A308616 -id:A308616 - cp's OEIS frontend

A365724 G.f. satisfies A(x) = 1 + x^3A(x)^3(1 + x*A(x)).

1, 0, 0, 1, 1, 0, 3, 7, 4, 12, 45, 55, 77, 286, 546, 728, 1960, 4760, 7548, 15504, 39729, 75582, 140448, 336490, 723327, 1366200, 2992990, 6758895, 13522275, 28094040, 63183315, 133231800, 273896532, 600805296, 1305229332, 2720740792, 5843241088, 12797739672

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Cf. A308616, A365723, A365725, A365726.
Cf. A001005, A001006, A365730.
Cf. A114997.

a(n) = sum(k=0, n\3, binomial(k, n-3*k)*binomial(n+1, k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(k,n-3*k) * binomial(n+1,k).

A365725 G.f. satisfies A(x) = 1 + x^3A(x)^4(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 4, 9, 5, 22, 78, 91, 175, 680, 1224, 1938, 6270, 14630, 24794, 63756, 166980, 322920, 720720, 1900080, 4125888, 8803008, 22151360, 51778804, 111882100, 267682272, 645736432, 1442390092, 3346519020, 8094247798, 18657762006, 42890295734

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Index entries for reversions of series

Cf. A308616, A365723, A365724, A365726.
Cf. A001002, A217358, A365731.
Cf. A054514.

a(n) = sum(k=0, n\3, binomial(k, n-3*k)*binomial(n+k+1, k)/(n+k+1));

a(n) = Sum_{k=0..floor(n/3)} binomial(k,n-3*k) * binomial(n+k+1,k) / (n+k+1).

G.f.: (1/x) * Series_Reversion( x*(1 - x^3*(1 + x)) ). - Seiichi Manyama, Sep 24 2024

A365723 G.f. satisfies A(x) = 1 + x^3A(x)^2(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 5, 3, 5, 21, 28, 26, 84, 180, 207, 385, 990, 1562, 2288, 5278, 10439, 15925, 30212, 64738, 112268, 192780, 396338, 761634, 1317840, 2512940, 5015867, 9146049, 16729462, 32927950, 62851525, 115101637, 220051350, 427898900, 800956600, 1505524800

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Cf. A308616, A365724, A365725, A365726.

Cf. A365694.

a(n) = sum(k=0, n\3, binomial(k, n-3*k)*binomial(n-k+1, k)/(n-k+1));

a(n) = Sum_{k=0..floor(n/3)} binomial(k,n-3*k) * binomial(n-k+1,k) / (n-k+1).

A365726 G.f. satisfies A(x) = 1 + x^3A(x)^5(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 5, 11, 6, 35, 120, 136, 336, 1330, 2310, 4301, 15456, 35100, 64701, 193662, 508921, 1023000, 2643432, 7298984, 16196682, 38795055, 105939288, 254015541, 596987183, 1575487320, 3959803694, 9418896773, 24081344034, 61781452530, 150293865540

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Cf. A308616, A365723, A365724, A365725.

Cf. A365695.

a(n) = sum(k=0, n\3, binomial(k, n-3*k)*binomial(n+2*k+1, k)/(n+2*k+1));

a(n) = Sum_{k=0..floor(n/3)} binomial(k,n-3*k) * binomial(n+2*k+1,k) / (n+2*k+1).

A344900 a(n) is the number of well-formed formulas (wffs) of zeroth-order logic containing n characters (see comments).

Original entry on oeis.org

1, 1, 13, 25, 37, 61, 97, 561, 1869, 4437, 9097, 17097, 54101, 194189, 583857, 1490017, 3371997, 8916485, 28974361, 94338361, 277239461, 728378813, 1938657473, 5839518033, 18961970605, 59883346869, 174804016553, 493085118121, 1460284207861, 4646560028141

