A243659 -id:A243659 - cp's OEIS frontend

A364747 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 / (1 - xA(x)).

1, 1, 5, 32, 234, 1854, 15490, 134380, 1198944, 10931761, 101412677, 954155059, 9083120975, 87326765375, 846709605539, 8269910074087, 81291388929027, 803592049667495, 7983612883739843, 79671910265120574, 798283229227457304, 8027625597750959053

Offset: 0

Seiichi Manyama, Aug 05 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..972

Cf. A000108, A001003, A108447, A364748, A378691, A378692.

Cf. A002294, A243659, A364765, A364792.

a(n) = if(n==0, 1, sum(k=0, n-1, 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(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(4*n-3*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)^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(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r). (End)

A336573 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = (-1)^n * Sum_{j=0..n} (-2)^j * binomial(n,j) * binomial(kn+j+1,n)/(kn+j+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 11, 8, 1, 1, 5, 21, 45, 16, 1, 1, 6, 34, 126, 197, 32, 1, 1, 7, 50, 267, 818, 903, 64, 1, 1, 8, 69, 484, 2279, 5594, 4279, 128, 1, 1, 9, 91, 793, 5105, 20540, 39693, 20793, 256, 1, 1, 10, 116, 1210, 9946, 56928, 192350, 289510, 103049, 512

Offset: 0

Seiichi Manyama, Jul 26 2020

T(n,k) is the number of Sylvester classes of k-packed words of degree n.

			Square array begins:
   1,   1,   1,    1,    1,    1, ...
   1,   1,   1,    1,    1,    1, ...
   2,   3,   4,    5,    6,    7, ...
   4,  11,  21,   34,   50,   69, ...
   8,  45, 126,  267,  484,  793, ...
  16, 197, 818, 2279, 5105, 9946, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
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.

Columns k = 0-5 are: A011782, A001003, A003168, A243659, A243667, A243668.
Main diagonal is A336495.
Cf. A336534, A336574, A336575.

T := (n,k) -> `if`(k=0, `if`(n=0, 1, 2^(n-1)), (-1)^n*(binomial(k*n+1, n)* hypergeom([-n, k*n+1], [(k-1)*n+2], 2)) / (k*n+1)):
seq(lprint(seq(simplify(T(n, k)), k=0..9)), n=0..6); # Peter Luschny, Jul 26 2020

T[n_, k_] := (-1)^n * Sum[(-2)^j * Binomial[n, j] * Binomial[k*n+j+1, n]/(k*n+j+1), {j, 0, n}]; Table[T[k, n-k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)

T(n, k) = (-1)^n*sum(j=0, n, (-2)^j*binomial(n, j)*binomial(k*n+j+1, n)/(k*n+j+1));

T(n, k) = my(A=1+x*O(x^n)); for(i=0, n, A=1-x*A^k*(1-2*A)); polcoeff(A, n);

T(n, k) = (-1)^n*sum(j=0, n, (-2)^(n-j)*binomial(k*n+1, j)*binomial((k+1)*n-j, n-j))/(k*n+1);

G.f. A_k(x) of column k satisfies A_k(x) = 1 - x * A_k(x)^k * (1 - 2 * A_k(x)).
T(n,k) = ( (-1)^n / (k*n+1) ) * Sum_{j=0..n} (-2)^(n-j) * binomial(k*n+1,j) * binomial((k+1)*n-j,n-j).
T(n,k) = (-1)^n*binomial(k*n+1, n)*hypergeom([-n, k*n+1], [(k-1)*n+2], 2)/(k*n+1) for k >= 1. - Peter Luschny, Jul 26 2020
T(n,k) = (1/n) * Sum_{j=0..n-1} binomial(n,j) * binomial((k+1)*n-j,n-1-j) for n > 0. - Seiichi Manyama, Aug 08 2023

A369012 Expansion of (1/x) * Series_Reversion( x * (1-x/(1-x))^3 ).

Original entry on oeis.org

1, 3, 18, 133, 1095, 9636, 88718, 843993, 8230671, 81841987, 826641816, 8457710604, 87472494564, 912995025912, 9604763388534, 101736967518497, 1084125909550959, 11614159795566489, 125011746270524690, 1351312626871871661, 14662950224977228047

Offset: 0

Seiichi Manyama, Jan 11 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..941
Index entries for reversions of series

Cf. A369013, A369014.
Cf. A243659, A378612.
Cf. A001003, A211789, A378610.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x/(1-x))^3)/x)

