A242091 -id:A242091 - cp's OEIS frontend

A011001 Binomial coefficient C(n,48).

1, 49, 1225, 20825, 270725, 2869685, 25827165, 202927725, 1420494075, 8996462475, 52179482355, 279871768995, 1399358844975, 6566222272575, 29078984349975, 122131734269895, 488526937079580, 1867897112363100, 6848956078664700, 24151581961607100

Offset: 48

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 48..1000
Index entries for linear recurrences with constant coefficients, signature (49, -1176, 18424, -211876, 1906884, -13983816, 85900584, -450978066, 2054455634, -8217822536, 29135916264, -92263734836, 262596783764, -675248872536, 1575580702584, -3348108992991, 6499270398159, -11554258485616, 18851684897584, -28277527346376, 39049918716424, -49699896548176, 58343356817424, -63205303218876, 63205303218876, -58343356817424, 49699896548176, -39049918716424, 28277527346376, -18851684897584, 11554258485616, -6499270398159, 3348108992991, -1575580702584, 675248872536, -262596783764, 92263734836, -29135916264, 8217822536, -2054455634, 450978066, -85900584, 13983816, -1906884, 211876, -18424, 1176, -49, 1).

Cf. A010999, A011000, A001787, A242091.

[Binomial(n, 48): n in [48..70]]; // Vincenzo Librandi, Jun 12 2013

seq(binomial(n,48),n=48..67); # Zerinvary Lajos, Dec 20 2008

Table[Binomial[n,48],{n,48,77}] (* Vladimir Joseph Stephan Orlovsky, May 16 2011 *)

a(n)=binomial(n,48)  \\ Charles R Greathouse IV, Jan 08 2013

A011001_list, m = [], [1]*49
for _ in range(10**2):
    A011001_list.append(m[-1])
    for i in range(48):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

G.f.: x^48/(1-x)^49. - Zerinvary Lajos, Dec 20 2008
From Amiram Eldar, Dec 15 2020: (Start)
Sum_{n>=48} 1/a(n) = 48/47.
Sum_{n>=48} (-1)^n/a(n) = A001787(48)*log(2) - A242091(48)/47! = 6755399441055744*log(2) - 21594096339911519462651644572315136 / 4611673369413685575 = 0.9803635237... (End)

A000580 a(n) = binomial coefficient C(n,7).

Original entry on oeis.org

1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 11440, 19448, 31824, 50388, 77520, 116280, 170544, 245157, 346104, 480700, 657800, 888030, 1184040, 1560780, 2035800, 2629575, 3365856, 4272048, 5379616, 6724520, 8347680, 10295472

Offset: 7

N. J. A. Sloane

Figurate numbers based on 7-dimensional regular simplex. According to Hyun Kwang Kim, it appears that every nonnegative integer can be represented as the sum of g = 15 of these numbers. - Jonathan Vos Post, Nov 28 2004
a(n) is the number of terms in the expansion of (Sum_{i=1..8} a_i)^n. - Sergio Falcon, Feb 12 2007
Product of seven consecutive numbers divided by 7!. - Artur Jasinski, Dec 02 2007
In this sequence there are no primes. - Artur Jasinski, Dec 02 2007
For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 6 elements, which is 3*C(n+1,7) (for n>=6), hence a(n) = 3*C(n+1,7) = 3*A000580(n+1). - Serhat Bulut, Mar 13 2015
Partial sums of A000579. In general, the iterated sums S(m, n) = Sum_{j=1..n} S(m-1, j) with input S(1, n) = A000217(n) = 1 + 2 + ... + n are S(m, n) = risefac(n, m+1)/(m+1)! = binomial(n+m, m+1) = Sum_{k = 1..n} risefac(k, m)/m!, with the rising factorials risefac(x, m):= Product_{j=0..m-1} (x+j), for m >= 1. Such iterated sums of arithmetic progressions have been considered by Narayana Pandit (see The MacTutor History of Mathematics archive link, and the Gottwald et al. reference, p. 338, where the name Narayana Daivajna is also used). - Wolfdieter Lang, Mar 20 2015
a(n) = fallfac(n,7)/7! = binomial(n, 7) is also the number of independent components of an antisymmetric tensor of rank 7 and dimension n >= 7 (for n=1..6 this becomes 0). Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
From Juergen Will, Jan 02 2016: (Start)
Number of compositions (ordered partitions) of n+1 into exactly 8 parts.
Number of weak compositions (ordered weak partitions) of n-7 into exactly 8 parts. (End)

