A073075 -id:A073075 - cp's OEIS frontend

A115593 Number of forests of rooted trees with total weight n, where a node at height k has weight 2^k (with root considered to be at height 0).

1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 13, 17, 22, 29, 38, 50, 64, 82, 107, 136, 175, 224, 288, 363, 465, 587, 748, 942, 1196, 1503, 1902, 2385, 3004, 3765, 4729, 5911, 7406, 9246, 11549, 14395, 17941, 22326, 27767, 34501, 42821, 53134, 65828, 81546, 100871, 124783

Offset: 0

Franklin T. Adams-Watters, Mar 09 2006

The sequence b(2n)=0, b(2n+1)=a(n) is the number of trees of weight n.

			a(3) = 2; one forest with 3 single-node trees and one with a single two-node tree (root node has weight 1, other node has weight 2).

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Hartosh Singh Bal, Combinatorics of a Class of Completely Additive Arithmetic Functions, arXiv:2308.00455 [math.CO], 2023.

Cf. A000081, A117356, A363336, A363337.

with(numtheory):
b:= proc(n) local r; `if`(irem(n, 2, 'r')=0, 0, a(r)) end:
a:= proc(n) option remember; `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 16 2013

b[n_] := b[n] = If[{q, r} = QuotientRemainder[n, 2]; r == 0, 0, a[q]]; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 23 2015, after Alois P. Heinz *)

{a(n)=my(A=1+x); for(i=1, n, A=exp(sum(m=1, n, subst(A,x,x^(2*m)+x*O(x^n))*x^m/m))); polcoeff(A,n)} /* Paul D. Hanna */

Euler transform of b(n), where b(2n+1) = a(n) and b(2n) = 0.
From Paul D. Hanna, Oct 26 2011: (Start)
G.f. satisfies: A(x) = exp( Sum_{n>=1} A(x^(2*n)) * x^n/n ).
G.f. satisfies: A(x)*A(-x) = A(x^2). (End)
Let b(n) = a(n-1) for n>=1, then sum(n>=1, b(n)*x^n ) = x / prod(n>=1, (1-x^(2*n-1))^b(n) ); compare to A000081, A004111, and A073075. - Joerg Arndt, Mar 04 2015
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k and d is odd} d * a(floor(d/2)) ) * a(n-k). - Seiichi Manyama, May 31 2023

A248869 Satisfies Sum_{n>=0} a(n)x^n = x Product_{n>=0} (1 + x^n + x^(2*n))^a(n).

Original entry on oeis.org

0, 1, 1, 2, 3, 7, 15, 34, 79, 190, 459, 1136, 2833, 7154, 18206, 46723, 120656, 313514, 818763, 2148434, 5660790, 14972103, 39734107, 105779291, 282403830, 755921733, 2028277115, 5454368549, 14697955778, 39682793675, 107330573239, 290783511134, 789032648219

Offset: 0

Joerg Arndt, Mar 04 2015

nonn

What kind of trees are counted by this sequence (compare with A000081, A004111, A073075, and A115593)?

a(n) is the number of rooted trees of n vertices that have everywhere at most 2 siblings with the same (i.e., isomorphic) subtree below. The g.f. assembles a(n) as a root with child subtrees from among the smaller a(), but takes only 0, 1 or 2 copies of any one of them. Compare asymmetric trees A004111 g.f. which takes 0 or 1 copies. Here the x^(2*n) term allows a 2nd copy. The siblings condition is equivalent to the condition that the tree automorphisms form a 2-group, i.e., group order some power 2^k. 2 same siblings are a swap. 3 same siblings would be an element of order 3 and hence factor 3 in the group order. a(n) >= A213920 since the latter limits same size siblings, whereas here only limits same size plus structure. - Kevin Ryde, Jul 11 2019

Alois P. Heinz, Table of n, a(n) for n = 0..2213

Cf. A000081, A004111, A073075, A115593, A213920, A248870, A309352.

