A242091 -id:A242091 - cp's OEIS frontend

A010974 a(n) = binomial(n,21).

1, 22, 253, 2024, 12650, 65780, 296010, 1184040, 4292145, 14307150, 44352165, 129024480, 354817320, 927983760, 2319959400, 5567902560, 12875774670, 28781143380, 62359143990, 131282408400, 269128937220, 538257874440, 1052049481860, 2012616400080, 3773655750150

Offset: 21

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 = 21..1000
Milan Janjic, Two Enumerative Functions.
Index entries for linear recurrences with constant coefficients, signature (22,-231,1540,-7315,26334,-74613,170544,-319770,497420,-646646,705432,-646646,497420,-319770,170544,-74613,26334,-7315,1540,-231,22,-1).

Pascal's triangle A007318. - Zerinvary Lajos, Aug 04 2008

Cf. A001787, A242091.

[ Binomial(n,21): n in [21..80]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,21),n=21..41); # Zerinvary Lajos, Aug 04 2008

Table[Binomial[n,21],{n,21,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

for(n=21, 50, print1(binomial(n,21), ", ")) \\ G. C. Greubel, Nov 23 2017

a(n) = 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)*(n+19)*(n+20) / 21!. - Artur Jasinski, Dec 02 2007
a(n) = n/(n-21) * a(n-1), n > 21. - Vincenzo Librandi, Mar 26 2011
From Amiram Eldar, Dec 11 2020: (Start)
Sum_{n>=21} 1/a(n) = 21/20.
Sum_{n>=21} (-1)^(n+1)/a(n) = A001787(21)*log(2) - A242091(21)/20! = 22020096*log(2) - 42299423848079/2771340 = 0.9580705153... (End)

A010976 Binomial coefficient C(n,23).

Original entry on oeis.org

1, 24, 300, 2600, 17550, 98280, 475020, 2035800, 7888725, 28048800, 92561040, 286097760, 834451800, 2310789600, 6107086800, 15471286560, 37711260990, 88732378800, 202112640600, 446775310800, 960566918220, 2012616400080, 4116715363800, 8233430727600

Offset: 23

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 23..1000
Index entries for linear recurrences with constant coefficients, signature (24,-276,2024,-10626,42504,-134596,346104,-735471,1307504,-1961256,2496144,-2704156,2496144,-1961256,1307504,-735471,346104,-134596,42504,-10626,2024,-276,24,-1).

Pascal's triangle A007318. [Zerinvary Lajos, Aug 04 2008]

Cf. A010970, A010971, A010972, A001787, A242091.

[Binomial(n,23): n in [23..90]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,23), n=23..43); # Zerinvary Lajos, Aug 04 2008

Table[Binomial[n,23],{n,23,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2011 *)

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

a(n) = n/(n-23) * a(n-1) for n > 23. - Vincenzo Librandi, Mar 26 2011
G.f.: x^23/(1-x)^24. - G. C. Greubel, Nov 23 2017
From Amiram Eldar, Dec 11 2020: (Start)
Sum_{n>=23} 1/a(n) = 23/22.
Sum_{n>=23} (-1)^(n+1)/a(n) = A001787(23)*log(2) - A242091(23)/22! = 96468992*log(2) - 1945773591174209/29099070 = 0.9613305695... (End)

A010977 a(n) = binomial coefficient C(n,24).

Original entry on oeis.org

1, 25, 325, 2925, 20475, 118755, 593775, 2629575, 10518300, 38567100, 131128140, 417225900, 1251677700, 3562467300, 9669554100, 25140840660, 62852101650, 151584480450, 353697121050, 800472431850, 1761039350070, 3773655750150, 7890371113950, 16123801841550

Offset: 24

N. J. A. Sloane

nonn

T. D. Noe, Table of n, a(n) for n = 24..1000
Index entries for linear recurrences with constant coefficients, signature (25, -300, 2300, -12650, 53130, -177100, 480700, -1081575, 2042975, -3268760, 4457400, -5200300, 5200300, -4457400, 3268760, -2042975, 1081575, -480700, 177100, -53130, 12650, -2300, 300, -25, 1).

Pascal's triangle A007318 diagonal. - Zerinvary Lajos, Aug 04 2008

Cf. A010970, A010971, A010972, A001787, A242091.

