A001287 -id:A001287 - cp's OEIS frontend

A008493 Expansion of (1-x^11) / (1-x)^11.

1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184756, 352715, 646635, 1144000, 1960970, 3267759, 5308732, 8428277, 13103662, 19986252, 29952637, 44167409, 64159524, 91914394, 129984074, 181618140, 250918096, 343018401, 464297471, 622622286

Offset: 0

N. J. A. Sloane

Coordination sequence for 10-dimensional cyclotomic lattice Z[zeta_11].

Growth series of the affine Weyl group of type A10. - Paul E. Gunnells, Jan 06 2017

R. Bott, The geometry and the representation theory of compact Lie groups, in: Representation Theory of Lie Groups, in: London Math. Soc. Lecture Note Ser., vol. 34, Cambridge University Press, Cambridge, 1979, pp. 65-90.

Colin Barker, Table of n, a(n) for n = 0..1000
M. Beck and S. 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 (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A001287.

Concatenation([1], List([1..40], n-> 11*n*(193248+152900*n^2 +16401*n^4 +330*n^6+n^8)/362880 )); # G. C. Greubel, Nov 07 2019

[1] cat [11*n*(193248+152900*n^2+16401*n^4+330*n^6+n^8)/362880: n in [1..40]]; // G. C. Greubel, Nov 07 2019

1, seq(11*n*(193248+152900*n^2+16401*n^4+330*n^6+n^8)/362880, n=1..40); # G. C. Greubel, Nov 07 2019

CoefficientList[(1-x^11)/(1-x)^11 + O[x]^30, x] (* Jean-François Alcover, Jan 09 2019 *)
Table[If[n==0,1, 11*n*(193248+152900*n^2+16401*n^4+330*n^6+n^8)/362880], {n,0,40}] (* G. C. Greubel, Nov 07 2019 *)

Vec((1-x^11)/(1-x)^11 + O(x^40)) \\ Colin Barker, Jan 06 2017

[1]+[11*n*(193248+152900*n^2+16401*n^4+330*n^6+n^8)/362880 for n in (1..40)] # G. C. Greubel, Nov 07 2019

a(n) = A001287(n+10) - A001287(n-1). - R. J. Mathar, Aug 10 2013
a(n) = 11*n*(193248 + 152900*n^2 + 16401*n^4 + 330*n^6 + n^8)/362880 for n>0. - Colin Barker, Jan 06 2017
E.g.f.: 1 + x*(3991680 + 7983360*x + 7318080*x^2 + 3160080*x^3 + 765072*x^4 + 105336*x^5 + 8712*x^6 + 396*x^7 + 11*x^8)*exp(x)/362880. - G. C. Greubel, Nov 07 2019
a(n) = 10*a(n-1)-45*a(n-2)+120*a(n-3)-210*a(n-4)+252*a(n-5)-210*a(n-6)+120*a(n-7)-45*a(n-8)+10*a(n-9)-a(n-10). - Wesley Ivan Hurt, Jun 07 2021

A104679 a(n) = C(n+5,5)*C(n+10,5).

Original entry on oeis.org

252, 2772, 16632, 72072, 252252, 756756, 2018016, 4900896, 11027016, 23279256, 46558512, 88884432, 162954792, 288304632, 494236512, 823727520, 1338557220, 2125943820, 3307023720, 5047562520, 7571343780, 11176745580, 16257084480, 23325382080, 33044291280

Offset: 0

Zerinvary Lajos, Apr 22 2005

			If n=0 then C(0+5,0+0)*C(0+10,5) = C(5,0)*C(10,5) = 1*252 = 252.
If n=4 then C(4+5,4+0)*C(4+10,5) = C(9,4)*C(14,5) = 126*2002 = 252252.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A062190.

