A349331 -id:A349331 - cp's OEIS frontend

A349332 G.f. A(x) satisfies A(x) = 1 + x * A(x)^5 / (1 - x).

1, 1, 6, 46, 406, 3901, 39627, 418592, 4551672, 50610692, 572807157, 6577068383, 76426719408, 897078662538, 10620634999318, 126676885170703, 1520759193166329, 18361269213121164, 222814883564042704, 2716125963857227904, 33244557641365865109

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..897

Cf. A002212, A307678, A349331, A349333, A349334, A349335.

Cf. A002294, A346647 (partial sums), A349361, A364748.

a:= n-> coeff(series(RootOf(1+x*A^5/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021

nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x]^5/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^5, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(5*k,k) / (4*k+1).
a(n) ~ 3381^(n + 1/2) / (25 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Nov 15 2021
Recurrence: 8*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*a(n) = (4405*n^4 - 10346*n^3 + 9575*n^2 - 4354*n + 840)*a(n-1) - 12*(n-2)*(1255*n^3 - 3957*n^2 + 4492*n - 1820)*a(n-2) + 2*(n-3)*(n-2)*(10655*n^2 - 32733*n + 26908)*a(n-3) - 4*(n-4)*(n-3)*(n-2)*(3445*n - 6986)*a(n-4) + 3381*(n-5)*(n-4)*(n-3)*(n-2)*a(n-5). - Vaclav Kotesovec, Nov 17 2021

A349333 G.f. A(x) satisfies A(x) = 1 + x * A(x)^6 / (1 - x).

Original entry on oeis.org

1, 1, 7, 64, 678, 7836, 95838, 1219527, 15979551, 214151601, 2921712145, 40444378948, 566634504256, 8019501351103, 114484746457075, 1646614155398872, 23837794992712680, 347081039681365623, 5079306905986689309, 74670702678690897079, 1102218694940440851877

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..836

Cf. A002212, A307678, A349331, A349332, A349334, A349335.

Cf. A002295, A346648 (partial sums), A349362.

a:= n-> coeff(series(RootOf(1+x*A^6/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021

nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x]^6/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^6, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(6*k,k) / (5*k+1).

a(n) ~ 49781^(n + 1/2) / (72 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

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

Original entry on oeis.org

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

A364987 E.g.f. satisfies A(x) = 1 + xexp(x)A(x)^4.

Original entry on oeis.org

1, 1, 10, 183, 5140, 196005, 9468486, 554425963, 38171336680, 3022130473065, 270537702834250, 27021535857472431, 2979254055371578524, 359411244032212931533, 47093111659782104431438, 6660135357832421444841555, 1011181346455643980818939856

Offset: 0

Seiichi Manyama, Aug 15 2023

nonn

Cf. A006153, A295238, A364983.

Cf. A002293, A349331.

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

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*k+1,k)/( (4*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A002293(k)/(n-k)!.

a(n) ~ 2*sqrt(1 + LambertW(27/256)) * n^(n-1) / (3^(3/2) * exp(n) * LambertW(27/256)^n). - Vaclav Kotesovec, Nov 11 2024

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

A349335 G.f. A(x) satisfies A(x) = 1 + x * A(x)^8 / (1 - x).

Original entry on oeis.org

1, 1, 9, 109, 1541, 23823, 390135, 6651051, 116798643, 2098313686, 38382509118, 712447023590, 13385500614902, 254065657922154, 4864482597112186, 93840443376075810, 1822169236520766546, 35586928273002974487, 698572561837366684479, 13775697096997873764647

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

In general, for m > 1, Sum_{k=0..n} binomial(n-1,k-1) * binomial(m*k,k) / ((m-1)*k+1) ~ (m-1)^(m/2 - 2) * (1 + m^m/(m-1)^(m-1))^(n + 1/2) / (sqrt(2*Pi) * m^((m-1)/2) * n^(3/2)). - Vaclav Kotesovec, Nov 15 2021

Seiichi Manyama, Table of n, a(n) for n = 0..756

Cf. A002212, A307678, A349331, A349332, A349333, A349334.