			For A={1,2,3,4,5,6,7}, subsets with 6 elements are {1,2,3,4,5,6}, {1,2,3,4,5,7}, {1,2,3,4,6,7}, {1,2,3,5,6,7}, {1,2,4,5,6,7}, {1,3,4,5,6,7}, {2,3,4,5,6,7}.
Sum of 2 smallest elements of each subset: a(7) = (1+2)+(1+2)+(1+2)+(1+2)+(1+2)+(1+3)+(2+3) = 24 = 3*C(7+1,7) = 3*A000580(7+1). - _Serhat Bulut_, Mar 13 2015

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
S. Gottwald, H.-J. Ilgauds and K.-H. Schlote (eds.), Lexikon bedeutender Mathematiker (in German), Bibliographisches Institut Leipzig, 1990.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 7..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Serhat Bulut and Oktay Erkan Temizkan, Subset Sum Problem, 2015
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Philippe A. J. G. Chevalier, On the discrete geometry of physical quantities, Preprint, 2012.
Philippe A. J. G. Chevalier, A "table of Mendeleev" for physical quantities?, Slides from a talk, May 14 2014, Leuven, Belgium.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 257.
Milan Janjic, Two Enumerative Functions.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901. - _Juergen Will_, Jan 02 2016
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
The MacTutor History of Mathematics archive, Narayana Pandit.
Eric Weisstein's World of Mathematics, Composition.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A053136, A053129, A000579, A000581, A000582, A000217, A000292, A000332, A000389, A104712, A001787, A242091.

[Binomial(n,7): n in [7..40]]; // Vincenzo Librandi, Mar 21 2015

ZL := [S, {S=Prod(B,B,B,B,B,B,B,B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=8..38); # Zerinvary Lajos, Mar 13 2007
A000580:=1/(z-1)**8; # Simon Plouffe in his 1992 dissertation, offset 0.
seq(binomial(n+7,7)*1^n,n=0..30); # Zerinvary Lajos, Jun 23 2008
G(x):=x^7*exp(x): f[0]:=G(x): for n from 1 to 38 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n]/7!,n=7..37); # Zerinvary Lajos, Apr 05 2009

Table[n(n + 1)(n + 2)(n + 3)(n + 4)(n + 5)(n + 6)/7!, {n, 1, 100}] (* Artur Jasinski, Dec 02 2007 *)
Binomial[Range[7,40],7] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{1,8,36,120,330,792,1716,3432},40] (* Harvey P. Dale, Nov 28 2011 *)
CoefficientList[Series[1 / (1-x)^8, {x, 0, 33}], x] (* Vincenzo Librandi, Mar 21 2015 *)

a(n)=binomial(n,7) \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x^7/(1-x)^8.
a(n) = (n^7 - 21*n^6 + 175*n^5 - 735*n^4 + 1624*n^3 - 1764*n^2 + 720*n)/5040.
a(n) = -A110555(n+1,7). - Reinhard Zumkeller, Jul 27 2005
Convolution of the nonnegative numbers (A001477) with the sequence A000579. Also convolution of the triangular numbers (A000217) with the sequence A000332. Also convolution of the sequence {1,1,1,1,...} (A000012) with the sequence A000579. Also self-convolution of the tetrahedral numbers (A000292). - Sergio Falcon, Feb 12 2007
a(n+4) = (1/3!)*(d^3/dx^3)S(n,x)|A049310.%20-%20_Wolfdieter%20Lang">{x=2}, n >= 3. One sixth of third derivative of Chebyshev S-polynomials evaluated at x=2. See A049310. - _Wolfdieter Lang, Apr 04 2007
a(n) = n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)/7!. - Artur Jasinski, Dec 02 2007, R. J. Mathar, Jul 07 2009
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) with a(7)=1, a(8)=8, a(9)=36, a(10)=120, a(11)=330, a(12)=792, a(13)=1716, a(14)=3432. - Harvey P. Dale, Nov 28 2011
a(n) = 3*C(n+1,7) = 3*A000580(n+1). - Serhat Bulut, Mar 13 2015
From Wolfdieter Lang, Mar 21 2015: (Start)
a(n) = A104712(n, 7), n >= 7.
a(n+6) = sum(A000579(j+5), j = 1..n), n >= 1. See the Mar 20 2015 comment above. (End)
Sum_{k >= 7} 1/a(k) = 7/6. - Tom Edgar, Sep 10 2015
Sum_{n>=7} (-1)^(n+1)/a(n) = A001787(7)*log(2) - A242091(7)/6! = 448*log(2) - 9289/30 = 0.8966035575... - Amiram Eldar, Dec 10 2020