[Binomial(n+5,n)*Binomial(n+10,5): n in [0..30]]; // G. C. Greubel, Nov 25 2017

Table[Binomial[n+5,n]Binomial[n+10,5],{n,0,20}] (* Harvey P. Dale, Feb 06 2015 *)

Vec(252/(1-x)^11 + O(x^100)) \\ Colin Barker, Feb 07 2015

G.f.: 252 / (1-x)^11. - Colin Barker, Feb 07 2015
a(n) = A000389(n+5)*A000389(n+10) = 252*A001287(n+11). - R. J. Mathar, Nov 29 2015
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 5/1134.
Sum_{n>=0} (-1)^n/a(n) = 1280*log(2)/63 - 447047/31752. (End)

Corrected and extended by Harvey P. Dale, Feb 06 2015

A008503 9-dimensional centered tetrahedral numbers.

Original entry on oeis.org

1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184755, 352705, 646580, 1143780, 1960255, 3265757, 5303727, 8416837, 13079352, 19937632, 29860259, 43999449, 63865594, 91416974

Offset: 0

N. J. A. Sloane

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

B:=Binomial;; List([0..30], n-> B(n+10,10)-B(n,10) ); # G. C. Greubel, Nov 09 2019

B:=Binomial; [B(n+10,10)-B(n,10): n in [0..30]]; // G. C. Greubel, Nov 09 2019

seq(binomial(n+10,10)-binomial(n,10), n=0..30); # G. C. Greubel, Nov 09 2019

Table[Binomial[n + 10, 10] - Binomial[n, 10], {n, 0, 23}] (* Bruno Berselli, Mar 22 2012 *)

vector(31, n, b=binomial; b(n+9,10) - b(n-1,10) ) \\ G. C. Greubel, Nov 09 2019

b=binomial; [b(n+10,10)-b(n,10) for n in (0..30)] # G. C. Greubel, Nov 09 2019

G.f.: (1-x^10)/(1-x)^11 = (1+x)*(1+x+x^2+x^3+x^4)*(1-x+x^2-x^3+x^4)/(1-x)^10.
a(n) = (2*n + 1)*(5*n^8 + 20*n^7 + 1370*n^6 + 4040*n^5 + 56549*n^4 + 106388*n^3 + 425916*n^2 + 373392*n + 362880)/362880. [Bruno Berselli, Mar 22 2012]
a(n) = A001287(n+10)-A001287(n). - R. J. Mathar, Aug 10 2013

A096754 Triangle read by rows giving coefficients of the trigonometric expansion of Cos(n*x).

Original entry on oeis.org

1, 1, 0, -1, 1, 0, -3, 1, 0, -6, 0, 1, 1, 0, -10, 0, 5, 1, 0, -15, 0, 15, 0, -1, 1, 0, -21, 0, 35, 0, -7, 1, 0, -28, 0, 70, 0, -28, 0, 1, 1, 0, -36, 0, 126, 0, -84, 0, 9, 1, 0, -45, 0, 210, 0, -210, 0, 45, 0, -1, 1, 0, -55, 0, 330, 0, -462, 0, 165, 0, -11, 1, 0, -66, 0, 495, 0, -924, 0, 495, 0, -66, 0, 1, 1, 0, -78, 0, 715

Offset: 1

Robert G. Wilson v, Jul 07 2004

T(n,k)=cos(n,k)*cos(pi*k/2) begins {1}, {1,0}, {1,0,-1}, {1,0,-3,0},... - Paul Barry, May 21 2006

			The trigonometric expansion of Cos(4x) = Cos[x]^4 - 6*Cos[x]^2*Sin[x]^2 + Sin[x]^4, therefore the fourth row is 1, 0, -6, 0, 1.
The trigonometric expansion of Cos(5x) = Cos[x]^5 - 10*Cos[x]^3*Sin[x]^2 + 5*Cos[x]*Sin[x]^4, therefore the fifth row of the triangle is 1, 0, -10, 0, 5
The table begins:
1
1 0 -1
1 0 -3
1 0 -6 0 1
1 0 -10 0 5
1 0 -15 0 15 0 -1
1 0 -21 0 35 0 -7
1 0 -28 0 70 0 -28 0 1

Clark Kimberling, Polynomials associated with reciprocation, JIS 12 (2009) 09.3.4, section 5.

