A364747 -id:A364747 - cp's OEIS frontend

A243659 Number of Sylvester classes of 3-packed words of degree n.

1, 1, 5, 34, 267, 2279, 20540, 192350, 1853255, 18252079, 182924645, 1859546968, 19127944500, 198725331588, 2082256791048, 21979169545670, 233495834018591, 2494624746580655, 26786319835972799, 288915128642169250, 3128814683222599331, 34007373443388857999

Offset: 0

N. J. A. Sloane, Jun 12 2014

See Novelli-Thibon (2014) for precise definition.

Seiichi Manyama, Table of n, a(n) for n = 0..941 (terms 1..100 from Lars Blomberg)
Paul Barry, The Triple Riordan Group, arXiv:2412.05461 [math.CO], 2024. See pp. 4, 10.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Eq. (185), p. 47 and Fig. 17.

Column k=3 of A336573.
Cf. A003168, A336539.
Cf. A002293, A349331, A364747, A364765.

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 1 else (4*(37604*n^5-158474*n^4+248391*n^3-178459*n^2+58042*n-6720)*a(n-1) - 3*(n-2)*(3*n-4)*(3*n-5)*(119*n^2-85*n+14)*a(n-2) )/ (12*n*(3*n-1)*(3*n+1)*(119*n^2-323*n+218)) fi; end:
seq(a(n), n = 0..20); # Peter Bala, Sep 08 2024

b[0] = 1; b[n_] := b[n] = 1/n Sum[Sum[2^(j-2i)(-1)^(i-j) Binomial[i, 3i-j] Binomial[i+j-1, i-1], {j, 0, 3i}] b[n-i], {i, 1, n}];
a[n_] := b[n+1];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 27 2018, after Vladimir Kruchinin *)

a(n):=if n=0 then 1 else 1/n*sum(sum(2^(j-2*i)*(-1)^(i-j)*binomial(i,3*i-j)*binomial(i+j-1,i-1),j,0,3*i)*a(n-i),i,1,n); /* Vladimir Kruchinin, Apr 07 2017 */

a(n) = if(n==0, 1, sum(i=1, n, a(n-i)*sum(j=0, 3*i, 2^(j-2*i)*(-1)^(i-j)*binomial(i,3*i-j)*binomial(i+j-1,i-1)))/n); \\ Seiichi Manyama, Jul 26 2020

a(n) = my(A=1+x*O(x^n)); for(i=0, n, A=1-x*A^3*(1-2*A)); polcoeff(A, n); \\ Seiichi Manyama, Jul 26 2020

a(n) = (-1)^n*sum(k=0, n, (-2)^k*binomial(n, k)*binomial(3*n+k+1, n)/(3*n+k+1)); \\ Seiichi Manyama, Jul 26 2020

a(n) = (-1)^n*sum(k=0, n, (-2)^(n-k)*binomial(3*n+1, k)*binomial(4*n-k, n-k))/(3*n+1); \\ Seiichi Manyama, Jul 26 2020

Novelli-Thibon give an explicit formula in Eq. (182).
a(0) = 1 and a(n) = (1/n) * Sum_{i=1..n} ( Sum_{j=0..3*i} (2^(j-2*i)*(-1)^(i-j) * binomial(i,3*i-j)*binomial(i+j-1,i-1)) *a(n-i) ) for n > 0. - Vladimir Kruchinin, Apr 09 2017
From Seiichi Manyama, Jul 26 2020: (Start)
G.f. A(x) satisfies: A(x) = 1 - x * A(x)^3 * (1 -  2 * A(x)).
a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(3*n+k+1,n)/(3*n+k+1).
a(n) = ( (-1)^n / (3*n+1) ) * Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+1,k) * binomial(4*n-k,n-k). (End)
a(n) ~ sqrt(24388 + 9221*sqrt(7)) * (316 + 119*sqrt(7))^(n - 1/2) / (sqrt(7*Pi) * n^(3/2) * 2^(n + 3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jul 31 2021
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 08 2023
P-recursive: 12*n*(3*n-1)*(3*n+1)*(119*n^2-323*n+218)*a(n) = 4*(37604*n^5-158474*n^4+248391*n^3-178459*n^2+58042*n-6720)*a(n-1) - (3*n-4)*(3*n-5)*(3*n-6)*(119*n^2-85*n+14)*a(n-2) with a(0) = a(1) = 1. - Peter Bala, Sep 08 2024

a(9)-a(21) from Lars Blomberg, Jul 12 2017

a(0)=1 inserted by Seiichi Manyama, Jul 26 2020

A364748 G.f. A(x) satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)).

