A032037 -id:A032037 - cp's OEIS frontend

A319138 Number of complete strict planar branching factorizations of n.

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 4, 1, 2, 2, 0, 1, 4, 1, 4, 2, 2, 1, 8, 0, 2, 0, 4, 1, 18, 1, 0, 2, 2, 2, 28, 1, 2, 2, 8, 1, 18, 1, 4, 4, 2, 1, 16, 0, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 84, 1, 2, 4, 0, 2, 18, 1, 4, 2, 18, 1, 112, 1, 2, 4, 4, 2, 18, 1, 16, 0, 2, 1

Offset: 1

Gus Wiseman, Sep 11 2018

nonn

A strict planar branching factorization of n is either the number n itself or a sequence of at least two strict planar branching factorizations, one of each factor in a strict ordered factorization of n. A strict planar branching factorization is complete if the leaves are all prime numbers.

			The a(12) = 4 trees: (2*(2*3)), (2*(3*2)), ((2*3)*2), ((3*2)*2).

Cf. A000108, A045778, A074206, A118376, A277130, A281113, A281118, A295279, A295281, A317144, A319122, A319123, A319136, A319137.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
sotfs[n_]:=Prepend[Join@@Table[Tuples[sotfs/@f],{f,Select[ordfacs[n],And[Length[#]>1,UnsameQ@@#]&]}],n];
Table[Length[Select[sotfs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,100}]

a(prime^n) = A000007(n - 1).

a(product of n distinct primes) = A032037(n).

A319136 Number of complete planar branching factorizations of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 9, 1, 2, 2, 11, 1, 9, 1, 9, 2, 2, 1, 44, 1, 2, 3, 9, 1, 18, 1, 45, 2, 2, 2, 66, 1, 2, 2, 44, 1, 18, 1, 9, 9, 2, 1, 225, 1, 9, 2, 9, 1, 44, 2, 44, 2, 2, 1, 132, 1, 2, 9, 197, 2, 18, 1, 9, 2, 18, 1, 450, 1, 2, 9, 9, 2, 18, 1, 225

Offset: 1

Gus Wiseman, Sep 11 2018

nonn

A planar branching factorization of n is either the number n itself or a sequence of at least two planar branching factorizations, one of each factor in an ordered factorization of n. A planar branching factorization is complete if the leaves are all prime numbers.

			The a(12) = 9 trees:
  (2*2*3), (2*3*2), (3*2*2),
  (2*(2*3)), (2*(3*2)), (3*(2*2)), ((2*2)*3), ((2*3)*2), ((3*2)*2).

Cf. A000108, A001055, A007716, A074206, A118376, A277130, A281118, A281119, A292505, A317144, A319122, A319138.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
otfs[n_]:=Prepend[Join@@Table[Tuples[otfs/@f],{f,Select[ordfacs[n],Length[#]>1&]}],n];
Table[Length[Select[otfs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,100}]

a(prime^n) = A001003(n - 1).

a(product of n distinct primes) = A032037(n).

A357367 Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.

Original entry on oeis.org

1, 0, 2, 0, 6, 12, 0, 24, 120, 120, 0, 120, 1080, 2520, 1680, 0, 720, 10080, 40320, 60480, 30240, 0, 5040, 100800, 604800, 1512000, 1663200, 665280, 0, 40320, 1088640, 9072000, 33264000, 59875200, 51891840, 17297280

Offset: 0

Peter Luschny, Sep 26 2022

T(n, k) is the cardinality of the set of all phylogenetic trees with linearly ordered children having n + 1 leaves and k internal vertices. (Proposition 4.16 in Deb and Sokal). - Peter Luschny, Aug 06 2025

			Triangle T(n, k) starts:
  [0] 1;
  [1] 0,     2;
  [2] 0,     6,      12;
  [3] 0,    24,     120,     120;
  [4] 0,   120,    1080,    2520,     1680;
  [5] 0,   720,   10080,   40320,    60480,    30240;
  [6] 0,  5040,  100800,  604800,  1512000,  1663200,   665280;
  [7] 0, 40320, 1088640, 9072000, 33264000, 59875200, 51891840, 17297280;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 0..150, flattened).
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 33.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See pp. 3, 5.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.

Cf. A032037 (row sums), A271703, A386789.

T := (n, k) -> add((-1)^(m + k) * binomial(n + k, n + m) * binomial(n + m - 1, m - 1) * (n + m)! / m!, m = 0..k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..8);
T := proc(n, k) option remember; if n = 0 and k = 0 then 1 elif k <= 0 or n < 0 then 0 else 2*(n + k - 1)*T(n-1, k-1) + (n + 2*k - 1)*T(n-1, k) fi end:
for n from 0 to 6 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Aug 06 2025

T[n_, k_] := Sum[(-1)^(m + k)*Binomial[n + k, n + m]*Binomial[n + m - 1, m - 1]*(n + m)!/m!, {m, 0, k}]; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 05 2025 *)

def Lah(n, k): return binomial(n, k) * falling_factorial(n - 1, n - k)
def T(n, k): return (sum((-1)^(m + k) * binomial(n + k, n + m) * Lah(n + m, m)
        for m in range(k + 1)))
for n in range(8): print([T(n, k) for k in range(n+1)])

T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * L(n + m, m), where L denotes the unsigned Lah numbers A271703.
T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * binomial(n + m - 1, m - 1) * (n + m)! / m!.
T(n, k) = (2*(n + k - 1))*T(n-1, k-1) + (n + 2*k - 1)*T(n-1, k) with suitable boundary conditions (from Deb and Sokal). - Peter Luschny, Aug 06 2025

New name using a formula of Deb and Sokal by Peter Luschny, Aug 06 2025

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A319138 Number of complete strict planar branching factorizations of n.

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A319136 Number of complete planar branching factorizations of n.

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A357367 Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.

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