A001979 -id:A001979 - cp's OEIS frontend

A067059 Square array read by antidiagonals of partitions which half fill an n*k box, i.e., partitions of floor(nk/2) or ceiling(nk/2) into up to n positive integers, each no more than k.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 1, 4, 6, 8, 6, 4, 1, 1, 1, 1, 4, 8, 12, 12, 8, 4, 1, 1, 1, 1, 5, 10, 18, 20, 18, 10, 5, 1, 1, 1, 1, 5, 13, 24, 32, 32, 24, 13, 5, 1, 1, 1, 1, 6, 15, 33, 49, 58, 49, 33, 15, 6, 1, 1, 1, 1, 6

Offset: 0

Henry Bottomley, Feb 17 2002

The number of partitions of m into up to n positive integers each no more than k is maximized for given n and k by m=floor(nk/2) or ceiling(nk/2) (and possibly some other values).

			Rows start:
1, 1, 1, 1, 1, 1, ...;
1, 1, 1, 1, 1, 1, ...;
1, 1, 2, 2, 3, 3, ...;
1, 1, 2, 3, 5, 6, ...;
1, 1, 3, 5, 8, 12, ...; etc.
T(4,5)=12 since 10 can be partitioned into
5+5, 5+4+1, 5+3+2, 5+3+1+1, 5+2+2+1, 4+4+2, 4+3+3,
4+4+1+1, 4+3+2+1, 4+2+2+2, 3+3+3+1, and 3+3+2+2.

As this is symmetric, rows and columns each include A000012 twice, A008619, A001971, A001973, A001975, A001977, A001979 and A001981. Diagonal is A029895. T(n, n*(n-1)) is the magic series A052456.

A067059 := proc(n,k)
    local m,a1,a2 ;
    a1 := 0 ;
    m := floor(n*k/2) ;
    for L in combinat[partition](m) do
        if nops(L) <= n then
            if max(op(L)) <= k then
                a1 := a1+1 ;
            end if ;
        end if;
    end do:
    a2 := 0 ;
    m := ceil(n*k/2) ;
    for L in combinat[partition](m) do
        if nops(L) <= n then
            if max(op(L)) <= k then
                a2 := a2+1 ;
            end if ;
        end if;
    end do:
    max(a1,a2) ;
end proc:
for d from 0 to 12 do
    for k from 0 to d do
        printf("%d,",A067059(d-k,k)) ;
    end do:
end do: # R. J. Mathar, Nov 13 2016

t[n_, k_] := Length[ IntegerPartitions[ Floor[n*k/2], n, Range[k]]]; Flatten[ Table[ t[n-k , k], {n, 0, 13}, {k, 0, n}]] (* Jean-François Alcover, Jan 02 2012 *)

def A067059(n, k):
    return Partitions((n*k)//2, max_length=n, max_part=k).cardinality()
for n in (0..9): [A067059(n,k) for k in (0..9)] # Peter Luschny, May 05 2014

A001980 Number of partitions of floor(7n/2)-1 into n nonnegative integers each no greater than 7.

Original entry on oeis.org

0, 1, 4, 10, 23, 48, 94, 166, 285, 464, 734, 1109, 1646, 2371, 3366, 4652, 6357, 8519, 11309, 14754, 19103, 24399, 30956, 38797, 48355, 59665, 73264, 89145, 108011, 129864, 155554, 185017, 219336, 258438, 303604, 354665, 413213, 479048, 554033

Offset: 0

N. J. A. Sloane

In Cayley's terminology, this is the number of literal terms of degree n and of weight floor(7n/2)-1 involving the letters a, b, c, d, e, f, g, h, having weights 0, 1, 2, 3, 4, 5, 6, 7 respectively. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 86 terms from Vincenzo Librandi)
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1, 2, -1, -2, 0, -1, 0, 1, 2, 2, -1, -2, 1, -4, 0, 4, -1, 2, 1, -2, -2, -1, 0, 1, 0, 2, 1, -2, 1, -1, -1, 1).

Cf. A001979.

f=1/((1-z)*(1-x*z)*(1-x^2*z)*(1-x^3*z)*(1-x^4*z)*(1-x^5*z)*(1-x^6*z)*(1-x^7*z)); n=400; p=subst(subst(f,x,x+x*O(x^n)),z,z+z*O(z^n)); for(d=0,60,w=floor(7*d/2)-1;print1(polcoeff(polcoeff(p,w),d)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

Coefficient of x^w*z^n in the expansion of 1/((1-z)(1-xz)(1-x^2z)(1-x^3z)(1-x^4z)(1-x^5z)(1-x^6z)(1-x^7z)), where w=floor(7n/2)-1. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

G.f.: -(x^24 +3*x^23 +5*x^22 +10*x^21 +17*x^20 +26*x^19 +33*x^18 +45*x^17 +55*x^16 +61*x^15 +63*x^14 +68*x^13 +67*x^12 +68*x^11 +63*x^10 +61*x^9 +55*x^8 +45*x^7 +33*x^6 +26*x^5 +17*x^4 +10*x^3 +5*x^2 +3*x +1)*x / ((x^4+x^3+x^2+x+1) *(x^4-x^2+1) *(x^2+x+1)^2 *(x^2-x +1)^2 *(x^2+1)^3 *(x+1)^5 *(x-1)^7). - Alois P. Heinz, Jul 25 2015

Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

a(0)=0 inserted by Alois P. Heinz, Jul 25 2015

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A067059 Square array read by antidiagonals of partitions which half fill an n*k box, i.e., partitions of floor(nk/2) or ceiling(nk/2) into up to n positive integers, each no more than k.

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A001980 Number of partitions of floor(7n/2)-1 into n nonnegative integers each no greater than 7.

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