Another version of the triangle in A034839. Cf. A095704.
First column is A000012 = C(n, 0), third column is A000217 = C(n, 2), fifth column is A000332 = C(n, 4), seventh column is A000579 = C(n, 6), ninth column is A000581 = C(n, 8).
A001287 = C(n, 10), A010965 = C(n, 12), A010967 = C(n, 14), A010969 = C(n, 16), A010971 = C(n, 18),
A010973 = C(n, 20), A010975 = C(n, 22), A010977 = C(n, 24), A010979 = C(n, 26), A010981 = C(n, 28),
A010983 = C(n, 30), A010985 = C(n, 32), A010987 = C(n, 34), A010989 = C(n, 36), A010991 = C(n, 38),
A010993 = C(n, 40), A010995 = C(n, 42), A010997 = C(n, 44), A010999 = C(n, 46), A011001 = C(n, 48),
A017714 = C(n, 50), A017716 = C(n, 52), A017718 = C(n, 54), A017720 = C(n, 56), etc.

Flatten[Table[ Plus @@ CoefficientList[ TrigExpand[ Cos[n*x]], { Cos[x], Sin[x]}], {n, 13}]]

A176566 Triangle T(n, k) = binomial(n*(n+1)/2 + k, k), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 4, 10, 20, 1, 7, 28, 84, 210, 1, 11, 66, 286, 1001, 3003, 1, 16, 136, 816, 3876, 15504, 54264, 1, 22, 253, 2024, 12650, 65780, 296010, 1184040, 1, 29, 435, 4495, 35960, 237336, 1344904, 6724520, 30260340, 1, 37, 703, 9139, 91390, 749398, 5245786, 32224114, 177232627, 886163135

Offset: 0

Roger L. Bagula, Apr 20 2010

			Square array of T(n, k):
  1,  1,   1,    1,     1,     1,      1 ...
  1,  1,   1,    1,     1,     1,      1 ... A000012;
  1,  2,   3,    4,     5,     6,      7 ... A000027;
  1,  4,  10,   20,    35,    56,     84 ... A000292;
  1,  7,  28,   84,   210,   462,    924 ... A000579;
  1, 11,  66,  286,  1001,  3003,   8008 ... A001287;
  1, 16, 136,  816,  3876, 15504,  54264 ... A010968;
  1, 22, 253, 2024, 12650, 65780, 296010 ... A010974;
Triangle begins as:
  1;
  1,  1;
  1,  2,   3;
  1,  4,  10,   20;
  1,  7,  28,   84,   210;
  1, 11,  66,  286,  1001,   3003;
  1, 16, 136,  816,  3876,  15504,   54264;
  1, 22, 253, 2024, 12650,  65780,  296010,  1184040;
  1, 29, 435, 4495, 35960, 237336, 1344904,  6724520,  30260340;
  1, 37, 703, 9139, 91390, 749398, 5245786, 32224114, 177232627, 886163135;

G. C. Greubel, Rows n = 0..50 of the triangle, flatten

Cf. A107868 (rows sums), A158498.

[Binomial(Binomial(n, 2) + k, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2021

T[n_, k_]= Binomial[Binomial[n, 2] + k, k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

row(n) = vector(n+1, k, k--; binomial(binomial(n,2) + k, k)); \\ Michel Marcus, Jul 10 2021

flatten([[binomial(binomial(n,2) +k, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 09 2021

T(n, k) = binomial(binomial(n, 2) + k, k).

Sum_{k=0..n} T(n, k) = A107868(n).

A270950 Number of distinct cardinalities of orbits of lattice points under the automorphism group of the n-dimensional integer lattice.

