A242091 -id:A242091 - cp's OEIS frontend

A010997 a(n) = binomial coefficient C(n,44).

1, 45, 1035, 16215, 194580, 1906884, 15890700, 115775100, 752538150, 4431613550, 23930713170, 119653565850, 558383307300, 2448296039700, 10142940735900, 39895566894540, 149608375854525, 536830054536825, 1849081298960175, 6131164307078475, 19619725782651120

Offset: 44

N. J. A. Sloane

nonn

T. D. Noe, Table of n, a(n) for n = 44..1000
Index entries for linear recurrences with constant coefficients, signature (45, -990, 14190, -148995, 1221759, -8145060, 45379620, -215553195, 886163135, -3190187286, 10150595910, -28760021745, 73006209045, -166871334960, 344867425584, -646626422970, 1103068603890, -1715884494940, 2438362177020, -3169870830126, 3773655750150, -4116715363800, 4116715363800, -3773655750150, 3169870830126, -2438362177020, 1715884494940, -1103068603890, 646626422970, -344867425584, 166871334960, -73006209045, 28760021745, -10150595910, 3190187286, -886163135, 215553195, -45379620, 8145060, -1221759, 148995, -14190, 990, -45, 1).

Cf. A010995, A010996, A001787, A242091.

[Binomial(n, 44): n in [44..70]]; // Vincenzo Librandi, Jun 12 2013

seq(binomial(n,44),n=44..67); # Zerinvary Lajos, Dec 20 2008

Table[Binomial[n,44],{n,44,70}] (* Vladimir Joseph Stephan Orlovsky, May 16 2011 *)

G.f.: x^44/(1-x)^45. - Zerinvary Lajos, Dec 20 2008
From Amiram Eldar, Dec 15 2020: (Start)
Sum_{n>=44} 1/a(n) = 44/43.
Sum_{n>=44} (-1)^n/a(n) = A001787(44)*log(2) - A242091(44)/43! = 387028092977152*log(2) - 7178888410874815560070307159852/26760193632961425 = 0.9786869603... (End)

A010998 a(n) = binomial coefficient C(n,45).

Original entry on oeis.org

1, 46, 1081, 17296, 211876, 2118760, 18009460, 133784560, 886322710, 5317936260, 29248649430, 148902215280, 707285522580, 3155581562280, 13298522298180, 53194089192720, 202802465047245, 739632519584070, 2588713818544245, 8719878125622720, 28339603908273840

Offset: 45

N. J. A. Sloane

nonn

T. D. Noe, Table of n, a(n) for n = 45..1000
Index entries for linear recurrences with constant coefficients, signature (46, -1035, 15180, -163185, 1370754, -9366819, 53524680, -260932815, 1101716330, -4076350421, 13340783196, -38910617655, 101766230790, -239877544005, 511738760544, -991493848554, 1749695026860, -2818953098830, 4154246671960, -5608233007146, 6943526580276, -7890371113950, 8233430727600, -7890371113950, 6943526580276, -5608233007146, 4154246671960, -2818953098830, 1749695026860, -991493848554, 511738760544, -239877544005, 101766230790, -38910617655, 13340783196, -4076350421, 1101716330, -260932815, 53524680, -9366819, 1370754, -163185, 15180, -1035, 46, -1).

Cf. A010996, A010997, A001787, A242091.

[Binomial(n, 45): n in [45..70]]; // Vincenzo Librandi, Jun 12 2013

seq(binomial(n,45),n=45..67); # Zerinvary Lajos, Dec 20 2008

Table[Binomial[n,45],{n,45,77}] (* Vladimir Joseph Stephan Orlovsky, May 16 2011 *)

G.f.: x^45/(1-x)^46. - Zerinvary Lajos, Dec 20 2008
From Amiram Eldar, Dec 15 2020: (Start)
Sum_{n>=45} 1/a(n) = 45/44.
Sum_{n>=45} (-1)^(n+1)/a(n) = A001787(45)*log(2) - A242091(45)/44! = 791648371998720*log(2) - 14357776821749657880334247281129/26165522663340060 = 0.9791324188... (End)

A010999 a(n) = binomial coefficient C(n,46).