Offset: 1

Christoph B. Kassir, Jun 01 2021

nonn

The characters of zeroth-order logic include the tilde (~), ampersand (&), wedge (∨), horseshoe (⊃), triple bar (≡), left and right parentheses, and variables (upper-case letters with or without subscripts.) However, since the set of upper-case letters with or without subscripts is infinitely large, it is then, for the sentences of zeroth-order logic containing k variables, restricted to the set {A1, ..., Ak}, with an additional restriction as follows: a sentence may only contain Ai iff it contains every Aj for j=1..i-1 (this gives a total of A000670(k-1) legal permutations for a sentence containing k variables.)
The rules for a well-formed formula (wff) of zeroth-order logic are defined recursively as follows (see M. Bergmann et al.):
1. Every variable is a wff.
2. If P is a wff, then so is ~P.
3. If P and Q are wffs, then so is (PxQ), where 'x' is any binary logical operator.
It is also customary to remove the outermost parentheses of a sentence.

			a(4) = 25, since the number of sentences of zeroth-order logic containing four characters are as follows: ~~~A, ~AxA, Ax~A, ~AxB, Bx~A, ~BxA, and Ax~B, where 'x' is any of the four binary logical operators.

Merrie Bergmann, James Moor, and Jack Nelson. The logic book. Vol. 2. New York: McGraw-Hill, 1990, p. 54.

Sean A. Irvine, Java program (github).

Cf. A300696, A000670.

Related sequences: A101248, A101273, A308616. - N. J. A. Sloane, Aug 17 2021

More terms from Sean A. Irvine, Jul 24 2021

A331106 Number of plane trees of total weight n of combinatorial class T=ZU + ZT^2/(1-T) with nodes Z of weight one and leaves U of weight three.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 5, 5, 0, 1, 9, 21, 14, 1, 14, 56, 84, 43, 20, 120, 300, 331, 159, 225, 825, 1486, 1322, 814, 1925, 5006, 7051, 5621, 5434, 14015, 28082, 32968, 27092, 39261, 91793, 149877, 156858, 152023, 276769, 558845, 778920, 786931, 953756

Offset: 1

Marko Riedel, Jan 09 2020

nonn

The underlying tree structure before the weights are applied (Z with weight one, U with weight three) is a series-reduced tree because a non-leaf node Z has at least two children.

			For n=4, the tree is Z-U, for n=9 the tree is
    Z-U
   /
  Z
   \
    Z-U.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009.

Marko Riedel et al., Math.StackExchange, Counting plane trees

Cf. A308616.

G.f.: (1 + z^4 - sqrt(z^8 - 4*z^5 - 2*z^4 +1))/(2*(z+1)).
a(n) = Sum_{k=floor(n/5)+1..floor((n-1)/4)} (1/(n-3*k)) * binomial(n-3*k,k) * binomial(k-2, n-4*k-1) for n >= 1, n <> 4. a(4) = 1.
D-finite with recurrence: n*a(n) +(n)*a(n-1) +(n-2)*a(n-2) +(n-2)*a(n-3) +2*(-n+6)*a(n-4) +6*(-n+7)*a(n-5) +2*(-3*n+23)*a(n-6) +6*(-n+9)*a(n-7) +(-3*n+26)*a(n-8) +(n-12)*a(n-9) +(n-14)*a(n-10) +(n-14)*a(n-11)=0. - R. J. Mathar, Jan 27 2020

cp's OEIS Frontend

A365724 G.f. satisfies A(x) = 1 + x^3*A(x)^3*(1 + x*A(x)).

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A365725 G.f. satisfies A(x) = 1 + x^3*A(x)^4*(1 + x*A(x)).

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A365723 G.f. satisfies A(x) = 1 + x^3*A(x)^2*(1 + x*A(x)).

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A365726 G.f. satisfies A(x) = 1 + x^3*A(x)^5*(1 + x*A(x)).

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A344900 a(n) is the number of well-formed formulas (wffs) of zeroth-order logic containing n characters (see comments).

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A331106 Number of plane trees of total weight n of combinatorial class T=Z*U + Z*T^2/(1-T) with nodes Z of weight one and leaves U of weight three.

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A365724 G.f. satisfies A(x) = 1 + x^3A(x)^3(1 + x*A(x)).

A365725 G.f. satisfies A(x) = 1 + x^3A(x)^4(1 + x*A(x)).

A365723 G.f. satisfies A(x) = 1 + x^3A(x)^2(1 + x*A(x)).

A365726 G.f. satisfies A(x) = 1 + x^3A(x)^5(1 + x*A(x)).

A331106 Number of plane trees of total weight n of combinatorial class T=ZU + ZT^2/(1-T) with nodes Z of weight one and leaves U of weight three.