Original entry on oeis.org

0, 1, 2, 5, 9, 12, 20, 29, 40, 53, 76, 99, 132, 172, 216, 270, 341, 424, 532, 660, 810, 983, 1210, 1446, 1750, 2111, 2508, 2975, 3569, 4197, 4948, 5807, 6817, 7963, 9351, 10863, 12604, 14598, 16892, 19439, 22472, 25780, 29588, 33892, 38800, 44206, 50463, 57297, 65086, 73919, 83842, 94510

Offset: 0

Philippe A.J.G. Chevalier, Mar 26 2016

A finite number of orbits partition hypercubic shells of infinity norm s in the n-dimensional integer lattice. The number of orbits is given by C(n+s-1,s). The number of distinct cardinalities of the orbits of lattice points under the automorphism group of the n-dimensional integer lattice is found under the condition that n <= s.

A new connection was discovered using the partition of the dimension 'n'. These partitions create a base set of cardinalities. Each of these cardinalities can be subjected to the process of prime factorization. The prime factorization yields the exponents of the primes that form lattice points in a new integer lattice of dimension 'n'. These lattice points become elements of a set A. The unique summands of a specific partition of 'n' give the multipliers of the base vector (1,0^n) that need to be subtracted from the specific partition representative element of set A. The cardinality of the set A increases until all the specific partitions of 'n' have been processed. This augmented set A* has the correct cardinality. This method is much faster than the brute force technique. - Philippe A.J.G. Chevalier, Jun 24 2022

			For n=0 the a(0)=0.
For n=3 we have the following distinct cardinalities of the orbits 6, 8, 12, 24, 48 and thus a(3)=5.
For n=4 we have the distinct cardinalities of the orbits 8, 16, 24, 32, 48, 64, 96, 192, 384 and thus a(4)=9.
For n=5 we have the distinct cardinalities of the orbits 10, 32, 40, 160, 240, 320, 480, 640, 960, 1920, 3840 and thus a(5)=12.

Philippe A.J.G. Chevalier, Table of n, a(n) for n = 0..54

Cf. A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582, A001287.

a(17) corrected and a(18)-a(51) from Philippe A.J.G. Chevalier, Jun 24 2022

A275514 Triangle read by rows: the coefficient [t^k] of the Ehrhart polynomial of the 2-hypersimplex in dimension n.

Original entry on oeis.org

1, 1, -1, 1, 0, 0, 1, 2, 1, 0, 1, 5, 5, 0, 0, 1, 9, 15, 1, 0, 0, 1, 14, 35, 7, 0, 0, 0, 1, 20, 70, 28, 1, 0, 0, 0, 1, 27, 126, 84, 9, 0, 0, 0, 0, 1, 35, 210, 210, 45, 1, 0, 0, 0, 0, 1, 44, 330, 462, 165, 11, 0, 0, 0, 0, 0, 1, 54, 495, 924, 495, 66, 1, 0, 0, 0, 0, 0, 1, 65, 715

Offset: 1

R. J. Mathar, Jul 31 2016

			The triangle starts in row n=1 with coefficients 0<=k<n as:
1;
1, -1;
1, 0, 0;
1, 2, 1, 0;
1, 5, 5, 0, 0;
1, 9, 15, 1, 0, 0;
1, 14, 35, 7, 0, 0, 0;
1, 20, 70, 28, 1, 0, 0, 0;
1, 27, 126, 84, 9, 0, 0, 0, 0;
1, 35, 210, 210, 45, 1, 0, 0, 0, 0;
1, 44, 330, 462, 165, 11, 0, 0, 0, 0, 0;
1, 54, 495, 924, 495, 66, 1, 0, 0, 0, 0, 0;
1, 65, 715, 1716, 1287, 286, 13, 0, 0, 0, 0, 0, 0;

Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 10.
Nan Li, Ehrhart h*-vectors of hypersimplices, Discr. Comp. Geom. 48 (2012) 847-878, Eq. (1.1)

