A077047 -id:A077047 - 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

A077045 Doubly restricted composition numbers: number of compositions of 1+2+3+...+n = n(n+1)/2 into exactly n positive integers each no more than n.

Original entry on oeis.org

1, 1, 2, 7, 44, 381, 4332, 60691, 1012664, 19610233, 432457640, 10701243741, 293661065788, 8851373201919, 290711372717976, 10334165623697259, 395320344293410544, 16192709833199300337, 707125993042984343136, 32795665902734099555845, 1609908874238209683872480

Offset: 0

Henry Bottomley, Oct 22 2002

			a(3) = 7 since the compositions of 1+2+3=6 into exactly 3 positive integers each no more than 3 are: 1+2+3, 1+3+2, 2+1+3, 2+2+2, 2+3+1, 3+1+2, 3+2+1.

Alois P. Heinz, Table of n, a(n) for n = 0..100
Index entries for sequences related to compositions

Cf. A077042, A077046, A077047, A077048, A362967.

b:= proc(n, h, t) option remember;
      `if`(t*h b(n*(n-1)/2, n-1, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 10 2012

f[n_] := Union[ CoefficientList[ Expand[ Sum[x^j, {j, n}]^n], x]][[-1]]; Join[{1}, Array[f, 18]] (* Robert G. Wilson v, Apr 06 2012 *)
f[n_] := Block[{ip = IntegerPartitions[n (n + 1)/2, {n}, Range@ n], k = 1, s = 0}, len = Length[ip] + 1; While[k < len, s = s + Length@ Permutations[ ip[[k]]]; k++]; s]; Array[f, 11, 0] (* CAUTION, very slow and requires a lot of resources beyond 10, Robert G. Wilson v, Apr 09 2012 *)
b[n_, h_, t_] := b[n, h, t] = If[t*h < n, 0, If[n == 0, 1, Sum[b[n-j, Min[h, n-j], t-1], {j, 0, Min[n, h]}]]]; a[n_] := b[n*(n-1)/2, n-1, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Sep 16 2013, after Alois P. Heinz *)
Table[Sum[Binomial[n, k] Binomial[n + Binomial[n, 2] - n k - 1, n - 1] (-1)^k, {k, 0, Floor[(n-1)/2] + If[n == 0, 1, 0]}], {n, 0, 100}] (* Emanuele Munarini, Jul 15 2016 *)

makelist(sum(binomial(n,k)*binomial(n+binomial(n,2)-n*k-1,n-1)*(-1)^k,k,0,floor((n-1)/2)), n, 1, 12); /* for n >= 1; Emanuele Munarini, Jul 15 2016 */

a(n) = if(n<1,n==0,sum(k=0, (n-1)\2, binomial(n,k)*binomial(n + binomial(n,2) - n*k - 1, n-1)*(-1)^k)); \\ Andrew Howroyd, Feb 27 2018

a(n) = A077042(n, n). Roughly n^(n-3/2)*sqrt(6/Pi) by the central limit theorem and something like n^n*sqrt(6/(Pi*(n^3+0.3*n^2-0.91*n+0.3))) seems to be even closer.
From Emanuele Munarini, Jul 15 2016: (Start)
a(n) = [x^binomial(n,2)](1+x+x^2+...+x^(n-1))^n.
a(n) = Sum_{k = 0..floor((n-1)/2)} binomial(n,k)*binomial(n + binomial(n,2) - n*k - 1, n-1)*(-1)^k for n >=1. (End)

A077046 Doubly restricted composition numbers: number of compositions of [n^2/2] into exactly n positive integers each strictly less than n.

Original entry on oeis.org

0, 1, 3, 19, 155, 1751, 24017, 398567, 7635987, 167729959, 4123838279, 112835748609, 3386455204288, 110976634957761, 3932912125462725, 150186639579545295, 6137695417757646851, 267654541150392845543, 12391407810082341898091, 607584722159224093306229

Offset: 1

Henry Bottomley, Oct 22 2002

nonn

			a(3) = 3 since the compositions of [3^2/2]=4 into exactly 3 positive integers each strictly less than 3 are: 1+1+2, 1+2+1 and 2+1+1.

