A000601 -id:A000601 - cp's OEIS frontend

A301739 The number of trees with 4 nodes labeled by positive integers, where each tree's label sum is n.

2, 4, 10, 17, 30, 44, 67, 91, 126, 163, 213, 265, 333, 403, 491, 582, 693, 807, 944, 1084, 1249, 1418, 1614, 1814, 2044, 2278, 2544, 2815, 3120, 3430, 3777, 4129, 4520, 4917, 5355, 5799, 6287, 6781, 7321, 7868, 8463, 9065, 9718, 10378, 11091, 11812, 12588, 13372, 14214, 15064

Offset: 4

R. J. Mathar, Mar 26 2018

Computed by the sum over the A000055(4)=2 shapes of the trees: the linear graph of the n-Butane, and the star graph of (1)-Methyl-Propane.

			a(4)=2 because there is a linear tree with all labels equal 1 and the star tree with all labels equal to 1.

Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1).

4th column of A303841.

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

a(n) = A005993(n-4)+A000601(n-4).

G.f.: x^4*(2+2*x+2*x^2+x^3+x^4)/((1+x)^2*(x-1)^4*(1+x+x^2) ).

A099770 Expansion of 1/((1-x)(1-x^2)(1-x^4)*(1-x^6)).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 11, 11, 16, 16, 23, 23, 31, 31, 41, 41, 53, 53, 67, 67, 83, 83, 102, 102, 123, 123, 147, 147, 174, 174, 204, 204, 237, 237, 274, 274, 314, 314, 358, 358, 406, 406, 458, 458, 514, 514, 575, 575, 640, 640, 710, 710, 785, 785, 865, 865, 950, 950, 1041, 1041

Offset: 0

G. Nebe (nebe(AT)math.rwth-aachen.de), Nov 10 2004

nonn

Molien series for symmetrized weight enumerators of Hermitian self-dual codes over the Galois ring GR(4,2).
Number of partitions of n into parts 1, 2, 4, and 6. - Joerg Arndt, May 05 2014
a(n) is equal to the number of partitions of degree at most n+6 of length 3 with even entries. - John M. Campbell, Jan 20 2016

			From _John M. Campbell_, Jan 20 2016: (Start)
Letting n=6, a(n) = 7 is equal to the number of partitions of n into parts 1, 2, 4, and 6, as illustrated below, and a(n) is equal to the number of partitions of degree at most n+6 of length 3 with even entries, as illustrated below. The arrows below illustrate a natural bijection between the set of partitions of the former form and the set of partitions of the latter form.
(2, 2, 2) <-> (1, 1, 1, 1, 1, 1)
(4, 2, 2) <-> (2, 1, 1, 1, 1)
(6, 2, 2) <-> (4, 1, 1)
(4, 4, 2) <-> (2, 2, 1, 1)
(8, 2, 2) <-> (6)
(6, 4, 2) <-> (4, 2)
(4, 4, 4) <-> (2, 2, 2)
(End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,0,0,-1,1,-1,1,1,-1).

Cf. A000601, A004526.

a:=[1,1,2,2,4,4,7,7,11,11,16,16,23];; for n in [14..65] do a[n]:= a[n-1]+a[n-2]-a[n-3]+a[n-4]-a[n-5]-a[n-8]+a[n-9]-a[n-10]+a[n-11]+a[n-12] -a[n-13]; od; a; # G. C. Greubel, Sep 04 2019

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( 1/((1-x)*(1-x^2)*(1-x^4)*(1-x^6)) )); // G. C. Greubel, Sep 04 2019

seq(coeff(series(1/((1-x)*(1-x^2)*(1-x^4)*(1-x^6)), x, n+1), x, n), n = 0 .. 65); # G. C. Greubel, Sep 04 2019

CoefficientList[Series[1/((1-x)*(1-x^2)*(1-x^4)*(1-x^6)), {x, 0, 65}], x] (* G. C. Greubel, Sep 04 2019 *)