Cf. A010054 (coefficients for the 1-hypersimplex), A258993, A000295 (row sums), A000096 (column k=1), A000332 (k=2), A000579 (k=3), A000581 (k=4), A001287 (k=5).

subki := proc(n,r,l)
    local i,t;
    add(t^i,i=0..l-1) ;
    %^n ;
    expand(%) ;
    coeff(%,t,r) ;
end proc:
hstard := proc(d,k,n)
    add((-1)^i*binomial(n,i)*subki(n, (k-i)*d-i,k-i) ,i=0..k-1) ;
end proc:
A275514 := proc(n,k)
    hstard(k,2,n)
end proc:

A303987 Triangle read by rows: T(n, k) = (binomial(n,k)*binomial(n+k,k))^2 = A063007(n, k)^2, for n >= 0, k = 0..n.

Original entry on oeis.org

1, 1, 4, 1, 36, 36, 1, 144, 900, 400, 1, 400, 8100, 19600, 4900, 1, 900, 44100, 313600, 396900, 63504, 1, 1764, 176400, 2822400, 9922500, 7683984, 853776, 1, 3136, 571536, 17640000, 133402500, 276623424, 144288144, 11778624, 1, 5184, 1587600, 85377600, 1200622500, 5194373184, 7070119056, 2650190400, 165636900

Offset: 0

Wolfdieter Lang, May 14 2018

The row sums of this triangle are b(n) = A005259(n), for n >= 0. This sequence b was used in R. Apéry's 1979 proof of the irrationality of Zeta(3). See A005259 for references and links.

Row polynomials R(n, x) := Sum_{k=0..n} T(n, k)*x^k = hypergeometric([-n, -n, n+1, n+1], [1, 1, 1], x), hence b(n) = hypergeometric([-n, -n, n+1, n+1], [1, 1, 1], 1) (see the formula in A005259 given by K. A. Penson. This is the solution to Exercise 2.14 of the Koepf reference given there, p. 29).

			The triangle T begins:
n\k  0    1       2        3          4          5          6          7 ...
0:   1
1:   1    4
2:   1   36      36
3:   1  144     900      400
4:   1  400    8100    19600       4900
5:   1  900   44100   313600     396900      63504
6:   1 1764  176400  2822400    9922500    7683984     853776
7:   1 3136  571536 17640000  133402500  276623424  144288144   11778624
----------------------------------------------------------------------------
row n = 8:   1 5184 1587600 85377600 1200622500 5194373184 7070119056 2650190400 165636900,
row n = 9: 1 8100 3920400 341510400 8116208100 63631071504 176752976400 169612185600 47869064100 2363904400,
row n = 10: 1 12100 8820900 1177862400 44188244100 572679643536 2828047622400 5446435737600 3877394192100 853369488400 34134779536.
...

Cf. A000984, A002894, A005259, A008459, A063007.

The column sequences (without zeros) are A000012, A035287(n+1) = 4*A000217(n)^2, 36*A288876, 400*A000579(n+6)^2, 4900*A000581(n+8)^2, 63504*A001287(n+10)^2, ...

Flat(List([0..10],n->List([0..n],k->(Binomial(n,k)*Binomial(n+k,k))^2))); # Muniru A Asiru, May 15 2018

T[n_, k_] := (Gamma[k + n + 1]/(Gamma[k + 1]^2*Gamma[-k + n + 1]))^2;
Flatten[Table[T[n, k], {n, 0, 8}, {k, 0, n}]] (* Peter Luschny, May 14 2018 *)

T(n, k) = (binomial(n,k)*binomial(n+k,k))^2 = A063007(n, k)^2, for n >= 0 and k = 0..n.
T(n, k) = (binomial(n+k, 2*k)*cbi(k))^2, with cbi(k) = A000984(k) = binomial(2*k, k), and cbi(k)^2 = A002894(k).
G.f. for column sequences (without leading zeros):
  cbi(k)^2*P2(2*k, x)/(1 - x)^(4*k+1), with the row polynomials of A008459 (Pascal entries squared) P2(2*k, x) = Sum_{j=0..2*k} A008459(2*k, j)*x^j. For a proof see the general comment in A288876 on the diagonals and columns of A008459.