Original entry on oeis.org

1, 47, 1128, 18424, 230300, 2349060, 20358520, 154143080, 1040465790, 6358402050, 35607051480, 184509266760, 891794789340, 4047376351620, 17345898649800, 70539987842520, 273342452889765, 1012974972473835, 3601688791018080, 12321566916640800, 40661170824914640

Offset: 46

N. J. A. Sloane

nonn

Coordination sequence for 46-dimensional cyclotomic lattice Z[zeta_47].

T. D. Noe, Table of n, a(n) for n = 46..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 (47, -1081, 16215, -178365, 1533939, -10737573, 62891499, -314457495, 1362649145, -5178066751, 17417133617, -52251400851, 140676848445, -341643774795, 751616304549, -1503232609098, 2741188875414, -4568648125690, 6973199770790, -9762479679106, 12551759587422, -14833897694226, 16123801841550, -16123801841550, 14833897694226, -12551759587422, 9762479679106, -6973199770790, 4568648125690, -2741188875414, 1503232609098, -751616304549, 341643774795, -140676848445, 52251400851, -17417133617, 5178066751, -1362649145, 314457495, -62891499, 10737573, -1533939, 178365, -16215, 1081, -47, 1).

Cf. A010997, A010998, A001787, A242091.

[Binomial(n, 46): n in [46..70]]; // Vincenzo Librandi, Jun 12 2013

seq(binomial(n,46),n=46..67); # Zerinvary Lajos, Dec 20 2008

Table[Binomial[n,46],{n,46,77}] (* Vladimir Joseph Stephan Orlovsky, May 16 2011 *)

G.f.: x^46/(1-x)^47. - Zerinvary Lajos, Dec 20 2008
From Amiram Eldar, Dec 15 2020: (Start)
Sum_{n>=46} 1/a(n) = 46/45.
Sum_{n>=46} (-1)^n/a(n) = A001787(46)*log(2) - A242091(46)/45! = 1618481116086272*log(2) - 14357776821749670963095578951159/12798353476633725 = 0.9795596119... (End)

A017714 Binomial coefficients C(n,50).

Original entry on oeis.org

1, 51, 1326, 23426, 316251, 3478761, 32468436, 264385836, 1916797311, 12565671261, 75394027566, 418094152866, 2160153123141, 10468434365991, 47855699958816, 207374699821536, 855420636763836, 3371363686069236, 12736262814039336, 46252743903616536

Offset: 50

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 50..10000

Cf. A001787, A242091.

[Binomial(n,50): n in [50..80]]; // G. C. Greubel, Nov 03 2018

Table[Binomial[n, 50], {n, 50, 5!}] (* Vladimir Joseph Stephan Orlovsky, Sep 25 2008 *)

for(n=50, 80, print1(binomial(n,50), ", ")) \\ G. C. Greubel, Nov 03 2018

A017714_list, m = [], [1]*51
for _ in range(10**2):
    A017714_list.append(m[-1])
    for i in range(50):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

[binomial(n, 50) for n in range(50,68)] # Zerinvary Lajos, May 23 2009

From G. C. Greubel, Nov 03 2018: (Start)
G.f.: x^50/(1-x)^51.
E.g.f.: x^50*exp(x)/50!. (End)
From Amiram Eldar, Dec 16 2020: (Start)
Sum_{n>=50} 1/a(n) = 50/49.
Sum_{n>=50} (-1)^n/a(n) = A001787(50)*log(2) - A242091(50)/49! = 28147497671065600*log(2) - 302317348758761320570288162183704329 / 15495222521229983532 = 0.9811065191... (End)

A017715 Binomial coefficients C(n,51).

Original entry on oeis.org

1, 52, 1378, 24804, 341055, 3819816, 36288252, 300674088, 2217471399, 14783142660, 90177170226, 508271323092, 2668424446233, 13136858812224, 60992558771040, 268367258592576, 1123787895356412, 4495151581425648

Offset: 51

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 51..10000

Cf. A001787, A242091.

