A260332 -id:A260332 - cp's OEIS frontend

A364398 G.f. satisfies A(x) = 1 + x/A(x)^3*(1 + 1/A(x)).

1, 2, -14, 162, -2270, 35234, -582958, 10076354, -179802046, 3287029698, -61246957902, 1158889656930, -22207636788894, 430106644358242, -8405699952109166, 165557885912786818, -3282954949273886590, 65487784219460233602, -1313225110482709157518

Offset: 0

Seiichi Manyama, Jul 22 2023

sign

Cf. A112478, A364394, A364396.
Cf. A364399, A364400.
Cf. A260332.

A364398 := proc(n)
    if n = 0 then
        1;
    else
        (-1)^(n-1)*add( binomial(n,k) * binomial(4*n+k-2,n-1),k=0..n)/n ;
    end if;
end proc:
seq(A364398(n),n=0..70); # R. J. Mathar, Jul 25 2023

nmax = 18; A[] = 1; Do[A[x] = 1+x/A[x]^3*(1+1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}]; CoefficientList[A[x], x] (* Jean-François Alcover, Oct 21 2023 *)

a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(4*n+k-2, n-1))/n);

a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(4*n+k-2,n-1) for n > 0.
D-finite with recurrence 2*n*(462919*n -714364)*(4*n-3) *(2*n-1)*(4*n-1)*a(n) +(625365036*n^5 -2723245780*n^4 +4202103460*n^3 -2471353250*n^2 +81675089*n +289227120)*a(n-1) +(-484851248*n^5 +5501638270*n^4 -25122933600*n^3 +57439557800*n^2 -65490996232*n +29691239955)*a(n-2) +(2*n-5)*(652184*n -1103659)*(4*n-13) *(n-3)*(4*n-11)*a(n-3)=0. - R. J. Mathar, Jul 25 2023
a(n) ~ c*(-1)^(n-1)*256^n*27^(-n)*2F1([1-n, 4*n], [3*n], -1)*n^(-3/2), with c = sqrt(3/(32*Pi)). - Stefano Spezia, Oct 21 2023

A365847 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1+x)^4 ).

Original entry on oeis.org

1, 8, 96, 1368, 21440, 356968, 6197408, 110947768, 2033381760, 37963483592, 719495148768, 13806129179928, 267693334199616, 5236670783633960, 103227182363423008, 2048451544990578552, 40888361539777714944, 820400146864231266184

Offset: 0

Seiichi Manyama, Sep 20 2023

nonn

Cf. A066357, A365754, A365846, A365848.
Cf. A032349, A365622, A365843.
Cf. A260332.

a(n) = sum(k=0, n, binomial(4*n+k+3, k)*binomial(4*(n+1), n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(4*n+k+3,k) * binomial(4*(n+1),n-k).

Conjecture: g.f.: B^4, where B is the g.f. of A260332.

A260331 Labelings of n diamond-shaped posets with 4 vertices per diamond where the labels follow the poset relations.

Original entry on oeis.org

1, 2, 280, 277200, 1009008000, 9777287520000, 207786914375040000, 8508874143657888000000, 611958228411875304960000000, 72094798889203029677337600000000, 13177487340968529764423766528000000000, 3577714168047637768100581459885056000000000, 1392303245637418713834022280928868392960000000000

Offset: 0

Manda Riehl, Jul 29 2015

nonn

By diamond-shaped poset with 4 vertices, we mean a poset on four elements with e_1 < e_2, e_1 < e_3, e_2 < e_4, e_3 < e_4, and no order relations between e_2 and e_3. In the Hasse diagram for such a poset, we have a least element, two elements in the level above, and one element in the top level, so the diagram resembles a diamond.

			For a single diamond (n=1) poset with 4 vertices, we must label the least element 1 and the greatest element 4, and the two central elements can be labeled either 2, 3 or 3, 2 respectively. Thus the associated permutations are 1234 and 1324.

M. Paukner, L. Pepin, M. Riehl, and J. Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015.
Manda Riehl, A labelling of a diamond with 4 vertices so that the labels follow the poset relations.

Cf. A260332, A260579.

Table[(4 n)!/12^n, {n, 0, 12}] (* Michael De Vlieger, Apr 06 2016 *)

a(n) = (4n)!/12^n.

More terms from Michael De Vlieger, Apr 06 2016

a(4) corrected by Georg Fischer, May 08 2021

A363380 G.f. satisfies A(x) = 1 + x * A(x)^4 * (1 + A(x)^2).

Original entry on oeis.org

1, 2, 20, 284, 4712, 85392, 1638112, 32699472, 672188768, 14133399744, 302535052160, 6570819330688, 144442463464704, 3207564324825600, 71848240540852224, 1621452789508328704, 36831997860270007808, 841470878382566444032

Offset: 0

Seiichi Manyama, May 29 2023

Cf. A219534, A346626, A363311.

Cf. A217360, A260332, A363304.

a(n) = sum(k=0, n, binomial(n, k)*binomial(4*n+2*k+1, n)/(4*n+2*k+1));

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*n+2*k+1,n)/(4*n+2*k+1).