A127157 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and 2k nodes of odd degree (not outdegree; 1 <= k <= ceiling(n/2)).

Original entry on oeis.org

1, 2, 3, 2, 4, 10, 5, 30, 7, 6, 70, 56, 7, 140, 252, 30, 8, 252, 840, 330, 9, 420, 2310, 1980, 143, 10, 660, 5544, 8580, 2002, 11, 990, 12012, 30030, 15015, 728, 12, 1430, 24024, 90090, 80080, 12376, 13, 2002, 45045, 240240, 340340, 111384, 3876, 14, 2730

Offset: 1

Emeric Deutsch, Feb 27 2007

Row n has ceiling(n/2) terms.
Row sums are the Catalan numbers (A000108).
T(n,1) = n;
T(n,2) = 2*binomial(n+1, 4) = 2*A000332(n+1);
T(n,3) = 7*binomial(n+2, 7) = 7*A000580(n+2);
T(n,4) = 30*binomial(n+3, 10) = 30*A001287(n+3);
T(n,5) = 143*binomial(n+4, 13) = 143*A010966(n+4);
T(2n-1,n) = A006013(n-1).
T(n,k) is the number of ordered trees (A000108) with n edges, exactly k of whose vertices possess at least one leaf child. [David Callan, Aug 22 2014]

			Triangle starts:
  1;
  2;
  3,  2;
  4, 10;
  5, 30,  7;
  6, 70, 56;

J.-C. Aval, A. Boussicault, B. Delcroix-Oger, F. Hivert, et al., Non-ambiguous trees: new results and generalization, arXiv preprint arXiv:1511.09455 [math.CO], 2015.
Bérénice Delcroix-Oger, Florent Hivert, Patxi Laborde-Zubieta, Jean-Christophe Aval, and Adrien Boussicault, Non-ambiguous trees: new results and generalisation, hal-03165269v2 [cs.DM] [math.CO], 2021.
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.

Cf. A000108, A000332, A000580, A001287, A010966, A006013.

T:=(n,k)->2*binomial(3*k-1,2*k)*binomial(n-1+k,3*k-2)/(3*k-1): for n from 1 to 15 do seq(T(n,k),k=1..ceil(n/2)) od;

m = 14(*rows*); G = 0; Do[G = Series[(1 + t^2 z - G^3 z^2 + G^2 z (2+z))/ (1+2z), {t, 0, m}, {z, 0, m}] // Normal // Expand, m]; Rest[ CoefficientList[#, t^2]]& /@ Rest[CoefficientList[G-1, z] ] // Flatten (* Jean-François Alcover, Jan 23 2019 *)

T(n,k) = 2*binomial(3k-1,2k)*binomial(n-1+k,3k-2)/(3k-1) (formula obtained only by inspection).

G.f.: G-1, where G=G(t,z) satisfies z^2*G^3 - z(z+2)G^2 + (1+2z)*G - t^2*z - 1 = 0.

cp's OEIS Frontend

A008493 Expansion of (1-x^11) / (1-x)^11.

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A104679 a(n) = C(n+5,5)*C(n+10,5).

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A008503 9-dimensional centered tetrahedral numbers.

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A096754 Triangle read by rows giving coefficients of the trigonometric expansion of Cos(n*x).

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A176566 Triangle T(n, k) = binomial(n*(n+1)/2 + k, k), read by rows.

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A270950 Number of distinct cardinalities of orbits of lattice points under the automorphism group of the n-dimensional integer lattice.

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A275514 Triangle read by rows: the coefficient [t^k] of the Ehrhart polynomial of the 2-hypersimplex in dimension n.

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