[Binomial(n,51): n in [51..80]]; // G. C. Greubel, Nov 03 2018

Table[Binomial[n,51],{n,51,77}] (* Vladimir Joseph Stephan Orlovsky, May 16 2011 *)

for(n=51, 80, print1(binomial(n,51), ", ")) \\ G. C. Greubel, Nov 03 2018

[binomial(n, 51) for n in range(51,69)] # Zerinvary Lajos, May 23 2009

From G. C. Greubel, Nov 03 2018: (Start)
G.f.: x^51/(1-x)^52.
E.g.f.: x^51*exp(x)/51!. (End)
From Amiram Eldar, Dec 16 2020: (Start)
Sum_{n>=51} 1/a(n) = 51/50.
Sum_{n>=51} (-1)^(n+1)/a(n) = A001787(51)*log(2) - A242091(51)/50! = 57420895248973824*log(2) - 60463469751752265663579884559739219 / 1519139462865684660 = 0.9814572990... (End)

A017716 Binomial coefficients C(n,52).

Original entry on oeis.org

1, 53, 1431, 26235, 367290, 4187106, 40475358, 341149446, 2558620845, 17341763505, 107518933731, 615790256823, 3284214703056, 16421073515280, 77413632286320, 345780890878896, 1469568786235308, 5964720367660956

Offset: 52

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 52..10000

Cf. A017713, A017715, A001787, A242091.

[Binomial(n,52): n in [52..80]]; // G. C. Greubel, Nov 03 2018

Table[Binomial[n,52],{n,52,80}] (* Vladimir Joseph Stephan Orlovsky, May 16 2011 *)

for(n=52, 80, print1(binomial(n,52), ", ")) \\ G. C. Greubel, Nov 03 2018

[binomial(n, 52) for n in range(52,70)] # Zerinvary Lajos, May 23 2009

From G. C. Greubel, Nov 03 2018: (Start)
G.f.: x^52/(1-x)^53.
E.g.f.: x^52*exp(x)/52!. (End)
From Amiram Eldar, Dec 16 2020: (Start)
Sum_{n>=52} 1/a(n) = 52/51.
Sum_{n>=52} (-1)^n/a(n) = A001787(52)*log(2) - A242091(52)/51! = 117093590311632896*log(2) - 120926939503504532846299231985163098 / 1489925242425959955 = 0.9817952764... (End)

A017760 Binomial coefficients C(n,96).

Original entry on oeis.org

1, 97, 4753, 156849, 3921225, 79208745, 1346548665, 19813501785, 257575523205, 3005047770725, 31853506369685, 309847743777845, 2788629694000605, 23381587434312765, 183712472698171725, 1359472297966470765, 9516306085765295355, 63255446334792845595

Offset: 96

N. J. A. Sloane

Michael De Vlieger, Table of n, a(n) for n = 96..10000

Cf. A017761, A017762, A001787, A242091.

[Binomial(n,96): n in [96..110]]; // G. C. Greubel, Nov 12 2018

Table[Binomial[n, 96], {n, 96, 120}] (* Vladimir Joseph Stephan Orlovsky, Jul 15 2011 *)

a(n)=binomial(n,96) \\ Charles R Greathouse IV, Jun 28 2012

[binomial(n, 96) for n in range(96,111)] # Zerinvary Lajos, May 23 2009

From G. C. Greubel, Nov 12 2018: (Start)
G.f.: x^96/(1-x)^97.
E.g.f.: x^96*exp(x)/96!. (End)
From Amiram Eldar, Dec 20 2020: (Start)
Sum_{n>=96} 1/a(n) = 96/95.
Sum_{n>=96} (-1)^n/a(n) = A001787(96)*log(2) - A242091(96)/95! = 3802951800684688204490109616128*log(2) - 9868088483131918614290277915496170231743780878869426700864981897216 / 3743576848674424246377459672213933825 = 0.9898949826... (End)

A017761 Binomial coefficients C(n,97).

Original entry on oeis.org

1, 98, 4851, 161700, 4082925, 83291670, 1429840335, 21243342120, 278818865325, 3283866636050, 35137373005735, 344985116783580, 3133614810784185, 26515202245096950, 210227674943268675, 1569699972909739440, 11086006058675034795, 74341452393467880390

Offset: 97

N. J. A. Sloane

Michael De Vlieger, Table of n, a(n) for n = 97..10000

Cf. A017762, A001787, A242091.

