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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. 7
 1, 1, 3, 12, 85, 780, 9331, 134512, 2306025, 45433800, 1018872811, 25506741084, 707972099627, 21518492021208, 712601187601395, 25491847538274240, 981272544393935569, 40392787067756440272, 1772592132899627652691 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..375 Index entries for sequences related to compositions FORMULA a(n) = A077042(n+1, n). a(n) ~ exp(1) * sqrt(6/Pi) * n^(n-3/2). - Vaclav Kotesovec, Mar 26 2016 EXAMPLE 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. 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 *) CROSSREFS Cf. A077042, A077045, A077046, A077048, A270918. Sequence in context: A355322 A070825 A232934 * A074505 A260912 A266788 Adjacent sequences: A077044 A077045 A077046 * A077048 A077049 A077050 KEYWORD nonn AUTHOR Henry Bottomley, Oct 22 2002 STATUS approved

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Last modified September 12 20:35 EDT 2024. Contains 375854 sequences. (Running on oeis4.)