A270918 -id:A270918 - 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

A077047 Doubly restricted composition numbers: number of compositions of floor(n(n+2)/2) into exactly n positive integers each no more than n+1.

Original entry on oeis.org

1, 1, 3, 12, 85, 780, 9331, 134512, 2306025, 45433800, 1018872811, 25506741084, 707972099627, 21518492021208, 712601187601395, 25491847538274240, 981272544393935569, 40392787067756440272, 1772592132899627652691

Offset: 0

Henry Bottomley, Oct 22 2002

nonn

a(n) is the maximum number of ordered partitions when using n numbers each ranging from 0 to n. This maximum occurs when partitioning n^2/2 for n even, or (n^2 - 1)/2 or (n^2 + 1)/2 for n odd. Example for a(3)=12: the partitions of 4 are (1,1,2) and (0,2,2), each having 3 ordered arrangements, and (0,1,3) having 6 arrangements; hence 3+3+6=12. For 5 the partitions are (1,2,2) and (1,1,3), with 3 ordered arrangements each, and (0,2,3) having 6 arrangements. - J. M. Bergot, Jul 11 2015

Largest coefficient of (1 + x + x^2 + ... + x^n)^n. - Vaclav Kotesovec, Mar 26 2016

			a(3)=12 since the compositions of [3*5/2]=7 into exactly 3 positive integers each no more than 4 are 1+2+4, 1+3+3, 1+4+2, 2+1+4, 2+2+3, 2+3+2, 2+4+1, 3+1+3, 3+2+2, 3+3+2, 4+1+2, 4+2+1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..375
Index entries for sequences related to compositions

Cf. A077042, A077045, A077046, A077048, A270918.

f:= proc(n) if n::odd then coeff(add(x^i,i=0..n)^n,x,(n^2-1)/2)
else coeff(add(x^i,i=0..n)^n,x,n^2/2) fi end proc:
map(f, [$0..40]); # Robert Israel, Nov 16 2016

Table[Max[CoefficientList[Expand[Sum[x^k, {k, 0, n}]^n], x]], {n, 0, 20}] (* Vaclav Kotesovec, Mar 26 2016 *)
Table[Max[CoefficientList[((x^(n+1)-1)/(x-1))^n, x]], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 16 2016 *)

a(n) = A077042(n+1, n).

a(n) ~ exp(1) * sqrt(6/Pi) * n^(n-3/2). - Vaclav Kotesovec, Mar 26 2016

A349933 Array read by ascending antidiagonals: the s-th column gives the central s-binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 20, 19, 4, 1, 1, 70, 141, 44, 5, 1, 1, 252, 1107, 580, 85, 6, 1, 1, 924, 8953, 8092, 1751, 146, 7, 1, 1, 3432, 73789, 116304, 38165, 4332, 231, 8, 1, 1, 12870, 616227, 1703636, 856945, 135954, 9331, 344, 9, 1, 1, 48620, 5196627, 25288120, 19611175, 4395456, 398567, 18152, 489, 10, 1

Offset: 0

Stefano Spezia, Dec 06 2021

			The array begins:
n\s |   0     1     2     3     4
----+----------------------------
  0 |   1     1     1     1     1 ...
  1 |   1     2     3     4     5 ...
  2 |   1     6    19    44    85 ...
  3 |   1    20   141   580  1751 ...
  4 |   1    70  1107  8092 38165 ...
  ...

William Linz, s-Catalan numbers and Littlewood-Richardson polynomials, arXiv:2110.12095 [math.CO], 2021. See p. 2.

Cf. A000984 (s=1), A082758 (s=2), A005721 (s=3), A349936 (s=4), A063419 (s=5), A270918 (n=s), A163269 (s>0).
Cf. A007318, A008287, A025012, A027907, A035343, A063260.
Cf. A349934, A349935.

T[n_,k_,s_]:=If[k==0,1,Coefficient[(Sum[x^i,{i,0,s}])^n,x^k]]; A[n_,s_]:=T[2n,s n,s]; Flatten[Table[A[n-s,s],{n,0,9},{s,0,n}]]

A(n, s) = T(2*n, s*n, s), where T(n, k, s) is the s-binomial coefficient defined as the coefficient of x^k in (Sum_{i=0..s} x^i)^n.

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A077047 Doubly restricted composition numbers: number of compositions of floor(n(n+2)/2) into exactly n positive integers each no more than n+1.

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A349933 Array read by ascending antidiagonals: the s-th column gives the central s-binomial coefficients.

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