Original entry on oeis.org

1, 1, 6, 47, 424, 4159, 43097, 464197, 5145475, 58313310, 672598269, 7869856070, 93183973405, 1114471042413, 13443614108307, 163372291277764, 1998239045199623, 24580340878055298, 303893356012560280, 3774099648814193998, 47061518776483143441

Offset: 0

Seiichi Manyama, Aug 05 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..890

Cf. A000108, A001003, A108447, A364747, A378691, A378692.

Cf. A002295, A243667, A365192, A365193, A365194.

a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(5*n-4*k, n-1-k))/n);

a(n, r=1, s=1, t=5, u=1) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 05 2024

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(5*n-4*k,n-1-k) for n > 0.
From Seiichi Manyama, Dec 05 2024: (Start)
G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^4/(1 - x*A(x))).
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r). (End)

A364765 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 / (1 - xA(x)^5).

Original entry on oeis.org

1, 1, 5, 36, 304, 2808, 27475, 279845, 2935987, 31511097, 344344868, 3818320487, 42855633210, 485923475563, 5557803724920, 64046876264292, 742908320701832, 8667090253409215, 101631581618367133, 1197190915359577973, 14160413911721178800

Offset: 0

Seiichi Manyama, Aug 06 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..909

Cf. A001006, A106228, A219537, A271469.
Cf. A349331, A364747.
Cf. A378952, A378954.

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

G.f. satisfies A(x) = 1 + x*A(x)^6 / (1 + x*A(x)^4).
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(4*n+k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(6*n-2*k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,k) * binomial(5*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024

A365180 G.f. satisfies A(x) = 1 + xA(x)^4(1 + x*A(x)).

Original entry on oeis.org

1, 1, 5, 31, 223, 1740, 14328, 122549, 1078197, 9695359, 88710199, 823247686, 7730244098, 73310150097, 701163085849, 6755544043969, 65506554804129, 638794412442172, 6260571309256152, 61632794482411367, 609197871548209907, 6043456939539775056

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002294, A365178, A365181, A365182, A365183.

Cf. A364747.

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

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

A378692 G.f. A(x) satisfies A(x) = 1 + xA(x)^7/(1 - xA(x)).

Original entry on oeis.org

1, 1, 8, 86, 1075, 14667, 211799, 3182454, 49243854, 779379652, 12558073022, 205312307834, 3397359326116, 56790504859929, 957574385205771, 16267419813629731, 278162968238908681, 4783813617177604232, 82691541747420586716, 1435895455224032519430, 25035634270828781060188

Offset: 0

Seiichi Manyama, Dec 04 2024

nonn

Cf. A000108, A001003, A108447, A364747, A364748, A378691.
Cf. A378686, A378690, A378693.
Cf. A378685, A378688, A378694.

a(n, r=1, s=1, t=7, u=1) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^6/(1 - x*A(x))).

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).

A364758 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 / (1 + xA(x)).

Original entry on oeis.org

1, 1, 3, 14, 76, 450, 2818, 18352, 123028, 843345, 5884227, 41650479, 298352365, 2158751879, 15754446893, 115830820439, 857147952469, 6379136387303, 47715901304501, 358529599468636, 2704884469806606, 20481615947325089, 155605509972859999, 1185779099027494848

Offset: 0

Seiichi Manyama, Aug 05 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A090192, A106228, A364759, A378919.
Cf. A001764, A219537, A364865, A365224.
Cf. A300048, A364747, A378889.