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

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(n-1,n-k).
D-finite with recurrence 96*(3*n+2)*(3*n+1)*(n+1)*a(n) +4*(-4121*n^3 +1922*n^2 -1273*n+124)*a(n-1) +4*(20588*n^3 -76648*n^2 +98677*n -43586)*a(n-2) +(-90073*n^3 +671565*n^2 -1665278*n +1375320)*a(n-3) +210*(n-4)*(3*n-7) *(3*n-8)*a(n-4)=0. - R. J. Mathar, Jan 25 2024
From Seiichi Manyama, Dec 02 2024: (Start)
G.f.: exp( Sum_{k>=1} A378612(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - x/(1 - x))^(3*(n+1)).
G.f.: B(x)^3 where B(x) is the g.f. of A243659.
a(n) = 3 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(3*n+k+3,n)/(3*n+k+3). (End)

A364923 G.f. satisfies A(x) = 1 + xA(x)^4 / (1 - 2x*A(x)^3).

Original entry on oeis.org

1, 1, 6, 48, 442, 4419, 46626, 511032, 5761650, 66394596, 778518552, 9258850440, 111417705702, 1354135251538, 16598001854700, 204945037918800, 2546849778687138, 31828936270676172, 399777371427582024, 5043824569861127808, 63892650400004356776

Offset: 0

Seiichi Manyama, Aug 12 2023

nonn

Cf. A007564, A364924.

Cf. A243659.

A364923 := proc(n)
    add( 3^k*(-2)^(n-k)*binomial(n,k)*binomial(3*n+k+1,n)/(3*n+k+1),k=0..n) ;
end proc:
seq(A364923(n),n=0..80); # R. J. Mathar, Aug 16 2023

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

a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n,k) * binomial(3*n+k+1,n) / (3*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^k * binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 3^(n-k) * binomial(n,k) * binomial(3*n,k-1) for n > 0.
D-finite with recurrence +270*n*(3*n-1)*(3*n+1)*a(n) +(-9463*n^3 -45948*n^2 +88297*n -35478)*a(n-1) +36*(-9017*n^3 +49691*n^2 -90408*n +54354)*a(n-2) +48*(53*n^3 +1724*n^2 -11161*n +16518)*a(n-3) +576*(3*n-10)*(3*n-11) *(n-4)*a(n-4)=0. - R. J. Mathar, Aug 16 2023

A336539 G.f. A(x) satisfies A(x) = 1 + x * A(x)^3 * (1 + 2 * A(x)).

Original entry on oeis.org

1, 3, 33, 498, 8691, 164937, 3305868, 68855862, 1475636055, 32327521077, 720713175441, 16298128820568, 372946723698516, 8619565476744156, 200920644131737992, 4718057697038124750, 111505342455507462207, 2650261296098965752669, 63308992564445668959795

Offset: 0

Seiichi Manyama, Jul 25 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..710

Column k=3 of A336574.

Cf. A243659, A336538.

a[n_] := Sum[2^k * Binomial[n, k] * Binomial[3*n + k + 1, n]/(3*n + k + 1), {k, 0, n}];  Array[a, 19, 0] (* Amiram Eldar, Jul 27 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);

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

a(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

a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(3*n+k+1,n)/(3*n+k+1).
a(n) = (1/(3*n+1)) * Sum_{k=0..n} 2^(n-k) * binomial(3*n+1,k) * binomial(4*n-k,n-k).
a(n) ~ sqrt(168 + 97*sqrt(3)) * (26 + 15*sqrt(3))^(n - 1/2) / (3*sqrt(Pi) * n^(3/2) * 2^(n + 3/2)). - Vaclav Kotesovec, Jul 31 2021
From Seiichi Manyama, Aug 10 2023: (Start)
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 3^(n-k) * binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 3^k * 2^(n-k) * binomial(n,k) * binomial(3*n,k-1) for n > 0. (End)
a(n) = binomial(1+3*n, n)*hypergeom([-n, 1+3*n], [2+2*n], -2)/(1 + 3*n). - Stefano Spezia, Aug 09 2025

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

A364825 G.f. satisfies A(x) = 1 - xA(x)^3 (1 - 3*A(x)).

Original entry on oeis.org

1, 2, 18, 222, 3166, 49098, 804138, 13686198, 239671590, 4290463698, 78160665666, 1444298971662, 27005948771886, 510024567278234, 9714561608833242, 186403770207998310, 3599812021110287862, 69914211761486437026, 1364692279095996581490

Offset: 0

Seiichi Manyama, Aug 09 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..757

Cf. A025192, A107841, A235347, A364826, A364827.

Cf. A144097, A243659.

