A132332 -id:A132332 - cp's OEIS frontend

A132333 G.f.: A(x) = (A_1)^2 where A_1 = 1/[1 - x(A_2)^2], A_2 = 1/[1 - x^2(A_3)^2], A_3 = 1/[1 - x^3(A_4)^2], ... A_n = 1/[1 - x^n(A_{n+1})^2] for n>=1.

1, 2, 3, 8, 17, 36, 85, 184, 405, 898, 1962, 4296, 9371, 20376, 44244, 95844, 207217, 447264, 963835, 2073900, 4456374, 9563620, 20499344, 43891176, 93877423, 200594560, 428231448, 913400192, 1946652868, 4145533218, 8821743618

Offset: 0

Paul D. Hanna, Aug 20 2007

nonn

Self-convolution of A132332.

Cf. A132332; A132335 (variant).

{a(n)=local(A=1+x*O(x^n)); for(j=0, n-1, A=1/(1-x^(n-j)*A^2 +x*O(x^n))); polcoeff(A^2, n)}

A132334 G.f.: A(x) = A_1 where A_1 = 1/[1 - x(A_2)^3], A_2 = 1/[1 - x^2(A_3)^3], A_3 = 1/[1 - x^3(A_4)^3], ... A_n = 1/[1 - x^n(A_{n+1})^3] for n>=1.

Original entry on oeis.org

1, 1, 1, 4, 7, 16, 43, 89, 216, 502, 1154, 2715, 6268, 14583, 33936, 78787, 183141, 425547, 988765, 2297533, 5338321, 12403697, 28819646, 66962219, 155583912, 361492693, 839915741, 1951499287, 4534218339, 10535031491, 24477592379

Offset: 0

Paul D. Hanna, Aug 20 2007

nonn

Self-convolution cube is A132335.

Cf. A132335; A132332 (variant).

{a(n)=local(A=1+x*O(x^n)); for(j=0, n-1, A=1/(1-x^(n-j)*A^3 +x*O(x^n))); polcoeff(A, n)}

A204387 Triangle read by rows: T(n,k) is number of noncrossing trees with k edges and path-length n, n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 4, 1, 0, 0, 3, 6, 1, 0, 0, 4, 10, 8, 1, 0, 0, 0, 12, 21, 10, 1, 0, 0, 0, 12, 32, 36, 12, 1, 0, 0, 0, 6, 45, 72, 55, 14, 1, 0, 0, 0, 8, 36, 119, 140, 78, 16, 1, 0, 0, 0, 0, 46, 144, 270, 244, 105, 18, 1, 0, 0, 0, 0, 32, 164, 416, 550, 392, 136, 20, 1

Offset: 1

N. J. A. Sloane, Jan 17 2012

The number of nodes is k + 1. The path-length is the sum of the distances of all nodes from the root node. - Andrew Howroyd, Nov 19 2024

			Triangle begins:
1
0 1
0 2 1
0 0 4 1
0 0 3 6 1
0 0 4 10 8 1
0 0 0 12 21 10 1
0 0 0 12 32 36 12 1

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
E. Deutsch and M. Noy, Statistics on non-crossing trees, Discrete Math., 254 (2002), 75-87 (see Th. 6).

Row sums are A132332.
Column sums are A001764.
Cf. A062236.

T(n)={my(g=1+O(x)); for(i=1, n, g=1/(1 - x*y*subst(g,y,x*y)^2)); [Vecrev(p/y) | p<-Vec(g-1)]}
{my(A=T(10)); for(i=1, #A, print(A[i]))} \\ Andrew Howroyd, Nov 19 2024

From Andrew Howroyd, Nov 19 2024: (Start)
G.f.: A(x,y) satisfies A(x,y) = 1/(1 - x*y*A(x,x*y)^2).
T(k*(k+1)/2, k) = 2^(k-1).
T(n,k) = 0 for n > k*(k+1)/2.
Sum_{n>=1} n*T(n,k) = A062236(k). (End)

a(34) corrected and a(42) onwards from Andrew Howroyd, Nov 19 2024

cp's OEIS Frontend

A132333 G.f.: A(x) = (A_1)^2 where A_1 = 1/[1 - x(A_2)^2], A_2 = 1/[1 - x^2(A_3)^2], A_3 = 1/[1 - x^3(A_4)^2], ... A_n = 1/[1 - x^n(A_{n+1})^2] for n>=1.

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A132334 G.f.: A(x) = A_1 where A_1 = 1/[1 - x(A_2)^3], A_2 = 1/[1 - x^2(A_3)^3], A_3 = 1/[1 - x^3(A_4)^3], ... A_n = 1/[1 - x^n(A_{n+1})^3] for n>=1.

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A204387 Triangle read by rows: T(n,k) is number of noncrossing trees with k edges and path-length n, n >= 1, 1 <= k <= n.

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cp's OEIS Frontend

A132333 G.f.: A(x) = (A_1)^2 where A_1 = 1/[1 - x*(A_2)^2], A_2 = 1/[1 - x^2*(A_3)^2], A_3 = 1/[1 - x^3*(A_4)^2], ... A_n = 1/[1 - x^n*(A_{n+1})^2] for n>=1.

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A132334 G.f.: A(x) = A_1 where A_1 = 1/[1 - x*(A_2)^3], A_2 = 1/[1 - x^2*(A_3)^3], A_3 = 1/[1 - x^3*(A_4)^3], ... A_n = 1/[1 - x^n*(A_{n+1})^3] for n>=1.

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PARI

A204387 Triangle read by rows: T(n,k) is number of noncrossing trees with k edges and path-length n, n >= 1, 1 <= k <= n.

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PARI

Formula

Extensions

A132333 G.f.: A(x) = (A_1)^2 where A_1 = 1/[1 - x(A_2)^2], A_2 = 1/[1 - x^2(A_3)^2], A_3 = 1/[1 - x^3(A_4)^2], ... A_n = 1/[1 - x^n(A_{n+1})^2] for n>=1.

A132334 G.f.: A(x) = A_1 where A_1 = 1/[1 - x(A_2)^3], A_2 = 1/[1 - x^2(A_3)^3], A_3 = 1/[1 - x^3(A_4)^3], ... A_n = 1/[1 - x^n(A_{n+1})^3] for n>=1.