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

a(n, r=1, s=-1, t=4, u=1) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 11 2024

a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(4*n-3*k,n-1-k) for n > 0.
From Seiichi Manyama, Dec 11 2024: (Start)
G.f. A(x) satisfies A(x)^3 = 1 + x*A(x) + x*A(x)^5 + x*A(x)^6.
G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^3/(1 + x*A(x))).
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r). (End)

A364792 G.f. satisfies A(x) = 1 + xA(x)^4 / (1 - xA(x)^2).

Original entry on oeis.org

1, 1, 5, 33, 250, 2054, 17800, 160183, 1482535, 14022415, 134943095, 1317046306, 13005842030, 129708875695, 1304588594925, 13217663310305, 134775670244250, 1382019265706377, 14242560597119165, 147435736533094415, 1532365596794307010

Offset: 0

Seiichi Manyama, Aug 08 2023

nonn

Cf. A002293, A243659, A349331, A364747, A364765.

A364792 := proc(n)
    if n = 0 then
        1;
    else
        add( binomial(n,k) * binomial(4*n-2*k,n-1-k),k=0..n-1) ;
        %/n ;
    end if ;
end proc:
seq(A364792(n),n=0..80); # R. J. Mathar, Aug 10 2023

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

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

D-finite with recurrence 3*n*(36653*n-48128)*(3*n-1)*(3*n+1)*a(n) +5*(-2160545*n^4 +5139476*n^3 -2463019*n^2 -1385144*n +913296)*a(n-1) +4*(-948403*n^4 +17991137*n^3 -77629283*n^2 +126107767*n -70578450)*a(n-2) +10*(n-3)*(599072*n^3 -5090881*n^2 +13501042*n -11263100)*a(n-3) -50*(6861*n-12886)*(n-3) *(n-4)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Aug 10 2023

A378691 G.f. A(x) satisfies A(x) = 1 + xA(x)^6/(1 - xA(x)).

Original entry on oeis.org

1, 1, 7, 65, 699, 8192, 101538, 1309007, 17373825, 235820907, 3258327727, 45676003435, 648019627185, 9286982935406, 134247731827970, 1955128344950960, 28659409029300490, 422517375650417841, 6260750426764454787, 93191618760715641120, 1392823412892172416996

Offset: 0

Seiichi Manyama, Dec 04 2024

nonn

Cf. A000108, A001003, A108447, A364747, A364748, A378692.

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

G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^5/(1 - x*A(x))).

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).

A371889 G.f. A(x) satisfies A(x) = 1 - x/A(x)^2 * (1 - A(x) - A(x)^3).

Original entry on oeis.org

1, 1, 2, 2, -1, -4, 7, 33, -5, -200, -151, 1185, 2202, -6069, -21799, 21791, 182718, 26520, -1349611, -1613331, 8674338, 21651795, -44750412, -217666394, 121538304, 1859974399, 1023915107, -13828122997, -23155237537, 86925632115, 282182920662

Offset: 0

Seiichi Manyama, Apr 11 2024

sign

Cf. A243156, A307678, A364747, A365192.

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

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

cp's OEIS Frontend

A243659 Number of Sylvester classes of 3-packed words of degree n.

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A364748 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)).

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A364765 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 / (1 - x*A(x)^5).

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A365180 G.f. satisfies A(x) = 1 + x*A(x)^4*(1 + x*A(x)).

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A378692 G.f. A(x) satisfies A(x) = 1 + x*A(x)^7/(1 - x*A(x)).

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A364758 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 / (1 + x*A(x)).

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A364792 G.f. satisfies A(x) = 1 + x*A(x)^4 / (1 - x*A(x)^2).

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A378691 G.f. A(x) satisfies A(x) = 1 + x*A(x)^6/(1 - x*A(x)).

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A364748 G.f. A(x) satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)).

A364765 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 / (1 - xA(x)^5).

A365180 G.f. satisfies A(x) = 1 + xA(x)^4(1 + x*A(x)).

A378692 G.f. A(x) satisfies A(x) = 1 + xA(x)^7/(1 - xA(x)).

A364758 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 / (1 + xA(x)).

A364792 G.f. satisfies A(x) = 1 + xA(x)^4 / (1 - xA(x)^2).

A378691 G.f. A(x) satisfies A(x) = 1 + xA(x)^6/(1 - xA(x)).