A077044 -id:A077044 - cp's OEIS frontend

A077042 Square array read by falling antidiagonals of central polynomial coefficients: largest coefficient in expansion of (1 + x + x^2 + ... + x^(n-1))^k = ((1-x^n)/(1-x))^k, i.e., the coefficient of x^floor(k(n-1)/2) and of x^ceiling(k(n-1)/2); also number of compositions of floor(k*(n+1)/2) into exactly k positive integers each no more than n.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 3, 1, 1, 0, 1, 6, 7, 4, 1, 1, 0, 1, 10, 19, 12, 5, 1, 1, 0, 1, 20, 51, 44, 19, 6, 1, 1, 0, 1, 35, 141, 155, 85, 27, 7, 1, 1, 0, 1, 70, 393, 580, 381, 146, 37, 8, 1, 1, 0, 1, 126, 1107, 2128, 1751, 780, 231, 48, 9, 1, 1, 0, 1, 252, 3139

Offset: 0

Henry Bottomley, Oct 22 2002

From Michel Marcus, Dec 01 2012: (Start)
A pair of numbers written in base n are said to be comparable if all digits of the first number are at least as big as the corresponding digit of the second number, or vice versa. Otherwise, this pair will be defined as uncomparable. A set of pairwise uncomparable integers will be called anti-hierarchic.
T(n,k) is the size of the maximal anti-hierarchic set of integers written with k digits in base n.
For example, for base n=2 and k=4 digits:
- 0 (0000) and 15 (1111) are comparable, while 6 (0110) and 9 (1001) are uncomparable,
- the maximal antihierarchic set is {3 (0011), 5 (0101), 6 (0110), 9 (1001), 10 (1010), 12 (1100)} with 6 elements that are all pairwise uncomparable. (End)

			Rows of square array start:
  1,    0,    0,    0,    0,    0,    0, ...
  1,    1,    1,    1,    1,    1,    1, ...
  1,    1,    2,    3,    6,   10,   20, ...
  1,    1,    3,    7,   19,   51,  141, ...
  1,    1,    4,   12,   44,  155,  580, ...
  1,    1,    5,   19,   85,  381, 1751, ...
  ...
Read by antidiagonals:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 1, 2, 1, 1;
  0, 1, 3, 3, 1, 1;
  0, 1, 6, 7, 4, 1, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Denis Bouyssou, Thierry Marchant, Marc Pirlot, The size of the largest antichains in products of linear orders, arXiv:1903.07569 [math.CO], 2019.
J. W. Sander, On maximal antihierarchic sets of integers, Discrete Mathematics, Volume 113, Issues 1-3, 5 April 1993, Pages 179-189.
Index entries for sequences related to compositions

Rows include A000007, A000012, A001405, A002426, A005190, A005191, A018901, A025012, A025013, A025014, A025015, A201549, A225779, A201550. Columns include A000012, A000012, A001477, A077043, A005900, A077044, A071816, A133458.
Central diagonal is A077045, with A077046 and A077047 either side. Cf. A067059, A270918, A201552.
Cf. A273975.

t[n_, k_] := Max[ CoefficientList[ Series[ ((1-x^n) / (1-x))^k, {x, 0, k*(n-1)}], x]]; t[0, 0] = 1; t[0, ] = 0; Flatten[ Table[ t[n-k, k], {n, 0, 12}, {k, n, 0, -1}]] (* _Jean-François Alcover, Nov 04 2011 *)

T(n,k)=if(n<1 || k<1,k==0,vecmax(Vec(((1-x^n)/(1-x))^k)))

By the central limit theorem, T(n,k) is roughly n^(k-1)*sqrt(6/(Pi*k)).

T(n,k) = Sum{j=0,h/n} (-1)^j*binomial(k,j)*binomial(k-1+h-n*j,k-1) with h=floor(k*(n-1)/2), k>0. - Michel Marcus, Dec 01 2012

A277950 Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^5.