[ Binomial(n,24): n in [24..90]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,24),n=24..41); # Zerinvary Lajos, Aug 04 2008

Table[Binomial[n,24],{n,24,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2011 *)

x='x+O('x^50); Vec(x^24/(1-x)^25) \\ G. C. Greubel, Nov 23 2017

G.f.: x^24/(1-x)^25. - Zerinvary Lajos, Aug 04 2008 [Corrected by Georg Fischer, May 19 2019]
a(n) = n/(n-24) * a(n-1), n > 24. - Vincenzo Librandi, Mar 26 2011
From Amiram Eldar, Dec 11 2020: (Start)
Sum_{n>=24} 1/a(n) = 24/23.
Sum_{n>=24} (-1)^n/a(n) = A001787(24)*log(2) - A242091(24)/23! = 201326592*log(2) - 15566188845789952/111546435 = 0.9627768409... (End)

A010995 Binomial coefficient C(n,42).

Original entry on oeis.org

1, 43, 946, 14190, 163185, 1533939, 12271512, 85900584, 536878650, 3042312350, 15820024220, 76223753060, 343006888770, 1451182990950, 5804731963800, 22057981462440, 79960182801345, 277508869722315, 925029565741050, 2969831763694950, 9206478467454345

Offset: 42

N. J. A. Sloane

nonn

Coordination sequence for 42-dimensional cyclotomic lattice Z[zeta_43].

T. D. Noe, Table of n, a(n) for n = 42..1000
Matthias Beck and Serkan Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
Index entries for linear recurrences with constant coefficients, signature (43, -903, 12341, -123410, 962598, -6096454, 32224114, -145008513, 563921995, -1917334783, 5752004349, -15338678264, 36576848168, -78378960360, 151532656696, -265182149218, 421171648758, -608359048206, 800472431850, -960566918220, 1052049481860, -1052049481860, 960566918220, -800472431850, 608359048206, -421171648758, 265182149218, -151532656696, 78378960360, -36576848168, 15338678264, -5752004349, 1917334783, -563921995, 145008513, -32224114, 6096454, -962598, 123410, -12341, 903, -43, 1).

Cf. A010992, A010994, A001787, A242091.

[Binomial(n, 42): n in [42..70]]; // Vincenzo Librandi, Jun 12 2013

seq(binomial(n,42),n=42..57); # Zerinvary Lajos, Dec 20 2008

Table[Binomial[n,42],{n,42,70}] (* Vladimir Joseph Stephan Orlovsky, May 16 2011 *)

G.f.: x^42/(1-x)^43. - Zerinvary Lajos, Dec 20 2008
From Amiram Eldar, Dec 15 2020: (Start)
Sum_{n>=42} 1/a(n) = 42/41.
Sum_{n>=42} (-1)^n/a(n) = A001787(42)*log(2) - A242091(42)/41! = 92358976733184*log(2) - 41737723319039140166101476641/651964850415450 = 0.9777363438... (End)

A017717 Binomial coefficients C(n,53).

Original entry on oeis.org

1, 54, 1485, 27720, 395010, 4582116, 45057474, 386206920, 2944827765, 20286591270, 127805525001, 743595781824, 4027810484880, 20448884000160, 97862516286480, 443643407165376, 1913212193400684, 7877932561061640

Offset: 53

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 53..10000

Cf. A017713, A017715, A017716, A001787, A242091.

[Binomial(n,53): n in [53..80]]; // G. C. Greubel, Nov 03 2018

Table[Binomial[n,53],{n,53,80}] (* Vladimir Joseph Stephan Orlovsky, May 16 2011 *)

for(n=53, 80, print1(binomial(n,53), ", ")) \\ G. C. Greubel, Nov 03 2018

[binomial(n, 53) for n in range(53,71)] # Zerinvary Lajos, May 23 2009

From G. C. Greubel, Nov 03 2018: (Start)
G.f.: x^53/(1-x)^54.
E.g.f.: x^53*exp(x)/53!. (End)
From Amiram Eldar, Dec 16 2020: (Start)
Sum_{n>=53} 1/a(n) = 53/52.
Sum_{n>=53} (-1)^(n+1)/a(n) = A001787(53)*log(2) - A242091(53)/52! = 238690780250636288*log(2) - 12818255587371480560673756439003166003 / 77476112606149917660 = 0.9821211403... (End)

A017719 Binomial coefficients C(n,55).

Original entry on oeis.org

1, 56, 1596, 30856, 455126, 5461512, 55525372, 491796152, 3872894697, 27540584512, 179013799328, 1074082795968, 5996962277488, 31368725759168, 154603005527328, 721480692460864, 3201570572795084, 13559593014190944

Offset: 55

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 55..10000

Cf. A001787, A242091.

