A002621 -id:A002621 - cp's OEIS frontend

A301740 The number of trees with 5 nodes labeled by positive integers, where each tree's label sum is n.

3, 9, 24, 50, 96, 164, 267, 408, 603, 856, 1186, 1598, 2115, 2742, 3505, 4411, 5489, 6746, 8215, 9904, 11849, 14059, 16573, 19401, 22586, 26138, 30103, 34493, 39357, 44707, 50596, 57037, 64086, 71757, 80109, 89157, 98964, 109545, 120966, 133244, 146448, 160595, 175758, 191955

Offset: 5

R. J. Mathar, Mar 26 2018

Computed by the sum over the A000055(5)=3 shapes of the trees: the linear graph of the n-Pentane, the branched 2-Methyl-Butane, and the star graph of (1,1)-Bimethyl-Propane.

			a(5)=3 because there is a linear tree with all labels equal 1, the branched tree with all labels equal to 1, and the star tree with all labels equal 1.

R. J. Mathar, Labeled Trees with Fixed Node Label sum, sequence v_5.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-2,2,0,1,0,-2,1).

Cf. A002620 (labeled trees with 3 nodes), A301739 (labeled trees with 4 nodes).

-x^5*(3+3*x+6*x^2+5*x^3+5*x^4+2*x^5+x^6)/(1+x^2)/(1+x+x^2)/(1+x)^2/(x-1)^5 ;
taylor(%,x=0,80) ;
gfun[seriestolist](%) ;

LinearRecurrence[{2,0,-1,0,-2,2,0,1,0,-2,1},{3,9,24,50,96,164,267,408,603,856,1186},50] (* Harvey P. Dale, Jun 16 2025 *)

a(n) = A005994(n-5)+A001752(n-5)+A002621(n-5).

A259324 Infinite square array read by antidiagonals: T(n,k) = number of ways of partitioning numbers <= n into k parts (n >= 0, k >= 1).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 4, 6, 5, 1, 2, 4, 7, 9, 6, 1, 2, 4, 7, 11, 12, 7, 1, 2, 4, 7, 12, 16, 16, 8, 1, 2, 4, 7, 12, 18, 23, 20, 9, 1, 2, 4, 7, 12, 19, 27, 31, 25, 10, 1, 2, 4, 7, 12, 19, 29, 38, 41, 30, 11, 1, 2, 4, 7, 12, 19, 30, 42, 53, 53, 36, 12, 1, 2, 4, 7, 12, 19, 30, 44, 60, 71, 67, 42, 13, 1, 2, 4, 7, 12, 19, 30, 45, 64, 83, 94, 83, 49, 14, 1, 2

Offset: 0

N. J. A. Sloane, Jun 24 2015

			The first few antidiagonals are:
1,
1,2,
1,2,3,
1,2,4,4,
1,2,4,6,5,
1,2,4,7,9,6,
1,2,4,7,11,12,7,
1,2,4,7,12,16,16,8,
...

E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. See Table I.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]

Columns give A002620, A000601, A002621, A002622.

Cf. A137679.

A259324 := proc(u,m)
    option remember;
    if u = 0 then
        1;
    elif u < 0 then
        0;
    elif m = 1 then
        u+1 ;
    else
        procname(u,m-1)+procname(u-m,m) ;
    end if;
end proc:
for d from 1 to 15 do
    for m from d to 1 by -1 do
        printf("%d,",A259324(d-m,m)) ;
    end do:
end do: # R. J. Mathar, Jul 14 2015

T[0, ] = 1; T[u /; u > 0, m_ /; m > 1] := T[u, m] = T[u, m - 1] + T[u - m, m]; T[u_, 1] := u + 1; T[, ] = 0;
Table[T[u - m, m], {u, 0, 14}, {m, u, 1, -1}] // Flatten (* Jean-François Alcover, Apr 05 2020 *)

T(u,m) = T(u,m-1)+T(u-m,m), with initial conditions T(0,m)=1, T(m,1)=u+1.

cp's OEIS Frontend

A301740 The number of trees with 5 nodes labeled by positive integers, where each tree's label sum is n.

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