Original entry on oeis.org

1, 1, 5, 10, 10, 5, 1, 1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1, 1, 5, 15, 35, 65, 101, 135, 155, 155, 135, 101, 65, 35, 15, 5, 1, 1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1

Offset: 1

Juan Pablo Herrera P., Nov 05 2016

Sum of n-th row is n^5. The n-th row contains 5n-4 entries. Largest coefficients of each row are listed in A077044.
The n-th row is the fifth row of the n-nomial triangle.  For example, row 2 (1,5,10,10,5,1) is the fifth row in the binomial triangle.
T(n,k) gives the number of possible ways of randomly selecting k cards from n-1 sets, each with five different playing cards. It is also the number of lattice paths from (0,0) to (5,k) using steps (1,0), (1,1), (1,2), ..., (1,n-1).

			Triangle starts:
1;
1, 5, 10, 10, 5, 1;
1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1;
1, 5, 15, 35, 65, 101, 135, 155, 155, 135, 101, 65, 35, 15, 5, 1;
1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1;

Juan Pablo Herrera P., Rows n=1..60 of the triangle, flattened

Cf. A000332, A004737, A077044, A109439, A154286, A277949, A277951.

Mentioned in: A273975.

Table[CoefficientList[Series[((x^n - 1)/(x - 1))^5, {x, 0, 5 n}], x], {n, 10}] // Flatten

row(n) = Vec(((1 - x^n)/(1 - x))^5); tabf(nn) = for (n=1, nn, print(row(n)));

T(n,k) = Sum_{i=k-n+1..k} A277949(T(n,i)).
From Juan Pablo Herrera P., Dec 20 2016: (Start)
T(n,k) = A000332(k+4) = (k+4)!/(k!*24) if 0 =< k < n.
T(n,k) = ((k+4)!/k!-5*(k-n+4)!/(k-n)!)/24 if n =< k < 2*n.
T(n,k) = ((k+4)!/k!-5*(k-n+4)!/(k-n)!+10*(k-2*n+4)!/(k-2*n)!)/24 if 2*n =< k < 3*n.
T(n,k) = ((5*n-k-1)!/(5*n-k-5)!-5*(4*n-k-1)!/(4*n-k-5)!)/24 if 3*n-4 =< k < 4*n-4.
T(n,k) = A000332(5*n-k-1) = (5*n-k-1)!/(5*n-k-5)!*24 4*n-4 =< k < 5*n-4. (End)

A133458 The size of the largest antichain in the 7-dimensional hypercubic lattice of size n; also the coefficient of x^floor(7*(n-1)/2) in (1 + x + ... + x^(n-1))^7.

Original entry on oeis.org

1, 35, 393, 2128, 8135, 24017, 60691, 134512, 273127, 512365, 908755, 1528688, 2473325, 3852919, 5832765, 8582336, 12354469, 17395119, 24072133, 32726960, 43874139, 57971221, 75715487, 97702640, 124853275, 157924585, 198105727

Offset: 1

Leonid Chindelevitch (leonidus(AT)mit.edu), Dec 22 2007

nonn

The middle coefficients for dimension d>=1 are in A000012, A000027, A077043, A005900, A077044, A071816, here, the d-th row in A077042.

For d=8 the sequence starts 1, 70, 1107, 8092, 38165, 135954, 398567, 1012664, 2306025, ... and for d=9 it starts 1, 126, 3139, 30276, 180325, 767394, 2636263, 7635987, 19610233, ... - R. J. Mathar, Sep 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
R. P. Stanley, Weyl groups, the Hard Lefschetz Theorem and the Sperner property, SIAM J. Alg. Disc. Meth. 1 (2) (1980) 168, see eq. (4).

Cf. A077043, A005900, A077044, A071816.