[Binomial(n,55): n in [55..80]]; // G. C. Greubel, Nov 03 2018

Table[Binomial[n,55],{n,55,80}] (* Vladimir Joseph Stephan Orlovsky, May 16 2011 *)

for(n=55, 80, print1(binomial(n,55), ", ")) \\ G. C. Greubel, Nov 03 2018

[binomial(n, 55) for n in range(55,73)] # Zerinvary Lajos, May 23 2009

From G. C. Greubel, Nov 03 2018: (Start)
G.f.: x^55/(1-x)^56.
E.g.f.: x^55*exp(x)/55!. (End)
From Amiram Eldar, Dec 16 2020: (Start)
Sum_{n>=55} 1/a(n) = 55/54.
Sum_{n>=55} (-1)^(n+1)/a(n) = A001787(55)*log(2) - A242091(55)/54! = 990791918021509120*log(2) - 25636511174742961236844310374211301851 / 37329399710235869418 = 0.9827390452... (End)

A017764 a(n) = binomial coefficient C(n,100).

Original entry on oeis.org

1, 101, 5151, 176851, 4598126, 96560646, 1705904746, 26075972546, 352025629371, 4263421511271, 46897636623981, 473239787751081, 4416904685676756, 38393094575497956, 312629484400483356, 2396826047070372396, 17376988841260199871, 119594570260437846171

Offset: 100

N. J. A. Sloane

More generally, the ordinary generating function for the binomial coefficients C(n,k) is x^k/(1 - x)^(k+1). - Ilya Gutkovskiy, Mar 21 2016

G. C. Greubel, Table of n, a(n) for n = 100..1100

Cf. similar sequences of the binomial coefficients C(n,k): A000012 (k = 0), A001477 (k = 1), A000217 (k = 2), A000292 (k = 3), A000332 (k = 4), A000389 (k = 5), A000579-A000582 (k = 6..9) A001287 (k = 10), A001288 (k = 11), A010965-A011001 (k = 12..48), A017713-A017763 (k = 49..99), this sequence (k = 100).

Cf. A001787, A242091.

[Binomial(n,100): n in [100..130]]; // G. C. Greubel, Nov 24 2017

Table[Binomial[n, 100], {n, 100, 5!}] (* Vladimir Joseph Stephan Orlovsky, Sep 25 2008 *)

a(n)=binomial(n,100) \\ Charles R Greathouse IV, Jun 28 2012

A017764_list, m = [], [1]*101
for _ in range(10**2):
    A017764_list.append(m[-1])
    for i in range(100):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

[binomial(n, 100) for n in range(100,115)] # Zerinvary Lajos, May 23 2009

G.f.: x^100/(1 - x)^101. - Ilya Gutkovskiy, Mar 21 2016
E.g.f.: x^100 * exp(x)/(100)!. - G. C. Greubel, Nov 24 2017
From Amiram Eldar, Dec 20 2020: (Start)
Sum_{n>=100} 1/a(n) = 100/99.
Sum_{n>=100} (-1)^n/a(n) = A001787(100)*log(2) - A242091(100)/99! = 63382530011411470074835160268800*log(2) - 1914409165727592211172313915606932788039791776845041612575266508424929 / 43575234518570298227833630584570189723 = 0.9902877001... (End)

A010973 a(n) = binomial(n,20).

Original entry on oeis.org

1, 21, 231, 1771, 10626, 53130, 230230, 888030, 3108105, 10015005, 30045015, 84672315, 225792840, 573166440, 1391975640, 3247943160, 7307872110, 15905368710, 33578000610, 68923264410, 137846528820, 269128937220, 513791607420, 960566918220, 1761039350070

Offset: 20

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 = 20..1000
Milan Janjic, Two Enumerative Functions.
Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Pascal's triangle A007318 diagonal. - Zerinvary Lajos, Aug 04 2008

Cf. A001787, A242091.

