A161797 -id:A161797 - cp's OEIS frontend

A321798 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^4).

1, 1, 5, 23, 117, 636, 3607, 21106, 126489, 772468, 4789844, 30075937, 190851839, 1222000222, 7885041530, 51222338580, 334720178969, 2198755865424, 14511029102232, 96169424666028, 639757737711300, 4270520564506069, 28595671605541357, 192025292117465445, 1292866976587651519

Offset: 0

Ludovic Schwob, Nov 19 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..199 from Ludovic Schwob)

Cf. A109081, A161797, A321799.

List([0..25],n->Sum([0..n],k->Binomial(n,k)/(n-k+1)*Binomial(n+3*k-1,n-k))); # Muniru A Asiru, Nov 24 2018

a[n_] := Sum[Binomial[n, k] * Binomial[n + 3k - 1, n - k]/(n - k + 1), {k, 0,
n}]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)

a(n) = sum(k=0, n, binomial(n,k)*binomial(n+3*k-1, n-k)/(n-k+1)); \\ Michel Marcus, Nov 19 2018

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

a(n) ~ sqrt((1 - r*s)*(1 + 3*r*s)/(8*Pi*(8*s - 3))) / (n^(3/2) * r^(n+1)), where r = 0.139684805934917057093949761392656080860096066578... and s = 1.76437708701490464570032194388560298744432681226... are real roots of the system of equations s*(1 - r/(1 - r*s)^4) = 1, 4*r^2*s^2 = (1 - r*s)^5. - Vaclav Kotesovec, Nov 21 2018

A161798 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^3)^2.

Original entry on oeis.org

1, 2, 9, 46, 262, 1590, 10081, 65986, 442518, 3024772, 20996141, 147603198, 1048747751, 7519252606, 54332565330, 395264527626, 2892666314150, 21281120904168, 157299607827727, 1167582500757800, 8699515577902203

Offset: 0

Paul D. Hanna, Jun 19 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Recurrence

Cf. A161797, A161799.

Table[Sum[Binomial[2*n-k+1,k]/(n-k+1)*Binomial[n+2*k-1,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Sep 18 2013 *)

{a(n,m=1)=sum(k=0,n,binomial(2*n-k+m,k)*m/(n-k+m)*binomial(n+2*k-1,n-k))}

a(n) = Sum_{k=0..n} C(2*n-k+1,k)/(n-k+1) * C(n+2*k-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n then
a(n,m) = Sum_{k=0..n} C(2*n-k+m,k)*m/(n-k+m) * C(n+2*k-1,n-k).
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 8.01957328653868383... is the root of the equation 3125 + 22356*d - 162432*d^2 - 361584*d^3 - 326592*d^4 + 46656*d^5 = 0 and c = 1.216730444416766043545857948227854793382399566... - Vaclav Kotesovec, Sep 18 2013

A321799 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^5).

Original entry on oeis.org

1, 1, 6, 31, 176, 1071, 6797, 44493, 298279, 2037550, 14131441, 99244564, 704360703, 5043969503, 36399930179, 264451303466, 1932650461883, 14198082537190, 104792195449688, 776681663951998, 5778226417888171, 43135097969972931, 323012620411650708, 2425745980876575899, 18264470545275495152

Offset: 0

Ludovic Schwob, Nov 19 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..199 from Ludovic Schwob)

Cf. A109081, A161797, A321798.

List([0..25],n->Sum([0..n],k->Binomial(n,k)/(n-k+1)*Binomial(n+4*k-1,n-k))); # Muniru A Asiru, Nov 24 2018

[1] cat [&+[(Binomial(n,k)/(n-k+1)) * Binomial(n+4*k-1,n-k):  k in [0..n]]: n in [1.. 25]]; // Vincenzo Librandi, Dec 08 2018

eq:= a - 1/(1-x/(1-x*a)^5):
S:= series(RootOf(numer(eq),a),x,31):
seq(coeff(S,x,j),j=0..30); # Robert Israel, Dec 10 2018

a[n_]:=Sum[ Binomial[n,k]/(n-k+1)*Binomial[n+4*k-1,n-k], {k,0,n}]; Array[a, 20, 0] (* Stefano Spezia, Nov 19 2018 *)
A[] = 0; Do[A[x] = 1/(1-x/(1-x*A[x])^5)+O[x]^25, {25}];
CoefficientList[A[x], x] (* Jean-François Alcover, Dec 30 2018 *)

a(n) = sum(k=0, n, binomial(n,k)*binomial(n+4*k-1, n-k)/(n-k+1)); \\ Michel Marcus, Nov 19 2018

[sum(binomial(n,k)*binomial(n+4*k-1,n-k)/(n-k+1) for k in (0..n)) for n in range(25)] # G. C. Greubel, Dec 14 2018

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

a(n) ~ sqrt((1 - r*s)*(1 + 4*r*s) / (5*Pi*(5*s - 2))) / (2 * n^(3/2) * r^(n+1)), where r = 0.124910212976238209867004924637837518925706044646... and s = 1.72708330560542094133450070142549940430523638921... are real roots of the system of equations s*(1 - r/(1 - r*s)^5) = 1, 5*r^2*s^2 = (1 - r*s)^6. - Vaclav Kotesovec, Nov 21 2018

A365113 G.f. satisfies A(x) = 1 + x / (1 - x*A(x))^3.

Original entry on oeis.org

1, 1, 3, 9, 31, 114, 438, 1739, 7077, 29364, 123756, 528324, 2279868, 9928679, 43580301, 192601419, 856317717, 3827501985, 17188943523, 77521747638, 350959738842, 1594390493067, 7266093316649, 33209221327752, 152182572790008, 699083290518817, 3218624408121555

Offset: 0

Seiichi Manyama, Aug 22 2023

nonn

Cf. A000108, A365114, A365115.

