A053129 -id:A053129 - cp's OEIS frontend

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

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

A128908 Riordan array (1, x/(1-x)^2).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 4, 10, 6, 1, 0, 5, 20, 21, 8, 1, 0, 6, 35, 56, 36, 10, 1, 0, 7, 56, 126, 120, 55, 12, 1, 0, 8, 84, 252, 330, 220, 78, 14, 1, 0, 9, 120, 462, 792, 715, 364, 105, 16, 1, 0, 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1

Offset: 0

Philippe Deléham, Apr 22 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,2,-1/2,1/2,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
Row sums give A088305. - Philippe Deléham, Nov 21 2007
Column k is C(n,2k-1) for k > 0. - Philippe Deléham, Jan 20 2012
From R. Bagula's comment in A053122 (cf. Damianou link p. 10), this array gives the coefficients (mod sign) of the characteristic polynomials for the Cartan matrix of the root system A_n. - Tom Copeland, Oct 11 2014
T is the convolution triangle of the positive integers (see A357368). - Peter Luschny, Oct 19 2022

			The triangle T(n,k) begins:
   n\k  0    1    2    3    4    5    6    7    8    9   10
   0:   1
   1:   0    1
   2:   0    2    1
   3:   0    3    4    1
   4:   0    4   10    6    1
   5:   0    5   20   21    8    1
   6:   0    6   35   56   36   10    1
   7:   0    7   56  126  120   55   12    1
   8:   0    8   84  252  330  220   78   14    1
   9:   0    9  120  462  792  715  364  105   16    1
  10:   0   10  165  792 1716 2002 1365  560  136   18    1
  ... reformatted by _Wolfdieter Lang_, Jul 31 2017
From _Peter Luschny_, Mar 06 2022: (Start)
The sequence can also be seen as a square array read by upwards antidiagonals.
   1, 1,   1,    1,    1,     1,     1,      1,      1, ...  A000012
   0, 2,   4,    6,    8,    10,    12,     14,     16, ...  A005843
   0, 3,  10,   21,   36,    55,    78,    105,    136, ...  A014105
   0, 4,  20,   56,  120,   220,   364,    560,    816, ...  A002492
   0, 5,  35,  126,  330,   715,  1365,   2380,   3876, ... (A053126)
   0, 6,  56,  252,  792,  2002,  4368,   8568,  15504, ... (A053127)
   0, 7,  84,  462, 1716,  5005, 12376,  27132,  54264, ... (A053128)
   0, 8, 120,  792, 3432, 11440, 31824,  77520, 170544, ... (A053129)
   0, 9, 165, 1287, 6435, 24310, 75582, 203490, 490314, ... (A053130)
    A27,A292, A389, A580,  A582, A1288, A10966, A10968, A165817       (End)

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv:1110.6620 [math.RT], 2014.

Cf. A002450, A007318, A034008, A053122, A078812, A084938, A088305.
Cf. Columns : A000007, A000027, A000292, A000389, A000580, A000582, A001288, A010966 ..(+2).. A011000, A017713 ..(+2).. A017763.
Cf. A000007, A001352, A008574, A054888, A084099, A084103, A163810, A357368.
Cf. A165817 (the main diagonal of the array).

# Computing the rows of the array representation:
S := proc(n,k) option remember;
if n = k then 1 elif k < 0 or k > n then 0 else
S(n-1, k-1) + 2*S(n-1, k) - S(n-2, k) fi end:
Arow := (n, len) -> seq(S(n+k-1, k-1), k = 0..len-1):
for n from 0 to 8 do Arow(n, 9) od; # Peter Luschny, Mar 06 2022
# Uses function PMatrix from A357368.
PMatrix(10, n -> n); # Peter Luschny, Oct 19 2022

With[{nmax = 10}, CoefficientList[CoefficientList[Series[(1 - x)^2/(1 - (2 + y)*x + x^2), {x, 0, nmax}, {y, 0, nmax}], x], y]] // Flatten (* G. C. Greubel, Nov 22 2017 *)

for(n=0,10, for(k=0,n, print1(if(n==0 && k==0, 1, if(k==0, 0, binomial(n+k-1,2*k-1))), ", "))) \\ G. C. Greubel, Nov 22 2017

from functools import cache
@cache
def A128908(n, k):
    if n == k: return 1
    if (k <= 0 or k > n): return 0
    return A128908(n-1, k-1) + 2*A128908(n-1, k) - A128908(n-2, k)
for n in range(10):
    print([A128908(n, k) for k in range(n+1)]) # Peter Luschny, Mar 07 2022

@cached_function
def T(k,n):
    if k==n: return 1
    if k==0: return 0
    return sum(i*T(k-1,n-i) for i in (1..n-k+1))
A128908 = lambda n,k: T(k,n)
for n in (0..10): print([A128908(n,k) for k in (0..n)]) # Peter Luschny, Mar 12 2016

T(n,0) = 0^n, T(n,k) = binomial(n+k-1, 2k-1) for k >= 1.
Sum_{k=0..n} T(n,k)*2^(n-k) = A002450(n) = (4^n-1)/3 for n>=1. - Philippe Deléham, Oct 19 2008
G.f.: (1-x)^2/(1-(2+y)*x+x^2). - Philippe Deléham, Jan 20 2012
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A001352(n), (-1)^(n+1)*A054888(n+1), (-1)^n*A008574(n), (-1)^n*A084103(n), (-1)^n*A084099(n), A163810(n), A000007(n), A088305(n) for x = -6, -5, -4, -3, -2, -1, 0, 1 respectively. - Philippe Deléham, Jan 20 2012
Riordan array (1, x/(1-x)^2). - Philippe Deléham, Jan 20 2012

A053130 Binomial coefficients C(2*n-7,8).