More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000

Some formulas that referred to other offsets corrected by R. J. Mathar, Jul 07 2009

A000581 a(n) = binomial coefficient C(n,8).

Original entry on oeis.org

1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, 43758, 75582, 125970, 203490, 319770, 490314, 735471, 1081575, 1562275, 2220075, 3108105, 4292145, 5852925, 7888725, 10518300, 13884156, 18156204, 23535820, 30260340, 38608020, 48903492, 61523748, 76904685

Offset: 8

N. J. A. Sloane

Figurate numbers based on 8-dimensional regular simplex. - Jonathan Vos Post, Nov 28 2004
Just as A005712 and A000574 are described as the coefficients of x^4 and x^5 in the expansion of (1+x+x^2)^n, so should this sequence be described as the coefficients of x^3 therein. - R. K. Guy, Oct 19 2007
Product of 8 consecutive numbers divided by 8!. - Artur Jasinski, Dec 02 2007
In this sequence there are no primes. - Artur Jasinski, Dec 02 2007
a(n) = number of (n-8)-digit numbers with nondescending digits. E.g., a(9) = 9 = {1,2,3,..,9}, a(10) = 45 = {11-19, 22-29, 33-39, ..., 99} [0 is counted as a zero-digit number rather than a 1-digit number]. - Toby Gottfried, Feb 14 2012
a(n) =fallfac(n, 8)/8! = binomial(n, 8) is also the number of independent components of an antisymmetric tensor of rank 8 and dimension n >= 8 (for n = 1..7 this becomes 0). Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
Number of compositions (ordered partitions) of n+1 into exactly 9 parts. - Juergen Will, Jan 02 2016
Number of weak compositions (ordered weak partitions) of n-8 into exactly 9 parts. - Juergen Will, Jan 02 2016
Partial sums of A000580. - Art Baker, Mar 26 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 8..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 9.
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 258.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901. - _Juergen Will_, Jan 02 2016
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Composition.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000217, A000292, A000332, A000389, A000579, A000580, A053130, A053137, A254142, A001787, A242091.

[Binomial(n,8): n in [8..100]]; // Vincenzo Librandi, Apr 08 2011

ZL := [S, {S=Prod(B, B, B, B, B, B, B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n+1), n=8..40); # Zerinvary Lajos, Mar 13 2007
A000581:=-1/(z-1)**9; # Simon Plouffe in his 1992 dissertation, with offset 0
seq(binomial(n,8),n=8..40); # Zerinvary Lajos, Jun 23 2008

Table[Binomial[n,8],{n,8,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

a(n)=binomial(n,8) \\ Charles R Greathouse IV, Feb 14 2012

G.f.: x^8/(1-x)^9.
a(n) = A110555(n+1,8). - Reinhard Zumkeller, Jul 27 2005
a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)/8!. - Artur Jasinski, Dec 02 2007
Sum_{k>=8} 1/a(k) = 8/7. - Tom Edgar, Sep 10 2015
Sum_{n>=8} (-1)^n/a(n) = A001787(8)*log(2) - A242091(8)/7! = 1024*log(2) - 74432/105 = 0.9065224171... - Amiram Eldar, Dec 10 2020

More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000
Some formulas referring to other offsets rewritten by R. J. Mathar, Jul 07 2009
3 more terms from William Boyles, Aug 06 2015

A000582 a(n) = binomial coefficient C(n,9).

Original entry on oeis.org

1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, 92378, 167960, 293930, 497420, 817190, 1307504, 2042975, 3124550, 4686825, 6906900, 10015005, 14307150, 20160075, 28048800, 38567100, 52451256, 70607460, 94143280, 124403620, 163011640, 211915132

Offset: 9

N. J. A. Sloane

Figurate numbers based on 9-dimensional regular simplex. - Jonathan Vos Post, Nov 28 2004
Product of 9 consecutive numbers divided by 9!. - Artur Jasinski, Dec 02 2007
In this sequence there are no primes. - Artur Jasinski, Dec 02 2007
a(9+n) gives the number of words with n letters over the alphabet {0,1,..,9} such that these letters are read from left to right in weakly increasing (nondecreasing) order. - R. J. Cano, Jul 20 2014
a(n) = fallfac(n, 9)/9! = binomial(n, 9) is also the number of independent components of an antisymmetric tensor of rank 9 and dimension n >= 9 (for n=1..8 this becomes 0). Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
From Juergen Will, Jan 23 2016: (Start)
Number of compositions (ordered partitions) of n+1 into exactly 10 parts.
Number of weak compositions (ordered weak partitions) of n-9 into exactly 10 parts. (End)
Number of integers divisible by 9 in the interval [0, 10^(n-8)-1]. - Miquel Cerda, Jul 02 2017

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 9..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 259
Milan Janjic, Two Enumerative Functions
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.
P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901.
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Ch. Stover and E. W. Weisstein, Composition. From MathWorld - A Wolfram Web Resource.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A053138, A053131, A000581, A035927, A001787, A242091.

[Binomial(n,9) : n in [9..50]]; // Wesley Ivan Hurt, Jul 20 2014

A000582 := n->binomial(n,9): seq(A000582(n), n=9..40);
A000582:=1/(z-1)**10; # Simon Plouffe in his 1992 dissertation (offset 0)
seq(binomial(n,9),n=0..29); # Zerinvary Lajos, Jun 23 2008, R. J. Mathar, Jul 07 2009

Table[n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)/9!,{n,100}] (* Artur Jasinski, Dec 02 2007 *)
Table[Binomial[n, 9], {n, 9, 50}] (* Wesley Ivan Hurt, Jul 20 2014 *)

a(n)=binomial(n,9) \\ Charles R Greathouse IV, Jul 21 2014

G.f.: x^9/(1-x)^10.
a(n) = -A110555(n+1, 9). - Reinhard Zumkeller, Jul 27 2005
a(n+8) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)/9!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
Sum_{k>=9} 1/a(k) = 9/8. - Tom Edgar, Sep 10 2015
Sum_{n>=9} (-1)^(n+1)/a(n) = A001787(9)*log(2) - A242091(9)/8! = 2304*log(2) - 446907/280 = 0.9146754386... - Amiram Eldar, Dec 10 2020

Formulas referring to other offsets rewritten by R. J. Mathar, Jul 07 2009

A001287 a(n) = binomial coefficient C(n,10).

Original entry on oeis.org

1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184756, 352716, 646646, 1144066, 1961256, 3268760, 5311735, 8436285, 13123110, 20030010, 30045015, 44352165, 64512240, 92561040, 131128140, 183579396, 254186856, 348330136, 472733756, 635745396

