A053138 -id:A053138 - cp's OEIS frontend

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

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

A258993 Triangle read by rows: T(n,k) = binomial(n+k,n-k), k = 0..n-1.

Original entry on oeis.org

1, 1, 3, 1, 6, 5, 1, 10, 15, 7, 1, 15, 35, 28, 9, 1, 21, 70, 84, 45, 11, 1, 28, 126, 210, 165, 66, 13, 1, 36, 210, 462, 495, 286, 91, 15, 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1, 55, 495, 1716, 3003, 3003, 1820, 680, 153, 19, 1, 66, 715, 3003, 6435, 8008, 6188, 3060, 969, 190, 21

Offset: 1

Reinhard Zumkeller, Jun 22 2015

T(n,k) = A085478(n,k) = A007318(A094727(n),A004736(k)), k = 0..n-1;
rounded(T(n,k)/(2*k+1)) = A258708(n,k);
rounded(sum(T(n,k)/(2*k+1)): k = 0..n-1) = A000967(n).

			.  n\k |  0  1    2    3     4     5     6     7    8    9  10 11
. -----+-----------------------------------------------------------
.   1  |  1
.   2  |  1  3
.   3  |  1  6    5
.   4  |  1 10   15    7
.   5  |  1 15   35   28     9
.   6  |  1 21   70   84    45    11
.   7  |  1 28  126  210   165    66    13
.   8  |  1 36  210  462   495   286    91    15
.   9  |  1 45  330  924  1287  1001   455   120   17
.  10  |  1 55  495 1716  3003  3003  1820   680  153   19
.  11  |  1 66  715 3003  6435  8008  6188  3060  969  190  21
.  12  |  1 78 1001 5005 12870 19448 18564 11628 4845 1330 231 23  .

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

If a diagonal of 1's is added on the right, this becomes A085478.
Essentially the same as A143858.
Cf. A007318, A004736, A094727.
Cf. A027941 (row sums), A117671 (central terms), A143858, A000967, A258708.
T(n,k): A000217 (k=1), A000332 (k=2), A000579 (k=3), A000581 (k=4), A001287 (k=5), A010965 (k=6), A010967 (k=7), A010969 (k=8), A010971 (k=9), A010973 (k=10), A010975 (k=11), A010977 (k=12), A010979 (k=13), A010981 (k=14), A010983 (k=15), A010985 (k=16), A010987 (k=17), A010989 (k=18), A010991 (k=19), A010993 (k=20), A010995 (k=21), A010997 (k=22), A010999 (k=23), A011001 (k=24), A017714 (k=25), A017716 (k=26), A017718 (k=27), A017720 (k=28), A017722 (k=29), A017724 (k=30), A017726 (k=31), A017728 (k=32), A017730 (k=33), A017732 (k=34), A017734 (k=35), A017736 (k=36), A017738 (k=37), A017740 (k=38), A017742 (k=39), A017744 (k=40), A017746 (k=41), A017748 (k=42), A017750 (k=43), A017752 (k=44), A017754 (k=45), A017756 (k=46), A017758 (k=47), A017760 (k=48), A017762 (k=49), A017764 (k=50).
T(n+k,n): A005408 (k=1), A000384 (k=2), A000447 (k=3), A053134 (k=4), A002299 (k=5), A053135 (k=6), A053136 (k=7), A053137 (k=8), A053138 (k=9), A196789 (k=10).
Cf. A165253.

Flat(List([1..12], n-> List([0..n-1], k-> Binomial(n+k,n-k) ))); # G. C. Greubel, Aug 01 2019

a258993 n k = a258993_tabl !! (n-1) !! k
a258993_row n = a258993_tabl !! (n-1)
a258993_tabl = zipWith (zipWith a007318) a094727_tabl a004736_tabl

[Binomial(n+k,n-k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Aug 01 2019

Table[Binomial[n+k,n-k], {n,1,12}, {k,0,n-1}]//Flatten (* G. C. Greubel, Aug 01 2019 *)

T(n,k) = binomial(n+k,n-k);
for(n=1, 12, for(k=0,n-1, print1(T(n,k), ", "))) \\ G. C. Greubel, Aug 01 2019

[[binomial(n+k,n-k) for k in (0..n-1)] for n in (1..12)] # G. C. Greubel, Aug 01 2019

T(n,k) = A085478(n,k) = A007318(A094727(n),A004736(k)), k = 0..n-1;
rounded(T(n,k)/(2*k+1)) = A258708(n,k);
rounded(sum(T(n,k)/(2*k+1)): k = 0..n-1) = A000967(n).

A196789 Binomial coefficients C(2*n+10,10).

Original entry on oeis.org

1, 66, 1001, 8008, 43758, 184756, 646646, 1961256, 5311735, 13123110, 30045015, 64512240, 131128140, 254186856, 472733756, 847660528, 1471442973, 2481256778, 4076350421, 6540715896, 10272278170, 15820024220, 23930713170, 35607051480, 52179482355, 75394027566

Offset: 0

Vincenzo Librandi, Oct 07 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A002299, A053134, A053135, A053136, A053137, A053138.

Magma
```
[Binomial(2*n+10,10): n in [0..30]];
```

a[n_] := Binomial[2*n + 10, 10]; Array[a, 20, 0] (* Amiram Eldar, Oct 21 2022 *)

G.f.: (11*x^5+165*x^4+462*x^3+330*x^2+55*x+1) / (1-x)^11.
From Amiram Eldar, Oct 21 2022: (Start)
Sum_{n>=0} 1/a(n) = 2560*log(2) - 148969/84.
Sum_{n>=0} (-1)^n/a(n) = 40*Pi - 80*log(2) - 5815/84. (End)

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

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A258993 Triangle read by rows: T(n,k) = binomial(n+k,n-k), k = 0..n-1.

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A196789 Binomial coefficients C(2*n+10,10).

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