A371676 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 40, 524, 7824, 126228, 2143544, 37750812, 683194912, 12628104740, 237388091208, 4524456276524, 87228274533040, 1698091537435444, 33332913873239640, 659038408936005692, 13112372856351746112, 262338658739430857796, 5274545338183090647656

Offset: 0

Seiichi Manyama, Apr 02 2024

nonn

Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.

Cf. A006319, A032349, A371675. A371677, A371678.

Cf. A260332.

a(n, r=2, t=4, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

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

a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(4*n+k+2,n)/(4*n+k+2).

A260579 Labelings of n diamond-shaped posets with 4 vertices per diamond where the labels follow the poset relations whose associated reading permutation avoids 321 in the classical sense.

Original entry on oeis.org

1, 2, 106, 5976, 387564, 27247446, 2020632046, 155622020610, 12327937844924, 998103225615208

Offset: 0

Manda Riehl, Jul 29 2015

By diamond-shaped poset with 4 vertices, we mean a poset on four elements with e_1 < e_2, e_1 < e_3, e_2 < e_4, e_3 < e_4, and no order relations between e_2 and e_3. In the Hasse diagram for such a poset, we have a least element, two elements in the level above, and one element in the top level, so the diagram resembles a diamond. The associated permutation is the permutation obtained by reading the labels of each poset by levels left to right, starting with the least element.

Additional terms were provided by David Bevan.

			For a single diamond (n=1) poset with 4 vertices, we must label the least element 1 and the greatest element 4, and the two central elements can be labeled either 2, 3 or 3, 2 respectively. Thus the associated permutations are 1234 and 1324.

M. Paukner, L. Pepin, M. Riehl, and J. Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015.
Manda Riehl, A 321-avoiding diamond with associated permutation 1234.

Cf. A260331, A260332.

A364826 G.f. satisfies A(x) = 1 - xA(x)^4 (1 - 3*A(x)).

Original entry on oeis.org

1, 2, 22, 338, 6038, 117570, 2420758, 51833106, 1142472150, 25749801986, 590737764118, 13748997055826, 323842714201622, 7704914865207362, 184899022770465558, 4470200057557410834, 108776308617293352534, 2662072268791363675650

Offset: 0

Seiichi Manyama, Aug 09 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..704

Cf. A025192, A107841, A235347, A364825, A364827.

Cf. A243667, A260332.

a(n) = (-1)^n*sum(k=0, n, (-3)^k*binomial(n, k)*binomial(4*n+k+1, n)/(4*n+k+1));

a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(4*n+k+1,n) / (4*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(5*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * 3^(n-k) * binomial(n,k) * binomial(4*n,k-1) for n > 0.

A371700 G.f. satisfies A(x) = 1 + x * A(x)^6 * (1 + A(x)).

Original entry on oeis.org

1, 2, 26, 482, 10450, 247554, 6208970, 162064322, 4356511138, 119788611458, 3353361311738, 95251219926690, 2738421518770546, 79531905952256642, 2329955712706784682, 68770993359030211458, 2043143866891345880898, 61050342965542475675906

Offset: 0

Seiichi Manyama, Apr 03 2024

nonn

Cf. A006318, A027307, A144097, A260332, A363006.

Cf. A371678.

a(n, r=1, t=6, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6*n+k+1,n)/(6*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(7*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * binomial(n,k) * binomial(6*n,k-1) for n > 0.

A378840 G.f. A(x) satisfies A(x) = ( 1 + x * A(x)^(4/3) * (1 + A(x)^(1/3)) )^3.

Original entry on oeis.org

1, 6, 66, 902, 13794, 225990, 3878946, 68854278, 1253647938, 23283474310, 439394508162, 8401507608966, 162413310158626, 3169029168475206, 62330703549363810, 1234503404283308038, 24599422679682518658, 492824963618477891334, 9920626149798702401730

Offset: 0

Seiichi Manyama, Dec 09 2024

nonn

Cf. A260332, A365847, A371676.

Cf. A006320, A033296, A365843.

a(n, r=3, t=4, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

a(n) = 3 * Sum_{k=0..n} binomial(n,k) * binomial(4*n+k+3,n)/(4*n+k+3).

Conjecture: g.f.: B(x)^3, where B(x) is the g.f. of A260332.

cp's OEIS Frontend

A364398 G.f. satisfies A(x) = 1 + x/A(x)^3*(1 + 1/A(x)).

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A365847 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1+x)^4 ).

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A260331 Labelings of n diamond-shaped posets with 4 vertices per diamond where the labels follow the poset relations.

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A363380 G.f. satisfies A(x) = 1 + x * A(x)^4 * (1 + A(x)^2).

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A371676 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x)^(1/2))^2.

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A260579 Labelings of n diamond-shaped posets with 4 vertices per diamond where the labels follow the poset relations whose associated reading permutation avoids 321 in the classical sense.

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A364826 G.f. satisfies A(x) = 1 - x*A(x)^4 * (1 - 3*A(x)).

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A371700 G.f. satisfies A(x) = 1 + x * A(x)^6 * (1 + A(x)).

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A378840 G.f. A(x) satisfies A(x) = ( 1 + x * A(x)^(4/3) * (1 + A(x)^(1/3)) )^3.

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A364826 G.f. satisfies A(x) = 1 - xA(x)^4 (1 - 3*A(x)).