Offset: 10

N. J. A. Sloane

Coordination sequence for 10-dimensional cyclotomic lattice Z[zeta_11].
Product of 10 consecutive numbers divided by 10!. - Artur Jasinski, Dec 02 2007
In this sequence only 11 is prime. - Artur Jasinski, Dec 02 2007
With a different offset, number of n-permutations (n>=10) of 2 objects: u,v, with repetition allowed, containing exactly 10 u's. Example: a(1)=11 because we have uuuuuuuuuuv, uuuuuuuuuvu, uuuuuuuuvuu, uuuuuuuvuuu, uuuuuuvuuuu, uuuuuvuuuuu, uuuuvuuuuuu, uuuvuuuuuuu, uuvuuuuuuuu, uvuuuuuuuuu and vuuuuuuuuuu. - Zerinvary Lajos, Aug 03 2008
a(9+k) is the number of times that each digit appears repeated inside a list made with all the possible base 10 numbers of k digits such that their digits are read in ascending order from left to right. - R. J. Cano Jul 20 2014
a(n) = fallfac(n,10)/10! = binomial(n, 10) is also the number of independent components of an antisymmetric tensor of rank 10 and dimension n >= 10 (for n=1..9 this becomes 0). Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 10..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Matthias Beck and Serkan Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 260.
Milan Janjic, Two Enumerative Functions.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A110555, A001787, A242091.

[Binomial(n,10): n in [10..40]]; // Vincenzo Librandi, Sep 11 2015

seq(binomial(n,10),n=10..31); # Zerinvary Lajos, Aug 06 2008

Table[n (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9)/10!, {n, 1, 100}] (* Artur Jasinski, Dec 02 2007 *)
Table[Binomial[n, 10], {n, 10, 20}] (* Zerinvary Lajos, Jan 31 2010 *)

a(n)=binomial(n,10) \\ Charles R Greathouse IV, Sep 24 2015

A001287_list, m = [], [1]*11
for _ in range(10**2):
    A001287_list.append(m[-1])
    for i in range(10):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

a(n) = A110555(n+1,10). - Reinhard Zumkeller, Jul 27 2005
a(n+9) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)/10!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
G.f.: x^10/(1-x)^11. - Zerinvary Lajos, Aug 06 2008; R. J. Mathar, Jul 07 2009
Sum_{k>=10} 1/a(k) = 10/9. - Tom Edgar, Sep 10 2015
Sum_{n>=10} (-1)^n/a(n) = A001787(10)*log(2) - A242091(10)/9! = 5120*log(2) - 447047/126 = 0.9215009748... - Amiram Eldar, Dec 10 2020

Formulas valid for different offsets rewritten by R. J. Mathar, Jul 07 2009

Extended by Ray Chandler, Oct 25 2011

A001288 a(n) = binomial(n,11).

Original entry on oeis.org

1, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, 705432, 1352078, 2496144, 4457400, 7726160, 13037895, 21474180, 34597290, 54627300, 84672315, 129024480, 193536720, 286097760, 417225900, 600805296, 854992152, 1203322288, 1676056044

Offset: 11

N. J. A. Sloane

Product of 11 consecutive numbers divided by 11!. - Artur Jasinski, Dec 02 2007
In this sequence there are no primes. - Artur Jasinski, Dec 02 2007
With a different offset, number of n-permutations (n>=11) of 2 objects: u,v, with repetition allowed, containing exactly (11) u's. Example: n=11, a(0)=1 because we have uuuuuuuuuuu n=12, a(1)=12 because we have uuuuuuuuuuuv, uuuuuuuuuuvu, uuuuuuuuuvuu, uuuuuuuuvuuu, uuuuuuuvuuuu, uuuuuuvuuuuu, uuuuuvuuuuuu, uuuuvuuuuuuu, uuuvuuuuuuuu, uuvuuuuuuuuu uvuuuuuuuuuu, vuuuuuuuuuuu. - Zerinvary Lajos, Aug 06 2008
Does not satisfy Benford's law (because n^11 does not, see Ross, 2012). - N. J. A. Sloane, Feb 09 2017

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=11..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
A. S. Chinchon, Mixing Benford, GoogleVis And On-Line Encyclopedia of Integer Sequences, 2014. Note: as of Feb 09 2017, the results in this page appear to be incorrect - N. J. A. Sloane, Feb 09 2017.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 261
Milan Janjic, Two Enumerative Functions.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
Ângela Mestre, José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Kenneth A. Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42.
Index entries for sequences related to Benford's law
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A110555, A001787, A242091.

seq(binomial(n,11),n=0..30); # Zerinvary Lajos, Aug 06 2008, R. J. Mathar, Jul 07 2009

