A301739 -id:A301739 - cp's OEIS frontend

A303841 Triangle read by rows: T(s,n) (s>=1 and 1<=n<=s) = number of weighted trees with n nodes and positive integer node labels with label sum s.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 4, 4, 3, 1, 3, 6, 10, 9, 6, 1, 3, 9, 17, 24, 20, 11, 1, 4, 12, 30, 50, 63, 48, 23, 1, 4, 16, 44, 96, 146, 164, 115, 47, 1, 5, 20, 67, 164, 315, 437, 444, 286, 106, 1, 5, 25, 91, 267, 592, 1022, 1300, 1204, 719, 235, 1, 6, 30, 126, 408, 1059, 2126, 3331, 3899, 3328, 1842, 551

Offset: 1

R. J. Mathar, May 01 2018

			The triangle starts
1;
1 1;
1 1  1;
1 2  2   2;
1 2  4   4    3;
1 3  6  10    9   6;
1 3  9  17   24   20    11;
1 4 12  30   50   63    48    23;
1 4 16  44   96  146   164   115    47;
1 5 20  67  164  315   437   444   286  106;
1 5 25  91  267  592  1022  1300  1204  719     235;
1 6 30 126  408 1059  2126  3331  3899 3328    1842    551;
1 6 36 163  603 1754  4098  7511 10781 11692   9233   4766  1301;
1 7 42 213  856 2805  7368 15619 26294 34844  35136  25865 12486  3159;
1 7 49 265 1186 4270 12590 30111 58485 91037 112036 105592 72734 32973 7741;

Andrew Howroyd, Table of n, a(n) for n = 1..1275
F. Harary, G. Prins, The number of homeomorphically irreducible trees and other species, Acta. Math. 101 (1959) 141, equation (9b).
R. J. Mathar, Labeled trees with fixed node label sum vixra:1805.0205 (2018).
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.

Cf. A036250 (row sums), A002620 (column 3), A301739 (column 4), A301740 (column 5), A000055 (diagonal), A000081 (subdiagonal), A303911 (rooted).

\\ here b is A303911
EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp(sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], v + EulerMT(y*v))); v}
seq(n)={my(g=x*Ser(y*b(n))); Vec(g - g^2/2 + substvec(g,[x,y],[x^2,y^2])/2)}
{my(A=seq(15)); for(n=1, #A, print(Vecrev(A[n]/y)))} \\ Andrew Howroyd, May 19 2018

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

Original entry on oeis.org

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).

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A303841 Triangle read by rows: T(s,n) (s>=1 and 1<=n<=s) = number of weighted trees with n nodes and positive integer node labels with label sum s.

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

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