Column k=2 of A318757.

h:= proc(n, m, t) option remember; `if`(m=0, binomial(n+t, t),
      `if`(n=0, 0, add(h(n-1, m-j, t+1), j=1..min(2, m))))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*h(a(i), j, 0), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, b(n-1$2)):
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 04 2018

h[n_, m_, t_] := h[n, m, t] = If[m == 0, Binomial[n + t, t], If[n == 0, 0, Sum[h[n - 1, m - j, t + 1], {j, 1, Min[2, m]}]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1]* h[a[i], j, 0], {j, 0, n/i}]]];
a[n_] := If[n < 2, n, b[n - 1, n - 1]];
a /@ Range[0, 32] (* Jean-François Alcover, Oct 02 2019, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = 2.8458470164106425911151048..., c = 0.41694347809945986693376... . - Vaclav Kotesovec, Mar 17 2015

a(n) = A004111(n) + A318859(n). - Kevin Ryde, Jul 11 2019

A306768 G.f. A(x) satisfies: A(x) = x*exp(-A(-x) + A(-x^2)/2 - A(-x^3)/3 + A(-x^4)/4 - A(-x^5)/5 + ...).

Original entry on oeis.org

0, 1, 1, -1, -2, 2, 6, -5, -18, 15, 59, -54, -215, 199, 813, -744, -3135, 2890, 12394, -11538, -50017, 46806, 204893, -192451, -849681, 800974, 3560927, -3367656, -15058478, 14279426, 64171736, -60992032, -275304665, 262199050, 1188070488, -1133572891, -5153913606

Offset: 0

Ilya Gutkovskiy, Apr 14 2019

sign

			G.f.: A(x) = x + x^2 - x^3 - 2*x^4 + 2*x^5 + 6*x^6 - 5*x^7 - 18*x^8 + 15*x^9 + 59*x^10 - 54*x^11 - 215*x^12 + ...

Cf. A000081, A004111, A045648, A049075, A073075, A115593, A307365, A307366, A307538.

terms = 36; A[] = 0; Do[A[x] = x Exp[Sum[(-1)^k A[-x^k]/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 + x^k)^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = 0; Table[a[n], {n, 0, 36}]

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} 1/(1 + x^n)^((-1)^n*a(n)).

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+d)*d*a(d) ) * a(n-k+1).

A248870 Satisfies Sum_{n>=0} a(n)*x^n = x / Product_{n>=0} (1 - x^n/(1 - x^n))^a(n).

Original entry on oeis.org

0, 1, 1, 3, 8, 23, 62, 181, 513, 1513, 4476, 13483, 40933, 125845, 389769, 1217590, 3828775, 12115966, 38546124, 123238296, 395725493, 1275733730, 4127339091, 13396443708, 43610621823, 142354979662, 465838195260, 1527905193504, 5022061115901, 16539625666670, 54571760414658

Offset: 0

Joerg Arndt, Mar 04 2015

nonn

Which kind of trees is counted by this sequence (compare to A000081, A004111, A073075, A248869 and A115593)?

Cf. A248869.

A307538 G.f. A(x) satisfies: A(x) = xexp(2A(-x) + 2A(-x^3)/3 + 2A(-x^5)/5 + 2A(-x^7)/7 + 2A(-x^9)/9 + ...).

Original entry on oeis.org

0, 1, -2, -2, 10, 14, -86, -126, 858, 1302, -9378, -14606, 108954, 172698, -1319966, -2119118, 16489594, 26731542, -210887998, -344490170, 2747510514, 4515757426, -36336187630, -60023827438, 486540793914, 807121753178, -6582918170714, -10959656342678, 89860260268098

Offset: 0

Ilya Gutkovskiy, Apr 14 2019

sign

			G.f.: A(x) = x - 2*x^2 - 2*x^3 + 10*x^4 + 14*x^5 - 86*x^6 - 126*x^7 + 858*x^8 + 1302*x^9 - 9378*x^10 - 14606*x^11 + ...