Cf. A161797, A365110.

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

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

A365086 G.f. satisfies A(x) = 1 + xA(x) / (1 + xA(x))^3.

Original entry on oeis.org

1, 1, -2, -2, 15, -4, -122, 204, 903, -3374, -4635, 43539, -13233, -475123, 873392, 4244591, -16906773, -24952174, 244162840, -74520792, -2901715074, 5483226036, 27740164293, -112969486284, -172903931727, 1714556657881, -513739179725, -21235809823325

Offset: 0

Seiichi Manyama, Aug 21 2023

sign

Cf. A090192, A365085, A365087, A365088.

Cf. A127896, A161797, A364736.

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

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

A161799 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^2)^3.

Original entry on oeis.org

1, 3, 12, 61, 345, 2085, 13182, 86106, 576543, 3936029, 27294390, 191722887, 1361291244, 9754412169, 70447946556, 512278417176, 3747570671685, 27561220671408, 203657352324178, 1511270129552163, 11257532921742528

Offset: 0

Paul D. Hanna, Jun 19 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Recurrence

Cf. A161797, A161798.

A161799 := proc(n)
    local s,t ;
    s := 2 ;
    t := 3;
    add( binomial(t*n-(t-1)*(k-1),k) * binomial(n+(s-1)*k-1,n-k) /(n-k+1) ,k=0..n) ;
end proc:
seq(A161799(n),n=0..40) ; # R. J. Mathar, May 12 2022

Table[Sum[Binomial[3*n-2*k+2,k]/(n-k+1)*Binomial[n+k-1,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Sep 18 2013 *)

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

a(n) = Sum_{k=0..n} C(3*n-2*k+2,k)/(n-k+1) * C(n+k-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n then
a(n,m) = Sum_{k=0..n} C(3*n-2*k+3*m-1,k)*m/(n-k+m) * C(n+k-1,n-k).
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 8.01957328653868383... is the root of the equation 3125 + 22356*d - 162432*d^2 - 361584*d^3 - 326592*d^4 + 46656*d^5 = 0 and c = 1.56703431595354192843152170651865561188... - Vaclav Kotesovec, Sep 18 2013

A367233 G.f. satisfies A(x) = 1 + xA(x)^3 / (1 - xA(x))^3.

Original entry on oeis.org

1, 1, 6, 39, 284, 2223, 18267, 155445, 1358073, 12111306, 109802183, 1009001571, 9376972698, 87978198364, 832223905371, 7928413841673, 76002832317437, 732578811761670, 7095717550127526, 69029297500888522, 674181392461483212, 6607910786529613248

Offset: 0

Seiichi Manyama, Nov 11 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..983

Cf. A161797, A365113, A365150.

Cf. A108447, A367232, A367235.

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

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

A349022 G.f. satisfies A(x) = 1/(1 - x/(1 - x*A(x))^3)^4.

Original entry on oeis.org

1, 4, 22, 152, 1161, 9460, 80550, 708172, 6379368, 58576168, 546215580, 5158542152, 49239812893, 474285453628, 4604149947276, 44999181550032, 442430807369519, 4372944634271688, 43425156714959956, 433049078716727332, 4334925824762251939

Offset: 0

Seiichi Manyama, Nov 06 2021

nonn

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

Cf. A006632, A161797, A161798.

A349022 := proc(n)
    add(binomial(4*n-3*(k-1),k)*binomial(n+2*k-1,n-k)/(n-k+1),k=0..n) ;
end proc:
seq(A349022(n),n=0..40) ; # R. J. Mathar, Jan 25 2023

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

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

A365135 G.f. satisfies A(x) = (1 + xA(x)/(1 - xA(x))^3)^2.

Original entry on oeis.org

1, 2, 11, 68, 467, 3418, 26133, 206264, 1667908, 13746476, 115050074, 975180582, 8354044986, 72215867960, 629139381448, 5518236646614, 48689379017014, 431868759238498, 3848616161600778, 34441553184113542, 309390614528633311, 2788841905397090626

Offset: 0

Seiichi Manyama, Aug 23 2023

nonn

Cf. A161797, A365136.

Cf. A006013.

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

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

A365136 G.f. satisfies A(x) = (1 + xA(x)/(1 - xA(x))^3)^3.

Original entry on oeis.org

1, 3, 21, 172, 1563, 15141, 153240, 1601160, 17140686, 187026210, 2072333697, 23255417925, 263757940688, 3018654757212, 34817822871933, 404324843585061, 4723248984803013, 55467143334798210, 654435356605769574, 7753961433310798095, 92220463998917459652

Offset: 0

Seiichi Manyama, Aug 23 2023

nonn

Cf. A161797, A365135.

Cf. A365122.

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

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

cp's OEIS Frontend

A321798 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^4).

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Mathematica

PARI

Formula

A161798 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^3)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A321799 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^5).

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A365113 G.f. satisfies A(x) = 1 + x / (1 - x*A(x))^3.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A365086 G.f. satisfies A(x) = 1 + x*A(x) / (1 + x*A(x))^3.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A161799 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^2)^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A367233 G.f. satisfies A(x) = 1 + x*A(x)^3 / (1 - x*A(x))^3.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A349022 G.f. satisfies A(x) = 1/(1 - x/(1 - x*A(x))^3)^4.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A365086 G.f. satisfies A(x) = 1 + xA(x) / (1 + xA(x))^3.

A367233 G.f. satisfies A(x) = 1 + xA(x)^3 / (1 - xA(x))^3.

A365135 G.f. satisfies A(x) = (1 + xA(x)/(1 - xA(x))^3)^2.

A365136 G.f. satisfies A(x) = (1 + xA(x)/(1 - xA(x))^3)^3.