[ Binomial(n,20): n in [20..80]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,20),n=20..40); # Zerinvary Lajos, Aug 04 2008

Table[Binomial[n,20],{n,20,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

for(n=20,50, print1(binomial(n,20), ", ")) \\ G. C. Greubel, Nov 23 2017

a(n+19) = 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)*(n+19)/20!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
G.f.: x^20/(1-x)^21. - Zerinvary Lajos, Aug 04 2008; R. J. Mathar, Jul 07 2009
a(n) = n/(n-20) * a(n-1), n > 20. - Vincenzo Librandi, Mar 26 2011
From Amiram Eldar, Dec 11 2020: (Start)
Sum_{n>=20} 1/a(n) = 20/19.
Sum_{n>=20} (-1)^n/a(n) = A001787(20)*log(2) - A242091(20)/19! = 10485760*log(2) - 21149710469086/2909907 = 0.9562240549... (End)

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

A010975 a(n) = binomial(n,22).

Original entry on oeis.org

1, 23, 276, 2300, 14950, 80730, 376740, 1560780, 5852925, 20160075, 64512240, 193536720, 548354040, 1476337800, 3796297200, 9364199760, 22239974430, 51021117810, 113380261800, 244662670200, 513791607420, 1052049481860, 2104098963720, 4116715363800

Offset: 22

N. J. A. Sloane

nonn

Coordination sequence for 22-dimensional cyclotomic lattice Z[zeta_23].

T. D. Noe, Table of n, a(n) for n = 22..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 (23, -253, 1771, -8855, 33649, -100947, 245157, -490314, 817190, -1144066, 1352078, -1352078, 1144066, -817190, 490314, -245157, 100947, -33649, 8855, -1771, 253, -23, 1).

Pascal's triangle A007318. - Zerinvary Lajos, Aug 04 2008

Cf. A001787, A242091.

[ Binomial(n,22): n in [22..80]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,22),n=22..42); # Zerinvary Lajos, Aug 04 2008

Binomial[Range[22,50],22]  (* Harvey P. Dale, Apr 02 2011 *)

for(n=22, 50, print1(binomial(n,22), ", ")) \\ G. C. Greubel, Nov 23 2017

a(n) = n/(n-22) * a(n-1), n > 22. - Vincenzo Librandi, Mar 26 2011
G.f.: x^22/(1-x)^23. - G. C. Greubel, Nov 23 2017
From Amiram Eldar, Dec 11 2020: (Start)
Sum_{n>=22} 1/a(n) = 22/21.
Sum_{n>=22} (-1)^n/a(n) = A001787(22)*log(2) - A242091(22)/21! = 46137344*log(2) - 42299425233749/1322685 = 0.9597667941... (End)

A010978 a(n) = binomial(n,25).

Original entry on oeis.org

1, 26, 351, 3276, 23751, 142506, 736281, 3365856, 13884156, 52451256, 183579396, 600805296, 1852482996, 5414950296, 15084504396, 40225345056, 103077446706, 254661927156, 608359048206, 1408831480056, 3169870830126, 6943526580276, 14833897694226, 30957699535776

Offset: 25

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 25..1000
Index entries for linear recurrences with constant coefficients, signature (26, -325, 2600, -14950, 65780, -230230, 657800, -1562275, 3124550, -5311735, 7726160, -9657700, 10400600, -9657700, 7726160, -5311735, 3124550, -1562275, 657800, -230230, 65780, -14950, 2600, -325, 26, -1).

Cf. A010970, A010971, A010972, A001787, A242091.

[Binomial(n, 25): n in [25..50]]; // Vincenzo Librandi, Jun 12 2013

seq(binomial(n,25),n=25..41); # Zerinvary Lajos, Aug 18 2008

Table[Binomial[n,25],{n,25,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2011 *)

x='x+O('x^50); Vec(x^25/(1-x)^26) \\ G. C. Greubel, Nov 23 2017

From Zerinvary Lajos, Aug 18 2008: (Start)
a(n) = C(n,25), n >= 25.
G.f.: x^25/(1-x)^26. (End) [G.f. corrected by Georg Fischer, May 19 2019]
From Amiram Eldar, Dec 11 2020: (Start)
Sum_{n>=25} 1/a(n) = 25/24.
Sum_{n>=25} (-1)^(n+1)/a(n) = A001787(25)*log(2) - A242091(25)/24! = 419430400*log(2) - 155661889015631695/535422888 = 0.9641184185... (End)

cp's OEIS Frontend

A010974 a(n) = binomial(n,21).

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A010976 Binomial coefficient C(n,23).

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A010977 a(n) = binomial coefficient C(n,24).

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A010995 Binomial coefficient C(n,42).

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A017717 Binomial coefficients C(n,53).

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A017719 Binomial coefficients C(n,55).

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A017764 a(n) = binomial coefficient C(n,100).

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