[Binomial(n,97): n in [97..110]]; // G. C. Greubel, Nov 12 2018

Table[Binomial[n, 97], {n, 97, 120}] (* Vladimir Joseph Stephan Orlovsky, Jul 15 2011 *)

a(n)=binomial(n,97) \\ Charles R Greathouse IV, Jun 28 2012

[binomial(n, 97) for n in range(97,112)] # Zerinvary Lajos, May 23 2009

From G. C. Greubel, Nov 12 2018: (Start)
G.f.: x^97/(1-x)^98.
E.g.f.: x^97*exp(x)/97!. (End)
From Amiram Eldar, Dec 20 2020: (Start)
Sum_{n>=97} 1/a(n) = 97/96.
Sum_{n>=97} (-1)^(n+1)/a(n) = A001787(97)*log(2) - A242091(97)/96! = 7685131763883640746573763182592*log(2) - 1914409165727592211172313915606620151912614909652567393556011239640929 / 359383377472744727652236128532537647200 = 0.9899961107... (End)

A017762 Binomial coefficients C(n,98).

Original entry on oeis.org

1, 99, 4950, 166650, 4249575, 87541245, 1517381580, 22760723700, 301579589025, 3585446225075, 38722819230810, 383707936014390, 3517322746798575, 30032524991895525, 240260199935164200, 1809960172844903640, 12895966231519938435, 87237418624987818825

Offset: 98

N. J. A. Sloane

Michael De Vlieger, Table of n, a(n) for n = 98..10000

Cf. A001787, A242091.

[Binomial(n,98): n in [98..115]]; // G. C. Greubel, Nov 12 2018

A017762:=n->binomial(n,98); seq(A017762(k), k=98..200); # Wesley Ivan Hurt, Nov 05 2013

Table[Binomial[n, 98], {n, 98, 120}] (* Vladimir Joseph Stephan Orlovsky, Jul 15 2011 *)

a(n)=binomial(n,98) \\ Charles R Greathouse IV, Jun 28 2012

[binomial(n, 98) for n in range(98,113)] # Zerinvary Lajos, May 23 2009

From G. C. Greubel, Nov 12 2018: (Start)
G.f.: x^98/(1-x)^99.
E.g.f.: x^98*exp(x)/98!. (End)
From Amiram Eldar, Dec 20 2020: (Start)
Sum_{n>=98} 1/a(n) = 98/97.
Sum_{n>=98} (-1)^n/a(n) = A001787(98)*log(2) - A242091(98)/97! = 15528719852795810168334614265856*log(2) - 1914409165727592211172313915606799843601351282016393511620277508464529 / 177858100075797135623810737079878325400 = 0.9900952340... (End)

A010969 a(n) = binomial(n,16).

Original entry on oeis.org

1, 17, 153, 969, 4845, 20349, 74613, 245157, 735471, 2042975, 5311735, 13037895, 30421755, 67863915, 145422675, 300540195, 601080390, 1166803110, 2203961430, 4059928950, 7307872110, 12875774670, 22239974430, 37711260990, 62852101650, 103077446706, 166509721602

Offset: 16

N. J. A. Sloane

nonn

a(n) = A110555(n+1,16). - Reinhard Zumkeller, Jul 27 2005
Coordination sequence for 16-dimensional cyclotomic lattice Z[zeta_17].
In this sequence only 17 is prime. - Artur Jasinski, Dec 02 2007

T. D. Noe, Table of n, a(n) for n = 16..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 (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

[ Binomial(n,16): n in [16..80]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,16),n=16..37); # Zerinvary Lajos, Aug 06 2008

Table[Binomial[n,16],{n,16,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

for(n=16, 50, print1(binomial(n,16), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n+15) = 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)/16!. - Artur Jasinski, Dec 02 2007
G.f.: x^16/(1-x)^17. - Zerinvary Lajos, Aug 06 2008; R. J. Mathar, Jul 07 2009
a(n) = n/(n-16) * a(n-1), n > 16. - Vincenzo Librandi, Mar 26 2011
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=16} 1/a(n) = 16/15.
Sum_{n>=16} (-1)^n/a(n) = A001787(16)*log(2) - A242091(16)/15! = 524288*log(2) - 16369704448/45045 = 0.9468480104... (End)

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

cp's OEIS Frontend

A010997 a(n) = binomial coefficient C(n,44).

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A010998 a(n) = binomial coefficient C(n,45).

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A010999 a(n) = binomial coefficient C(n,46).

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A017714 Binomial coefficients C(n,50).

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A017715 Binomial coefficients C(n,51).

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A017716 Binomial coefficients C(n,52).

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A017760 Binomial coefficients C(n,96).

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