A243667 -id:A243667 - cp's OEIS frontend

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

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)

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

A365192 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)^2).

Original entry on oeis.org

1, 1, 6, 48, 443, 4445, 47107, 518835, 5880223, 68130860, 803369481, 9609294542, 116310009888, 1421951861817, 17533301767624, 217796367181117, 2722942699583650, 34236790400004432, 432649744252128084, 5492060945760586212, 69998993052214823013

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002295, A243667, A349332, A364748, A365193, A365194.

Cf. A365186.

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

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

A365188 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x*A(x)^4).

Original entry on oeis.org

1, 1, 6, 49, 465, 4807, 52533, 596936, 6981798, 83497115, 1016367737, 12550853210, 156845913315, 1979870172453, 25207383853375, 323325558146400, 4174108907656633, 54195445136831670, 707225283913589280, 9270735916525207605, 122020617365557674605

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002295, A365184, A365185, A365186, A365187, A365189.

Cf. A243667.

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

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

A365193 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)^3).

Original entry on oeis.org

1, 1, 6, 49, 463, 4760, 51702, 583712, 6781774, 80555066, 973813974, 11941861079, 148191437719, 1857464450449, 23481830726334, 299056887494427, 3833349330581255, 49416395972195630, 640256115370243620, 8332835556325119938, 108890550249605779116

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002295, A243667, A349332, A364748, A365192, A365194.

Cf. A365187.

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

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

A243668 Number of Sylvester classes of 5-packed words of degree n.

Original entry on oeis.org

1, 1, 7, 69, 793, 9946, 131993, 1822288, 25904165, 376601883, 5573626462, 83692267478, 1271883556731, 19525467196176, 302346907361688, 4716814859429384, 74065892877777885, 1169701519598447641, 18566836447453815317, 296053851068485920563, 4739945317989532651858

Offset: 0

N. J. A. Sloane, Jun 14 2014

nonn

See Novelli-Thibon (2014) for precise definition.

Seiichi Manyama, Table of n, a(n) for n = 0..812
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=5 of A336573.

Cf. A243667.

P[n_, m_, x_] := 1/(m n + 1) Sum[Binomial[m n + 1, k] Binomial[(m + 1) n - k, n - k] (1 - x)^k x^(n - k), {k, 0, n}];
a[n_] := P[n, 5, 2];
a /@ Range[20] (* Jean-François Alcover, Jan 28 2020 *)

a(n) = my(A=1+x*O(x^n)); for(i=0, n, A=1-x*A^5*(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(5*n+k+1, n)/(5*n+k+1)); \\ Seiichi Manyama, Jul 26 2020

a(n) = (-1)^n*sum(k=0, n, (-2)^(n-k)*binomial(5*n+1, k)*binomial(6*n-k, n-k))/(5*n+1); \\ Seiichi Manyama, Jul 26 2020

Novelli-Thibon give an explicit formula in Eq. (182).
From Seiichi Manyama, Jul 26 2020: (Start)
G.f. A(x) satisfies: A(x) = 1 - x * A(x)^5 * (1 -  2 * A(x)).
a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(5*n+k+1,n)/(5*n+k+1).
a(n) = ( (-1)^n / (5*n+1) ) * Sum_{k=0..n} (-2)^(n-k) * binomial(5*n+1,k) * binomial(6*n-k,n-k). (End)
a(n) ~ sqrt(27851068 + 7443921*sqrt(14)) * 5^(5*n - 13/2) / (sqrt(7*Pi) * n^(3/2) * 2^(2*(1 + n)) * (108007 - 28854*sqrt(14))^(n - 1/2)). - Vaclav Kotesovec, Jul 31 2021
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(6*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 08 2023

More terms from Jean-François Alcover, Jan 28 2020

a(0)=1 prepended by Seiichi Manyama, Jul 25 2020

A365194 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)^6).

Original entry on oeis.org

1, 1, 6, 52, 529, 5889, 69462, 853013, 10791018, 139659604, 1840435530, 24611295075, 333132371248, 4555465710569, 62839303262352, 873363902976309, 12218178082489873, 171918448407833112, 2431415226089290680, 34544425914499450493, 492807213597429920649

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002295, A243667, A349332, A364748, A365192, A365193.

Cf. A219537, A271469, A364765.

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

a(n) = Sum_{k=0..n} binomial(6*n-k+1,k) * binomial(n-1,n-k)/(6*n-k+1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(5*n+2*k+1,k) * binomial(n-1,n-k)/(5*n+2*k+1).
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,k) * binomial(6*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Dec 26 2024

A378610 Expansion of (1/x) * Series_Reversion( x * (1 - x/(1 - x))^4 ).

Original entry on oeis.org

1, 4, 30, 276, 2825, 30884, 353108, 4170500, 50485764, 623084056, 7810707894, 99175174284, 1272856327470, 16486135484248, 215212582153840, 2828658852385572, 37401956484705132, 497174193516767600, 6640063367021736728, 89058042321373540912, 1199031374607501831273

Offset: 0

Seiichi Manyama, Dec 01 2024

nonn

Index entries for reversions of series

Cf. A001003, A211789, A369012.

Cf. A243667, A371486, A378613.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x/(1-x))^4)/x)

a(n, s=1, t=4, u=-4) = 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);

G.f.: exp( Sum_{k>=1} A378613(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - x/(1 - x))^(4*(n+1)).
a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(4*n+k+3,k) * binomial(n-1,n-k).
G.f.: B(x)^4 where B(x) is the g.f. of A243667.
a(n) = 4 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(4*n+k+4,n)/(4*n+k+4).

A336572 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 * (1 + 2 * A(x)).

Original entry on oeis.org

1, 3, 42, 822, 18708, 464115, 12175368, 332156784, 9328004700, 267870927324, 7829893576878, 232189300430454, 6968123350684692, 211232335919261178, 6458598626291716128, 198949096401788859636, 6168233789851179030684, 192334850789654814053700, 6027727888877572168027368

Offset: 0

Seiichi Manyama, Jul 25 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..655

Column k=4 of A336574.

Cf. A243667, A336540.

a[n_] := Sum[2^k * Binomial[n, k] * Binomial[4*n + k + 1, n]/(4*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^4*(1+2*A)); polcoeff(A, n);

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

a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n+1, k)*binomial(5*n-k, n-k))/(4*n+1); \\ Seiichi Manyama, Jul 26 2020

a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(4*n+k+1,n)/(4*n+k+1).
a(n) = (1/(4*n+1)) * Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k) * binomial(5*n-k,n-k).
a(n) ~ sqrt(95781603 + 7199237*sqrt(177))*(69845 + 5251*sqrt(177))^(n - 1/2) / (sqrt(59*Pi) * n^(3/2) * 2^(12*n + 9/2)). - 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(5*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(4*n,k-1) for n > 0. (End)
a(n) = binomial(1+4*n, n)*hypergeom([-n, 1+4*n], [2+3*n], -2)/(1 + 4*n). - Stefano Spezia, Aug 09 2025

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.

cp's OEIS Frontend

A364748 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5 / (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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A365192 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)^2).

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A365188 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x*A(x)^4).

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A365193 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)^3).

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A243668 Number of Sylvester classes of 5-packed words of degree n.

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A365194 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)^6).

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A378610 Expansion of (1/x) * Series_Reversion( x * (1 - x/(1 - x))^4 ).

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

A365192 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)^2).

A365188 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x*A(x)^4).

A365193 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)^3).

A365194 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)^6).

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