A364825 := proc(n)
    (-1)^n*add( (-3)^k*binomial(n,k) * binomial(3*n+k+1,n)/(3*n+k+1),k=0..n) ;
end proc:
seq(A364825(n),n=0..80); # R. J. Mathar, Aug 10 2023

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

a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(3*n+k+1,n) / (3*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(4*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(3*n,k-1) for n > 0.
D-finite with recurrence +2079*n*(3*n-1)*(3*n+1)*a(n) +(-347173*n^3 +395007*n^2 -41030*n -43092)*a(n-1) +18*(-59207*n^3 +325826*n^2 -590255*n +352406)*a(n-2) +3*(-3299*n^3 +35998*n^2 -125399*n +141144)*a(n-3) +9*(3*n-10)*(3*n-11) *(n-4)*a(n-4)=0. - R. J. Mathar, Aug 10 2023

A378685 G.f. A(x) satisfies A(x) = 1 + xA(x)^7/(1 - xA(x)^3).

Original entry on oeis.org

1, 1, 8, 88, 1126, 15716, 232069, 3564835, 56382489, 912031280, 15018257510, 250913307393, 4242722219425, 72470224174650, 1248608968982903, 21673752440979879, 378677335852165297, 6654158090059397480, 117523324766568499072, 2085095374834405245007

Offset: 0

Seiichi Manyama, Dec 04 2024

nonn

Cf. A243659, A300048.
Cf. A108447, A365192.
Cf. A378686.

a(n, r=1, s=1, t=7, u=3) = 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)^3)).

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).

A379522 Expansion of (1/x) * Series_Reversion( x / ( (1+x)^3 * (1+2*x)^3 ) ).

Original entry on oeis.org

1, 9, 114, 1683, 27111, 462060, 8192078, 149541975, 2791795695, 53056724409, 1023021616920, 19963667407572, 393536736830724, 7824888965728584, 156750391932619254, 3160558799674447167, 64092227061832430895, 1306327265854324847595, 26746550927141536784370

Offset: 0

Seiichi Manyama, Dec 24 2024

nonn

Index entries for reversions of series

Cf. A243659, A379521.

my(N=20, x='x+O('x^N)); Vec(serreverse(x/((1+x)^3*(1+2*x)^3))/x)

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

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

a(n) = (1/(n+1)) * [x^n] ( (1+x) * (1+2*x) )^(3*(n+1)).

A378670 G.f. A(x) satisfies A(x) = 1/( 1 - xA(x)^(3/2)/(1 - xA(x)^(3/2)) )^2.

Original entry on oeis.org

1, 2, 11, 78, 627, 5432, 49464, 466726, 4522871, 44747874, 450127999, 4589821576, 47333631828, 492836382192, 5173697858508, 54700317431958, 581946708333055, 6225343630256678, 66921440314606905, 722546760572660030, 7832054418695360555, 85198490262065775840

Offset: 0

Seiichi Manyama, Dec 02 2024

nonn

Cf. A243659, A369012.
Cf. A010683, A211789, A378668.
Cf. A378612.

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

a(n, r=2, s=1, t=4, u=3) = 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.: exp( 2/3 * Sum_{k>=1} A378612(k) * x^k/k ).
G.f.: B(x)^2 where B(x) is the g.f. of A243659.
a(n) = 2 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(3*n+k+2,n)/(3*n+k+2).
a(n) = 2 * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(n-1,n-k)/(3*n+k+2).
G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^2/(1 - x*A(x)^(3/2)) )^2.

cp's OEIS Frontend

A364747 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 / (1 - x*A(x)).

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A336573 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = (-1)^n * Sum_{j=0..n} (-2)^j * binomial(n,j) * binomial(k*n+j+1,n)/(k*n+j+1).

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A369012 Expansion of (1/x) * Series_Reversion( x * (1-x/(1-x))^3 ).

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A364923 G.f. satisfies A(x) = 1 + x*A(x)^4 / (1 - 2*x*A(x)^3).

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A336539 G.f. A(x) satisfies A(x) = 1 + x * A(x)^3 * (1 + 2 * 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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A364825 G.f. satisfies A(x) = 1 - x*A(x)^3 * (1 - 3*A(x)).

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A378685 G.f. A(x) satisfies A(x) = 1 + x*A(x)^7/(1 - x*A(x)^3).

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A364747 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 / (1 - xA(x)).

A336573 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = (-1)^n * Sum_{j=0..n} (-2)^j * binomial(n,j) * binomial(kn+j+1,n)/(kn+j+1).

A364923 G.f. satisfies A(x) = 1 + xA(x)^4 / (1 - 2x*A(x)^3).

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

A364825 G.f. satisfies A(x) = 1 - xA(x)^3 (1 - 3*A(x)).

A378685 G.f. A(x) satisfies A(x) = 1 + xA(x)^7/(1 - xA(x)^3).

A378670 G.f. A(x) satisfies A(x) = 1/( 1 - xA(x)^(3/2)/(1 - xA(x)^(3/2)) )^2.