Table[n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)/11!,{n,1,100}] (* Artur Jasinski, Dec 02 2007 *)
Binomial[Range[11,50],11] (* Harvey P. Dale, Oct 02 2012 *)

for(n=11, 50, print1(binomial(n,11), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n) = -A110555(n+1,11). - Reinhard Zumkeller, Jul 27 2005
a(n+10) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)/11!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
G.f.: x^11/(1-x)^12. a(n) = binomial(n,11). - Zerinvary Lajos, Aug 06 2008; R. J. Mathar, Jul 07 2009
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=11} 1/a(n) = 11/10.
Sum_{n>=11} (-1)^(n+1)/a(n) = A001787(11)*log(2) - A242091(11)/10! = 11264*log(2) - 491821/63 = 0.9273021446... (End)

Some formulas for other offsets corrected by R. J. Mathar, Jul 07 2009

A010965 a(n) = binomial(n,12).

Original entry on oeis.org

1, 13, 91, 455, 1820, 6188, 18564, 50388, 125970, 293930, 646646, 1352078, 2704156, 5200300, 9657700, 17383860, 30421755, 51895935, 86493225, 141120525, 225792840, 354817320, 548354040, 834451800, 1251677700, 1852482996, 2707475148, 3910797436, 5586853480

Offset: 12

N. J. A. Sloane

nonn

Coordination sequence for 12-dimensional cyclotomic lattice Z[zeta_13].

In this sequence only 13 is prime. - Artur Jasinski, Dec 02 2007

T. D. Noe, Table of n, a(n) for n = 12..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 (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A000581, A010966, A010967, A001787, A242091.

[Binomial(n, 12): n in [12..100]]; // Vincenzo Librandi, Apr 22 2011

seq(binomial(n,12),n=12..36); # Zerinvary Lajos, Aug 06 2008

Table[Binomial[n,12],{n,12,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

for(n=12, 50, print1(binomial(n,12), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n) = A110555(n+1,12). - Reinhard Zumkeller, Jul 27 2005
a(n+11) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)(n+11)/12!. - Artur Jasinski, Dec 02 2007, R. J. Mathar, Jul 07 2009
G.f.: x^12/(1-x)^13. - Zerinvary Lajos, Aug 06 2008, R. J. Mathar, Jul 07 2009
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=12} 1/a(n) = 12/11.
Sum_{n>=12} (-1)^n/a(n) = A001787(12)*log(2) - A242091(12)/11! = 24576*log(2) - 3934820/231 = 0.9322955884... (End)

Some formulas referring to other offsets corrected by R. J. Mathar, Jul 07 2009

A010966 a(n) = binomial(n,13).

Original entry on oeis.org

1, 14, 105, 560, 2380, 8568, 27132, 77520, 203490, 497420, 1144066, 2496144, 5200300, 10400600, 20058300, 37442160, 67863915, 119759850, 206253075, 347373600, 573166440, 927983760, 1476337800, 2310789600, 3562467300, 5414950296, 8122425444, 12033222880

Offset: 13

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 = 13..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 (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Cf. A000581, A001787, A110555, A242091.

[ Binomial(n,13): n in [13..50]]; // Vincenzo Librandi, Mar 26 2011

seq(binomial(n,13),n=13..36); # Zerinvary Lajos, Aug 06 2008

Table[Binomial[n,13],{n,13,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

for(n=13, 50, print1(binomial(n,13), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n) = -A110555(n+1,13). - Reinhard Zumkeller, Jul 27 2005
a(n+12) = 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)/13!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
G.f.: x^13/(1-x)^14. - Zerinvary Lajos, Aug 06 2008
a(n) = n/(n-13) * a(n-1), n > 13. - Vincenzo Librandi, Mar 26 2011
From Amiram Eldar, Dec 10 2020: (Start)
Sum_{n>=13} 1/a(n) = 13/12.
Sum_{n>=13} (-1)^(n+1)/a(n) = A001787(13)*log(2) - A242091(13)/12! = 53248*log(2) - 102308323/2772 = 0.9366404415... (End)

Some formulas for different offsets rewritten by R. J. Mathar, Jul 07 2009

A017713 Binomial coefficients C(n,49).