Cf. A000081, A004111, A045648, A049075, A073075, A115593, A306768, A307365, A307366.

terms = 28; A[] = 0; Do[A[x] = x Exp[Sum[2 A[-x^(2 k - 1)]/(2 k - 1), {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = SeriesCoefficient[x Product[((1 + x^k)/(1 - x^k))^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = 0; Table[a[n], {n, 0, 28}]

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} ((1 + x^n)/(1 - x^n))^((-1)^n*a(n)).

Recurrence: a(n+1) = (2/n) * Sum_{k=1..n} ( Sum_{d|k, k/d odd} (-1)^d*d*a(d) ) * a(n-k+1).

A299021 G.f.: A(x) = Sum_{n>=1} a(n)x^n = xProduct_{n>=1} (1 + a(n)x^n)/(1 - a(n)x^n).

Original entry on oeis.org

1, 2, 6, 22, 86, 358, 1558, 6966, 31894, 148918, 705062, 3380054, 16381158, 80056550, 394266950, 1955139942, 9749771926, 48873487942, 246160229782, 1244801094742, 6318514387638, 32184084454166, 164425969781062, 842429440124854, 4327629345403078, 22283328480744070

Offset: 1

Ilya Gutkovskiy, Jun 18 2018

nonn

			G.f.: A(x) = x + 2*x^2 + 6*x^3 + 22*x^4 + 86*x^5 + ... = x * ((1 + x) * (1 + 2*x^2) * (1 + 6*x^3) * (1 + 22*x^4) * (1 + 86*x^5) * ...) / ((1 - x) * (1 - 2*x^2) * (1 - 6*x^3) * (1 - 22*x^4) * (1 - 86*x^5) * ...).

Cf. A032305, A073075, A093637.

a[n_] := a[n] = SeriesCoefficient[x Product[(1 + a[k] x^k)/(1 - a[k] x^k), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 26}]
a[n_] := a[n] = SeriesCoefficient[x Exp[Sum[Sum[(1 + (-1)^(k + 1)) a[j]^k x^(j k)/k, {j, 1, n - 1}], {k, 1, n - 1}]], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 26}]

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x*exp(Sum_{k>=1} Sum_{n>=1} (1 + (-1)^(k+1))*a(n)^k*x^(n*k)/k).

A307722 G.f. A(x) satisfies: A(x) = xexp(2Sum_{n>=1} Sum_{k>=1} na(n)x^(n(2k-1))/(2*k - 1)).

Original entry on oeis.org

0, 1, 2, 10, 78, 794, 9870, 143610, 2382350, 44266538, 909575170, 20468012850, 500542618118, 13218631046786, 374965272837542, 11372416113131346, 367296622702990270, 12587154399475110546, 456238999451039779510, 17440439387336903608866, 701272672299320517560470

Offset: 0

Ilya Gutkovskiy, Apr 24 2019

nonn

			G.f.: A(x) = x + 2*x^2 + 10*x^3 + 78*x^4 + 794*x^5 + 9870*x^6 + 143610*x^7 + 2382350*x^8 + 44266538*x^9 + ...

Cf. A073075, A307708, A307709.

a[n_] := a[n] = SeriesCoefficient[x Exp[2 Sum[Sum[j a[j] x^(j (2 k - 1))/(2 k - 1), {k, 1, n - 1}], {j, 1, n - 1}]], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 20}]
a[n_] := a[n] = SeriesCoefficient[x Product[((1 + x^k)/(1 - x^k))^(k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 20}]

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} ((1 + x^n)/(1 - x^n))^(n*a(n)).

A308152 G.f.: x * Product_{j>=1, k>=1} ((1 + x^(jk))/(1 - x^(jk)))^a(j).