Original entry on oeis.org

9, 165, 1287, 6435, 24310, 75582, 203490, 490314, 1081575, 2220075, 4292145, 7888725, 13884156, 23535820, 38608020, 61523748, 95548245, 145008513, 215553195, 314457495, 450978066, 636763050, 886322710, 1217566350, 1652411475, 2217471399, 2944827765, 3872894697

Offset: 8

Wolfdieter Lang

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

Vincenzo Librandi, Table of n, a(n) for n = 8..200
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].
Milan Janjić, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A053123, A053129.

[Binomial(2*n-7, 8): n in [8..50]]; // Vincenzo Librandi, Apr 07 2011

Table[Binomial[2*n-7,8], {n,8,50}] (* G. C. Greubel, Aug 26 2018 *)

for(n=8,50, print1(binomial(2*n-7,8), ", ")) \\ G. C. Greubel, Aug 26 2018

a(n) = binomial(2*n-7, 8) if n >= 8 else 0.
G.f.: (9+84*x+126*x^2+36*x^3+x^4)/(1-x)^9.
a(n) = A053123(n,8), n >= 8; a(n) := 0, n=0..7, (ninth column of shifted Chebyshev's S-triangle, decreasing order).
From Amiram Eldar, Oct 21 2022: (Start)
Sum_{n>=8} 1/a(n) = 37276/105 - 512*log(2).
Sum_{n>=8} (-1)^n/a(n) = 592/21 - 16*Pi + 32*log(2). (End)

A053136 Binomial coefficients C(2*n+7,7).

Original entry on oeis.org

1, 36, 330, 1716, 6435, 19448, 50388, 116280, 245157, 480700, 888030, 1560780, 2629575, 4272048, 6724520, 10295472, 15380937, 22481940, 32224114, 45379620, 62891499, 85900584, 115775100, 154143080, 202927725, 264385836, 341149446, 436270780, 553270671, 696190560

Offset: 0

Wolfdieter Lang

Even-indexed members of eighth column of Pascal's triangle A007318.

Number of standard tableaux of shape (2n+1,1^7). - Emeric Deutsch, May 30 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Milan Janjić, Two Enumerative Functions.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A007318, A053135, A000580, A053129.

[Binomial(2*n+7,7): n in [0..30]]; // Vincenzo Librandi, Oct 07 2011

Table[Binomial[2*n+7, 7], {n, 0, 30}] (* G. C. Greubel, Sep 03 2018 *)

a(n)=binomial(2*n+7,7) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = binomial(2*n+7, 7) = A000580(2*n+7).
G.f.: (1 + 28*x + 70*x^2 + 28*x^3 + x^4)/(1-x)^8.
E.g.f.: (630 + 22050*x + 81585*x^2 + 87465*x^3 + 36960*x^4 + 6888*x^5 + 560*x^6 + 16*x^7)*exp(x)/630. - G. C. Greubel, Sep 03 2018
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=0} 1/a(n) = 224*log(2) - 4627/30.
Sum_{n>=0} (-1)^n/a(n) = 28*log(2) - 553/30. (End)

A000973 Fermat coefficients.

Original entry on oeis.org

1, 15, 99, 429, 1430, 3978, 9690, 21318, 43263, 82225, 148005, 254475, 420732, 672452, 1043460, 1577532, 2330445, 3372291, 4790071, 6690585, 9203634, 12485550, 16723070, 22137570, 28989675, 37584261, 48275865, 61474519

Offset: 8

N. J. A. Sloane

a(n) = A258708(n,n-8). - Reinhard Zumkeller, Jun 23 2015

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).

Vincenzo Librandi, Table of n, a(n) for n = 8..1000
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
P. A. Piza, Fermat coefficients, Math. Mag., 27 (1954), 141-146.
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
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A053129.

Cf. A258708.

a000973 n = a258708 n (n - 8)  -- Reinhard Zumkeller, Jun 23 2015

[Binomial(2*n-8, 7)/8: n in [8..40]]; // Vincenzo Librandi, Apr 10 2012

A000973:=(z+1)*(z**2+6*z+1)/(z-1)**8; # conjectured by Simon Plouffe in his 1992 dissertation
A000973:=n->binomial(2*n-8, 7)/8; seq(A000973(n), n=8..40); # Wesley Ivan Hurt, Apr 16 2014

CoefficientList[Series[(1+7*x+7*x^2+x^3)/(1-x)^8,{x,0,40}],x] (* Vincenzo Librandi, Apr 10 2012 *)

a(n) = binomial(2*n-8, 7)/8.
G.f.: (x^8)*(1+7*x+7*x^2+x^3)/(1-x)^8.
G.f.: A(x)= (1+7*x+7*x^2+x^3)/(x-1)^8 = 1 + 45*x/(G(0)-45*x), |x|<1; if |x|>1, G(0)=45*x;
G(k) = (k+1)*(2*k+3) + x*(k+5)*(2*k+9) - x*(k+1)*(k+6)*(2*k+3)*(2*k+11)/G(k+1); (continued fraction Euler's 1st kind, 1-step ). - Sergei N. Gladkovskii, Jun 15 2012

More terms from David W. Wilson, Oct 11 2000

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A128908 Riordan array (1, x/(1-x)^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Sage

Formula

A053130 Binomial coefficients C(2*n-7,8).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A053136 Binomial coefficients C(2*n+7,7).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A000973 Fermat coefficients.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Formula

Extensions

A196790 Binomial coefficients C(2*n-9,10).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica