cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Henry Bottomley, Oct 22 2002

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • Maxima
    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 */
  • PARI
    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

Formula

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)

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

Views

Author

Henry Bottomley, Oct 22 2002

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

a(n) = A077042(n+1, n).
a(n) ~ exp(1) * sqrt(6/Pi) * n^(n-3/2). - Vaclav Kotesovec, Mar 26 2016

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

Views

Author

Henry Bottomley, Oct 22 2002

Keywords

Examples

			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.

Links

Crossrefs

Cf. A077042, A077045, A077047, A077048.

Formula

a(n) = A077042(n-1, n).
a(n) ~ exp(-1)*sqrt(6/Pi)*n^(n-3/2). - Vaclav Kotesovec, Mar 26 2016
Showing 1-3 of 3 results.

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