[-25*(-1)^n/512 +2261*n^2/23040 +25/512 +5887*n^6/11520 -77*(-1)^n*n^4/1536 +847*n^4/4608 -91*(-1)^n*n^2/1536 : n in [1..40]]; // Vincenzo Librandi, Sep 07 2011

f:=(L,d)->(sum(x^k,k=0..L-1))^d; A:=[seq(coeff(f(j,7),x,floor(7*(j-1)/2)),j=1..25)];
A133458 := proc(n) -25/512*(-1)^n +2261/23040*n^2 -91/1536*(-1)^n*n^2 -77/1536*(-1)^n*n^4 +847/4608*n^4 +5887/11520*n^6 +25/512 ; end proc: # R. J. Mathar, Sep 05 2011

From R. J. Mathar, Feb 19 2010: (Start)
a(n)= 2*a(n-1) +4*a(n-2) -10*a(n-3) -5*a(n-4) +20*a(n-5) -20*a(n-7) +5*a(n-8) +10*a(n-9) -4*a(n-10) -2*a(n-11) +a(n-12).
G.f.: x*(1+33*x +319*x^2 +1212*x^3 +2662*x^4 +3320*x^5 +2662*x^6 +1212*x^7 +319*x^8 +33*x^9 +x^10)/ ((1+x)^5 * (1-x)^7).
a(n) = -25*(-1)^n/512 +2261*n^2/23040 +25/512 +5887*n^6/11520 -77*(-1)^n*n^4/1536 +847*n^4/4608 -91*(-1)^n*n^2/1536. (End)

More terms from R. J. Mathar, Feb 19 2010

A175197 Array A(k,n) of the number of points of the A_k lattice with maximum infinity norm n, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 2, 0, 1, 18, 12, 2, 0, 1, 50, 66, 18, 2, 0, 1, 140, 330, 146, 24, 2, 0, 1, 392, 1610, 1070, 258, 30, 2, 0, 1, 1106, 7742, 7580, 2500, 402, 36, 2, 0, 1, 3138, 37058, 52556, 23330, 4850, 578, 42, 2, 0, 1, 8952, 177186, 360402, 212436, 56252, 8350, 786

Offset: 0

R. J. Mathar, Mar 02 2010

The values are computed starting with an auxiliary array which places the centered trinomial numbers A002426, the centered pentanomial numbers A005191, the centered 7-nomial numbers A025012 etc. into separate columns:
.1,....1,......1,.......1,........1,........1,.........1,.........1,.........1
.1,....3,......5,.......7,........9,.......11,........13,........15,........17
.1,....7,.....19,......37,.......61,.......91,.......127,.......169,.......217
.1,...19,.....85,.....231,......489,......891,......1469,......2255,......3281
.1,...51,....381,....1451,.....3951,.....8801,.....17151,.....30381,.....50101
.1,..141,...1751,....9331,....32661,....88913,....204763,....418503,....782153
.1,..393,...8135,...60691,...273127,...908755,...2473325,...5832765,..12354469
.1,.1107,..38165,..398567,..2306025,..9377467,..30162301,..82073295,.197018321
.1,.3139,.180325,.2636263,.19610233,.97464799,.370487485,1163205475,3164588407
This is a subarray of A077042. Rows are A005408, A003215, A063496, A083669 (see A077044) etc. The array A(k,n) is the first differences along each row of this auxiliary array.

			A(k,n) starts in row k=0, column n=0 as:
1,....0,......0,.......0,........0,........0,.........0,.........0,.........0
1,....2,......2,.......2,........2,........2,.........2,.........2,.........2
1,....6,.....12,......18,.......24,.......30,........36,........42,........48
1,...18,.....66,.....146,......258,......402,.......578,.......786,......1026
1,...50,....330,....1070,.....2500,.....4850,......8350,.....13230,.....19720
1,..140,...1610,....7580,....23330,....56252,....115850,....213740,....363650
1,..392,...7742,...52556,...212436,...635628,...1564570,...3359440,...6521704
1,.1106,..37058,..360402,..1907458,..7071442,..20784834,..51910994,.114945026
1,.3138,.177186,.2455938,.16973970,.77854566,.273022686,.792717990,2001382932

R. J. Mathar, Point counts of D_k and some A_k and E_K integer.., arXiv:1002.3844 [math .GT], variable A_k^s(n).

Cf. A008458 (row k=2), A010006 (row k=3), A110907.

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A277950 Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^5.

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A133458 The size of the largest antichain in the 7-dimensional hypercubic lattice of size n; also the coefficient of x^floor(7*(n-1)/2) in (1 + x + ... + x^(n-1))^7.

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A175197 Array A(k,n) of the number of points of the A_k lattice with maximum infinity norm n, read by antidiagonals.

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