, 32): n in [32..70]]; // Vincenzo Librandi, Jun 12 2013

seq(binomial(n,32),n=32..55); # Zerinvary Lajos, Dec 19 2008

Table[Binomial[n,32],{n,32,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2011 *)

x='x+O('x^50); Vec(x^32/(1-x)^33) \\ G. C. Greubel, Nov 23 2017

G.f.: x^32/(1-x)^33. - Zerinvary Lajos, Dec 19 2008; adapted to offset by Enxhell Luzhnica, Jan 21 2017
From Amiram Eldar, Dec 12 2020: (Start)
Sum_{n>=32} 1/a(n) = 32/31.
Sum_{n>=32} (-1)^n/a(n) = A001787(32)*log(2) - A242091(32)/31! = 68719476736*log(2) - 214947899422115237851136/4512611027925 = 0.9713417027... (End)

A010986 Binomial coefficient C(n,33).

Original entry on oeis.org

1, 34, 595, 7140, 66045, 501942, 3262623, 18643560, 95548245, 445891810, 1917334783, 7669339132, 28760021745, 101766230790, 341643774795, 1093260079344, 3348108992991, 9847379391150, 27900908274925, 76360380541900, 202355008436035, 520341450264090

Offset: 33

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 33..1000
Index entries for linear recurrences with constant coefficients, signature (34, -561, 5984, -46376, 278256, -1344904, 5379616, -18156204, 52451256, -131128140, 286097760, -548354040, 927983760, -1391975640, 1855967520, -2203961430, 2333606220, -2203961430, 1855967520, -1391975640, 927983760, -548354040, 286097760, -131128140, 52451256, -18156204, 5379616, -1344904, 278256, -46376, 5984, -561, 34, -1).

Cf. A010984, A010985, A001787, A242091.

[Binomial(n, 33): n in [33..70]]; // Vincenzo Librandi, Jun 12 2013

seq(binomial(n,33),n=33..55); # Zerinvary Lajos, Dec 19 2008

Table[Binomial[n,33],{n,33,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2011 *)

for(n=33, 60, print1(binomial(n,33), ", ")) \\ G. C. Greubel, Nov 23 2017

G.f.: x^33/(1-x)^34. - Zerinvary Lajos, Dec 19 2008; adapted to offset by Enxhell Luzhnica, Jan 23 2017
From Amiram Eldar, Dec 12 2020: (Start)
Sum_{n>=33} 1/a(n) = 33/32.
Sum_{n>=33} (-1)^(n+1)/a(n) = A001787(33)*log(2) - A242091(33)/32! = 141733920768*log(2) - 429895798848743086730197/4375865239200 = 0.9721422619... (End)

A010987 Binomial coefficient C(n,34).

Original entry on oeis.org

1, 35, 630, 7770, 73815, 575757, 3838380, 22481940, 118030185, 563921995, 2481256778, 10150595910, 38910617655, 140676848445, 482320623240, 1575580702584, 4923689695575, 14771069086725, 42671977361650, 119032357903550, 321387366339585, 841728816603675

Offset: 34

N. J. A. Sloane

nonn

T. D. Noe, Table of n, a(n) for n = 34..1000
Index entries for linear recurrences with constant coefficients, signature (35, -595, 6545, -52360, 324632, -1623160, 6724520, -23535820, 70607460, -183579396, 417225900, -834451800, 1476337800, -2319959400, 3247943160, -4059928950, 4537567650, -4537567650, 4059928950, -3247943160, 2319959400, -1476337800, 834451800, -417225900, 183579396, -70607460, 23535820, -6724520, 1623160, -324632, 52360, -6545, 595, -35, 1).

Cf. A010984, A010985, A010986, A001787, A242091.

[Binomial(n, 34): n in [34..70]]; // Vincenzo Librandi, Jun 12 2013

seq(binomial(n,34),n=34..55); # Zerinvary Lajos, Dec 19 2008

Table[Binomial[n,34],{n,34,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2011 *)

x='x+O('x^50); Vec(x^34/(1-x)^35) \\ G. C. Greubel, Nov 23 2017

G.f.: x^34/(1-x)^35 . - Zerinvary Lajos, Dec 19 2008; adapted to offset by Enxhell Luzhnica, Jan 23 2017
From Amiram Eldar, Dec 12 2020: (Start)
Sum_{n>=34} 1/a(n) = 34/33.
Sum_{n>=34} (-1)^n/a(n) = A001787(34)*log(2) - A242091(34)/33! = 292057776128*log(2) - 429895798850931019349797/2123581660200 = 0.9728992064... (End)

cp's OEIS Frontend

A010967 a(n) = binomial coefficient C(n,14).

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A010972 a(n) = binomial(n,19).

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A010968 a(n) = binomial(n,15).

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A010970 a(n) = binomial(n,17).

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A010971 a(n) = binomial(n,18).

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A010980 a(n) = binomial(n,27).

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