Original entry on oeis.org

1, 2, 8, 32, 138, 612, 2864, 13712, 67416, 337482, 1716208, 8837392, 45997032, 241571408, 1278625480, 6813568656, 36524390042, 196820310100, 1065583770168, 5793299764208, 31615962617272, 173131117881312, 951040865156928, 5239171609158304, 28937688613453048

Offset: 1

Ilya Gutkovskiy, May 14 2019

nonn

Cf. A073075, A179467, A308060.

a[n_] := a[n] = SeriesCoefficient[x Product[Product[((1 + x^(j k))/(1 - x^(j k)))^a[j], {k, 1, n - 1}], {j, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 25}]
terms = 25; A[] = 0; Do[A[x] = x Exp[2 Sum[Sum[A[x^(i (2 j - 1))]/(2 j - 1), {j, 1, terms}], {i, 1,terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]

G.f. A(x) satisfies: A(x) = x * exp(2 * Sum_{i>=1} Sum_{j>=1} A(x^(i*(2*j-1)))/(2*j - 1)).

A308227 G.f.: (x/(1 - x)) * Product_{k>=1} ((1 + x^k)/(1 - x^k))^a(k).

Original entry on oeis.org

1, 3, 11, 47, 217, 1065, 5453, 28789, 155633, 857207, 4793103, 27136555, 155249971, 896133487, 5212477023, 30522169103, 179777122393, 1064411910393, 6331361864657, 37817265028841, 226731778956181, 1363993567341257, 8231111557650837, 49812263080757845

Offset: 1

Ilya Gutkovskiy, May 16 2019

nonn

Cf. A029856, A036249, A052829, A073075.

a[n_] := a[n] = SeriesCoefficient[x/(1 - x) Product[((1 + x^k)/(1 - x^k))^a[k], {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 24}]
terms = 24; A[] = 0; Do[A[x] = x Exp[Sum[2 A[x^(2 k - 1)]/(2 k - 1) + x^k/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x * exp(Sum_{k>=1} 2*A(x^(2*k-1))/(2*k - 1) + x^k/k).

cp's OEIS Frontend

A115593 Number of forests of rooted trees with total weight n, where a node at height k has weight 2^k (with root considered to be at height 0).

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A248869 Satisfies Sum_{n>=0} a(n)*x^n = x * Product_{n>=0} (1 + x^n + x^(2*n))^a(n).

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A306768 G.f. A(x) satisfies: A(x) = x*exp(-A(-x) + A(-x^2)/2 - A(-x^3)/3 + A(-x^4)/4 - A(-x^5)/5 + ...).

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A248870 Satisfies Sum_{n>=0} a(n)*x^n = x / Product_{n>=0} (1 - x^n/(1 - x^n))^a(n).

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A307538 G.f. A(x) satisfies: A(x) = x*exp(2*A(-x) + 2*A(-x^3)/3 + 2*A(-x^5)/5 + 2*A(-x^7)/7 + 2*A(-x^9)/9 + ...).

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A299021 G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x*Product_{n>=1} (1 + a(n)*x^n)/(1 - a(n)*x^n).

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A307722 G.f. A(x) satisfies: A(x) = x*exp(2*Sum_{n>=1} Sum_{k>=1} n*a(n)*x^(n*(2*k-1))/(2*k - 1)).

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A308152 G.f.: x * Product_{j>=1, k>=1} ((1 + x^(j*k))/(1 - x^(j*k)))^a(j).

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A308227 G.f.: (x/(1 - x)) * Product_{k>=1} ((1 + x^k)/(1 - x^k))^a(k).

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A248869 Satisfies Sum_{n>=0} a(n)x^n = x Product_{n>=0} (1 + x^n + x^(2*n))^a(n).

A307538 G.f. A(x) satisfies: A(x) = xexp(2A(-x) + 2A(-x^3)/3 + 2A(-x^5)/5 + 2A(-x^7)/7 + 2A(-x^9)/9 + ...).

A299021 G.f.: A(x) = Sum_{n>=1} a(n)x^n = xProduct_{n>=1} (1 + a(n)x^n)/(1 - a(n)x^n).

A307722 G.f. A(x) satisfies: A(x) = xexp(2Sum_{n>=1} Sum_{k>=1} na(n)x^(n(2k-1))/(2*k - 1)).

A308152 G.f.: x * Product_{j>=1, k>=1} ((1 + x^(jk))/(1 - x^(jk)))^a(j).