Original entry on oeis.org

1, 50, 1275, 22100, 292825, 3162510, 28989675, 231917400, 1652411475, 10648873950, 62828356305, 342700125300, 1742058970275, 8308281242850, 37387265592825, 159518999862720, 648045936942300, 2515943049305400

Offset: 49

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 49..10000

Cf. A011000, A011001, A001787, A242091.

[Binomial(n,49): n in [49..80]]; // G. C. Greubel, Nov 03 2018

Table[Binomial[n,49],{n,49,77}] (* Vladimir Joseph Stephan Orlovsky, May 16 2011 *)

for(n=49, 80, print1(binomial(n,49), ", ")) \\ G. C. Greubel, Nov 03 2018

A017713_list, m = [], [1]*50
for _ in range(10**2):
    A017713_list.append(m[-1])
    for i in range(49):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

[binomial(n, 49) for n in range(49,67)] # Zerinvary Lajos, May 23 2009

From G. C. Greubel, Nov 03 2018: (Start)
G.f.: x^49/(1-x)^50.
E.g.f.: x^49*exp(x)/49!. (End)
From Amiram Eldar, Dec 16 2020: (Start)
Sum_{n>=49} 1/a(n) = 49/48.
Sum_{n>=49} (-1)^(n+1)/a(n) = A001787(49)*log(2) - A242091(49)/48! = 13792273858822144*log(2) - 302317348758761304758836609908210929 / 31622903104550986800 = 0.9807421943... (End)

A011000 a(n) = binomial coefficient C(n,47).

Original entry on oeis.org

1, 48, 1176, 19600, 249900, 2598960, 22957480, 177100560, 1217566350, 7575968400, 43183019880, 227692286640, 1119487075980, 5166863427600, 22512762077400, 93052749919920, 366395202809685, 1379370175283520, 4981058966301600, 17302625882942400, 57963796707857040

Offset: 47

N. J. A. Sloane

nonn

T. D. Noe, Table of n, a(n) for n = 47..1000
Index entries for linear recurrences with constant coefficients, signature (48, -1128, 17296, -194580, 1712304, -12271512, 73629072, -377348994, 1677106640, -6540715896, 22595200368, -69668534468, 192928249296, -482320623240, 1093260079344, -2254848913647, 4244421484512, -7309837001104, 11541847896480, -16735679449896, 22314239266528, -27385657281648, 30957699535776, -32247603683100, 30957699535776, -27385657281648, 22314239266528, -16735679449896, 11541847896480, -7309837001104, 4244421484512, -2254848913647, 1093260079344, -482320623240, 192928249296, -69668534468, 22595200368, -6540715896, 1677106640, -377348994, 73629072, -12271512, 1712304, -194580, 17296, -1128, 48, -1).

Cf. A010998, A010999, A001787, A242091.

[Binomial(n, 47): n in [47..70]]; // Vincenzo Librandi, Jun 12 2013

seq(binomial(n,47),n=47..67); # Zerinvary Lajos, Dec 20 2008

Table[Binomial[n,47],{n,47,77}] (* Vladimir Joseph Stephan Orlovsky, May 16 2011 *)

G.f.: x^47/(1-x)^48. - Zerinvary Lajos, Dec 20 2008
From Amiram Eldar, Dec 15 2020: (Start)
Sum_{n>=47} 1/a(n) = 47/46.
Sum_{n>=47} (-1)^(n+1)/a(n) = A001787(47)*log(2) - A242091(47)/46! = 3307330976350208*log(2) - 1349631021244469672053597823194021/588724259925151350 = 0.9799696418... (End)

cp's OEIS Frontend

A011001 Binomial coefficient C(n,48).

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A000580 a(n) = binomial coefficient C(n,7).

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A000581 a(n) = binomial coefficient C(n,8).

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A000582 a(n) = binomial coefficient C(n,9).

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A001287 a(n) = binomial coefficient C(n,10).

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A001288 a(n) = binomial(n,11).

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