Cf. A007556, A346650 (partial sums), A349364.

a:= n-> coeff(series(RootOf(1+x*A^8/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..19);  # Alois P. Heinz, Nov 15 2021

nmax = 19; A[] = 0; Do[A[x] = 1 + x A[x]^8/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^8, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(8*k,k) / (7*k+1).

a(n) ~ 17600759^(n + 1/2) / (2048 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A349334 G.f. A(x) satisfies A(x) = 1 + x * A(x)^7 / (1 - x).

Original entry on oeis.org

1, 1, 8, 85, 1051, 14197, 203064, 3022909, 46347534, 726894786, 11606936525, 188060979332, 3084087347910, 51094209834068, 853859480938095, 14376597494941454, 243649099741045190, 4153091242153905838, 71152973167920086796, 1224593757045581062444

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..791

Cf. A002212, A307678, A349331, A349332, A349333, A349335.

Cf. A002296, A346649 (partial sums), A349363.

a:= n-> coeff(series(RootOf(1+x*A^7/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021

nmax = 19; A[] = 0; Do[A[x] = 1 + x A[x]^7/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 19}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^7, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

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

a(n) ~ 870199^(n + 1/2) / (343 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A371483 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^3.

Original entry on oeis.org

1, 3, 18, 124, 933, 7446, 61943, 531348, 4666425, 41751325, 379230711, 3487769871, 32414437521, 303950138604, 2872137458010, 27322233357964, 261446381792670, 2514851398148595, 24303030755342128, 235841264063844258, 2297278004837062317

Offset: 0

Seiichi Manyama, Mar 25 2024

nonn

Cf. A002212, A270386, A371486.

Cf. A006632, A349331.

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

a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(4*k+2,k)/(k+1).
G.f.: A(x) = B(x/(1-x)), where B(x) = (1/x) * Series_Reversion( x*(1-x)^3 ).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A349331.

A371517 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^4.

Original entry on oeis.org

1, 4, 26, 188, 1459, 11892, 100444, 871528, 7722557, 69590628, 635807180, 5876094308, 54836925779, 516029817620, 4891147100886, 46653935716492, 447490869463145, 4313492172957396, 41763413498670702, 405968522259130636, 3960526930400038404

Offset: 0

Seiichi Manyama, Mar 26 2024

nonn

Cf. A045868, A371516, A371520, A371521.
Cf. A349331, A371483, A371518.
Cf. A371486.

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

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

G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A349331.

A378326 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(nk,k) / ((n-1)k+1).

Original entry on oeis.org

1, 1, 3, 19, 219, 3901, 95838, 3022909, 116798643, 5350403737, 283728025998, 17104314563843, 1155635807408096, 86513627563199279, 7109252862969177287, 636268582522962837475, 61610670571434193189443, 6418044336586421956746033, 715718717341021991299583730

Offset: 0

Vaclav Kotesovec, Nov 23 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..338

Cf. A002212, A307678, A349331, A349332, A349333, A349334, A349335.

Cf. A377098, A378325, A378327.

Table[Sum[Binomial[n-1, k-1]*Binomial[n*k, k]/((n-1)*k+1), {k, 0, n}], {n, 0, 20}]

a(n) ~ exp(n + exp(-1) - 1/2) * n^(n - 5/2) / sqrt(2*Pi).

cp's OEIS Frontend

A349332 G.f. A(x) satisfies A(x) = 1 + x * A(x)^5 / (1 - x).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A349333 G.f. A(x) satisfies A(x) = 1 + x * A(x)^6 / (1 - x).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

PARI

PARI

PARI

Formula

Extensions

A364987 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^4.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A349335 G.f. A(x) satisfies A(x) = 1 + x * A(x)^8 / (1 - x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A349334 G.f. A(x) satisfies A(x) = 1 + x * A(x)^7 / (1 - x).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A364987 E.g.f. satisfies A(x) = 1 + xexp(x)A(x)^4.

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

A378326 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(nk,k) / ((n-1)k+1).