Alois P. Heinz, Table of n, a(n) for n = 1..70
Index entries for sequences related to compositions

Cf. A077042, A077045, A077047, A077048.

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

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

A270918 Largest coefficient of (1+x+...+x^n)^(2*n).

Original entry on oeis.org

1, 2, 19, 580, 38165, 4395456, 786588243, 202384723528, 70886845397481, 32458256583753952, 18832730699014127291, 13507852690353224821652, 11738630472138500287398379, 12155701820213424461220851360, 14790850878997102285050287114419

Offset: 0

Vaclav Kotesovec, Mar 26 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..210

Cf. A077047.

Cf. A002426, A005190, A005191, A018901, A025012, A025013, A025014, A025015.

Table[Max[CoefficientList[Expand[Sum[x^k, {k, 0, n}]^(2n)], x]], {n, 0, 20}]

a(n) = vecmax(Vec((sum(k=0,n,x^k))^(2*n))); \\ Michel Marcus, Apr 01 2016

a(n) ~ exp(2) * sqrt(3/Pi) * n^(2*n - 3/2).

Typo in formula corrected by Vaclav Kotesovec, Dec 10 2021

A077048 Doubly restricted composition numbers: number of compositions of [n^2/2] into up to n positive integers each no more than n.

Original entry on oeis.org

1, 1, 2, 6, 44, 350, 4292, 57036, 996088, 18661152, 424241102, 10266964015, 287927722315, 8541887839459, 285144784116376, 10016752944160536, 388044155110294856, 15749295719801313452, 694726457246902921502

Offset: 0

Henry Bottomley, Oct 22 2002

nonn

			a(3)=6 since the number of compositions of [3^2/2]=4 into up to 3 positive integers each no more than 3 are: 1+1+2, 1+2+1, 1+3, 2+1+1, 2+2, 3+1.

Index entries for sequences related to compositions

Cf. A029895, A077045, A077046, A077047.

Similar in magnitude to A077045.

A369767 Maximal coefficient of Product_{i=1..n} Sum_{j=0..n} x^(i*j).

Original entry on oeis.org

1, 1, 2, 6, 31, 231, 2347, 29638, 449693, 7976253, 162204059, 3722558272, 95221978299, 2687309507102, 82967647793153, 2782190523572392, 100715040802229833, 3914979746952224303, 162662679830709439637, 7194483479557973730982, 337519906320930133470189

Offset: 0

Ilya Gutkovskiy, Jan 31 2024

nonn

Cf. A000041, A025591, A077047.

a:= n-> max(coeffs(expand(mul(add(x^(i*j), j=0..n), i=1..n)))):
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 31 2024

Table[Max[CoefficientList[Product[Sum[x^(i j), {j, 0, n}], {i, 1, n}], x]], {n, 0, 20}]

a(n) = vecmax(Vec(prod(i=1, n, sum(j=0, n, x^(i*j))))); \\ Michel Marcus, Jan 31 2024

from collections import Counter
def A369767(n):
    c = {j:1 for j in range(n+1)}
    for i in range(2,n+1):
        d = Counter()
        for k in c:
            for j in range(0,i*n+1,i):
                d[j+k] += c[k]
        c = d
    return max(c.values()) # Chai Wah Wu, Jan 31 2024

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A077045 Doubly restricted composition numbers: number of compositions of 1+2+3+...+n = n(n+1)/2 into exactly n positive integers each no more than n.

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A077046 Doubly restricted composition numbers: number of compositions of [n^2/2] into exactly n positive integers each strictly less than n.

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A270918 Largest coefficient of (1+x+...+x^n)^(2*n).

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A077048 Doubly restricted composition numbers: number of compositions of [n^2/2] into up to n positive integers each no more than n.

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A369767 Maximal coefficient of Product_{i=1..n} Sum_{j=0..n} x^(i*j).

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