Vec(1/((1-x)*(1-x^2)*(1-x^4)*(1-x^6)) + O(x^80)) \\ Michel Marcus, Jan 21 2016

def A099770_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)*(1-x^2)*(1-x^4)*(1-x^6))).list()
A099770_list(65) # G. C. Greubel, Sep 04 2019

a(n) ~ 1/288*n^3. - Ralf Stephan, Apr 29 2014
a(n) = (2*n^3 +39*n^2 +241*n +372 +3*(n^2 +13*n +40) * (-1)^n -84*(-1)^((2*n +3 +(-1)^n)/4) -192*floor(((2*n +9 +(-1)^n-6*(-1)^((2*n+3+(-1)^n)/4))/24)))/576. - Luce ETIENNE, May 05 2014
a(n) = A000601(A004526(n)). - Hoang Xuan Thanh, Jun 21 2025

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.

A282044 Reduced Kronecker coefficients for the case a=2, b=3, i=4.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 9, 16, 24, 37, 51, 71, 93

Offset: 0

N. J. A. Sloane, Feb 21 2017

Table 3 of Colmenarejo (2016) shows this sequence as the missing member of the family A266769, A000601, A006918, A014126, A282044, A175287.

It would be nice to have a g.f.

Laura Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016.

Cf. A266769, A000601, A006918, A014126, A175287.

Conjectured g.f.: x^4*(1+x^2)/(1-2*x+x^3+3*x^5-4*x^6). - Jean-François Alcover, Feb 18 2019.

A164679 Convolve A001399 with sequences which map to 2,3,5,7,11,13,17... A000040 then, by bending when needed, summarize the results in a triangular array.

Original entry on oeis.org

1, 2, 1, 5, 4, 2, 10, 9, 7, 3, 19, 18, 16, 11, 4, 33, 32, 30, 25, 16, 5, 57

Offset: 1

Alford Arnold, Sep 05 2009

Apparently the terms can be constructed by fixing the generating function of the diagonal g_0(x) = 1/(1-x)/(1-x^2)/(1-x^3), A001399, and deriving the generating function of the i-th subdiagonal by g_i(x) = g_{i-1}(x)/(1-x^i), i>=1. - R. J. Mathar, May 17 2016

			1;
2, 1;
5, 4, 2;
10, 9, 7, 3;
19, 18, 16, 11, 4;
33, 32, 30, 25, 16, 5;
57

Cf. A000098 (first column), A164678 (a similar triangle). Diagonals are A001399, A000601, A097701, A117485, ...

A259325 Infinite square array T(n,k) read by antidiagonals, defined by T(n,k) = T(n,k-1)+T(n-k,k), T(0,k)=1 (n >= 0, k >= 1).

Original entry on oeis.org

1, 1, 3, 1, 3, 7, 1, 3, 8, 13, 1, 3, 8, 16, 22, 1, 3, 8, 17, 30, 34, 1, 3, 8, 17, 33, 50, 50, 1, 3, 8, 17, 34, 58, 80, 70, 1, 3, 8, 17, 34, 61, 97, 120, 95

Offset: 0

N. J. A. Sloane, Jun 24 2015

			The first few antidiagonals are:
1
1,3,
1,3,7
1,3,8,13
1,3,8,16,22
1,3,8,17,30,34
1,3,8,17,33,50,50
1,3,8,17,34,58,80,70
1,3,8,17,34,61,97,120,95
...

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

Columns give A002623, A002624, A002625, A002626.

cp's OEIS Frontend

A301739 The number of trees with 4 nodes labeled by positive integers, where each tree's label sum is n.

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A099770 Expansion of 1/((1-x)*(1-x^2)*(1-x^4)*(1-x^6)).

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

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A282044 Reduced Kronecker coefficients for the case a=2, b=3, i=4.

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A164679 Convolve A001399 with sequences which map to 2,3,5,7,11,13,17... A000040 then, by bending when needed, summarize the results in a triangular array.

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A259325 Infinite square array T(n,k) read by antidiagonals, defined by T(n,k) = T(n,k-1)+T(n-k,k), T(0,k)=1 (n >= 0, k >= 1).

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A099770 Expansion of 1/((1-x)(1-x^2)(1-x^4)*(1-x^6)).