A109081 -id:A109081 - cp's OEIS frontend

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

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

A366049 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1-x+x^2) ).

Original entry on oeis.org

1, 2, 8, 39, 211, 1218, 7349, 45790, 292361, 1902834, 12577737, 84205212, 569788192, 3890728052, 26775751320, 185525538183, 1293171205833, 9061500578178, 63794947215218, 451028012126797, 3200898741338041, 22794860112273294, 162841330273522907

Offset: 0

Seiichi Manyama, Sep 27 2023

nonn

Cf. A005043, A109081, A366050.

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

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

A262440 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k-1,n-k).

Original entry on oeis.org

1, 1, 5, 22, 101, 476, 2282, 11075, 54245, 267592, 1327580, 6617128, 33110090, 166215895, 836761343, 4222640822, 21354409445, 108193910000, 549084400088, 2790744368660, 14203023709276, 72371208424880, 369170645788840, 1885051297844624

Offset: 0

Vladimir Kruchinin, Sep 23 2015

nonn

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

Cf. A109081, A246437, A262441, A262442.

[&+[Binomial(n, k)*Binomial(n+k-1, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 13 2015

Join[{1}, Table[Sum[ Binomial[n,k] Binomial[n+k-1, n-k], {k, n}], {n, 25}]] (* Vincenzo Librandi, Sep 23 2015 *)

a(n):=sum(binomial(n,k)*binomial(n+k-2,n-k-1),k,0,n-1)/n;
A(x):=sum(a(n)*x^n,n,1,30);
taylor(diff(A(x),x)/A(x)*x,x,0,10);

a(n)=sum(k=0,n,(binomial(n,k)*binomial(n+k-1,n-k))) \\ Anders Hellström, Sep 23 2015

G.f.: x*A'(x)/A(x), where A(x) is g.f. of A109081.
Recurrence: 2*n*(2*n-1)*(38*n^3 - 210*n^2 + 377*n - 219)*a(n) = 2*(380*n^5 - 2480*n^4 + 5998*n^3 - 6598*n^2 + 3219*n - 540)*a(n-1) + 2*(n-2)*(76*n^4 - 382*n^3 + 572*n^2 - 300*n + 45)*a(n-2) + 3*(n-3)*(n-2)*(38*n^3 - 96*n^2 + 71*n - 14)*a(n-3). - Vaclav Kotesovec, Sep 23 2015
a(n) = n^2*hypergeom([1-n, 1-n, n+1], [3/2, 2], 1/4) for n >= 1. - Peter Luschny, Mar 06 2022
a(n) = [x^n] ( (1 - x + x^2) / (1 - x)^2 )^n. - Seiichi Manyama, Apr 29 2024
a(n) ~ sqrt((513 - 67*sqrt(57))^(1/3) + (513 + 67*sqrt(57))^(1/3)) * (10 + (1261 - 57*sqrt(57))^(1/3) + (1261 + 57*sqrt(57))^(1/3))^n / (19^(1/3) * sqrt(Pi*n) * 2^(n + 5/6) * 3^(n + 1/3)). - Vaclav Kotesovec, Apr 30 2024

A367232 G.f. satisfies A(x) = 1 + xA(x)^3 / (1 - xA(x))^2.

Original entry on oeis.org

1, 1, 5, 29, 189, 1325, 9757, 74429, 583037, 4662653, 37911037, 312457469, 2604534269, 21919435517, 185992729085, 1589480795133, 13668519794685, 118188894992381, 1026965424910333, 8962634482450429, 78528344593006589, 690502653622083581

Offset: 0

Seiichi Manyama, Nov 11 2023

nonn

Cf. A000108, A011270, A109081.

a(n, s=2, 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).

A262442 a(n) = Sum_{k=0..n}(binomial(n-1,n-k)*binomial(n+k-1,n-k)).

Original entry on oeis.org

1, 1, 3, 12, 53, 244, 1152, 5531, 26875, 131760, 650492, 3229368, 16105344, 80624935, 404913225, 2039146908, 10293657125, 52071019600, 263888886528, 1339540863092, 6809667715812, 34663102092960, 176655038497000, 901269559693104

Offset: 0

Vladimir Kruchinin, Sep 23 2015

nonn

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

Cf. A109081, A262440, A262441.

[&+[Binomial(n-1, n-k)*Binomial(n+k-1, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 23 2015

Join[{1}, Table[Sum[Binomial[n-1, n-k] Binomial[ n+k-1, n-k], {k, n}], {n, 25}]] (* Vincenzo Librandi, Sep 23 2015 *)

a(n):=sum(binomial(n, k)*binomial(n+k-2, n-k-1), k, 0, n-1)/n;
A(x):=sum(a(n)*x^n, n, 1, 30);
taylor(x*diff(A(x),x)/A(x)-x^2*diff(1/x-1/A(x),x),x,0,10);

a(n) = sum(k=0,n,(binomial(n-1,n-k)*binomial(n+k-1,n-k))) \\ Anders Hellström, Sep 23 2015

G.f.: 1+A'(x)*(x*A(x)-x^2)/A(x)^2, where A(x) is g.f. of A109081.
Recurrence: 2*(n-1)*(2*n - 1)*(38*n^2 - 162*n + 163)*a(n) = 2*(380*n^4 - 2380*n^3 + 5200*n^2 - 4676*n + 1431)*a(n-1) + 2*(n-2)*(76*n^3 - 362*n^2 + 502*n - 189)*a(n-2) + 3*(n-3)*(n-2)*(38*n^2 - 86*n + 39)*a(n-3). - Vaclav Kotesovec, Sep 23 2015
a(n) = n*hypergeom([1 - n, 1 - n, n + 1], [1, 3/2], 1/4) for n >= 1. - Peter Luschny, Mar 07 2022

A365133 G.f. satisfies A(x) = (1 + xA(x)/(1 - xA(x))^2)^2.

Original entry on oeis.org

1, 2, 9, 48, 284, 1792, 11816, 80446, 561186, 3990398, 28815594, 210746538, 1557834174, 11620294376, 87357498949, 661194915408, 5034368831334, 38534430714502, 296341243824737, 2288568585083816, 17741278361562738, 138006870242288796, 1076905750814353045

Offset: 0

Seiichi Manyama, Aug 23 2023

nonn

Cf. A109081, A365134.

Cf. A365120.

a(n, s=2, 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).

A366081 Expansion of (1/x) * Series_Reversion( x*(1-x)^2/(1-x-x^2) ).

Original entry on oeis.org

1, 1, 1, 0, -5, -22, -68, -165, -285, -96, 1892, 10574, 38436, 107175, 217063, 165232, -1150565, -7780744, -31173680, -94537100, -212903852, -239418048, 788015576, 6734057510, 29396759220, 95418332383, 233697161887, 334222633632, -514863450175, -6299672869750

Offset: 0

Seiichi Manyama, Sep 28 2023

sign

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

Cf. A007440, A108623, A366082, A366083.

Cf. A109081.

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

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

A215067 Number of Motzkin n-paths avoiding odd-numbered steps that are up steps.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 21, 37, 80, 146, 322, 602, 1347, 2563, 5798, 11181, 25512, 49720, 114236, 224540, 518848, 1027038, 2384538, 4748042, 11068567, 22150519, 51817118, 104146733, 244370806, 493012682, 1159883685, 2347796965, 5536508864, 11239697816, 26560581688, 54061835288

Offset: 0

David Scambler, Aug 02 2012

nonn

This sequence interleaves the counts of the closely related sequences A109081 and A106228.

a(n) is the number of (peakless) Motzkin paths of length n where every pair of matching up and down edges occupies positions of the same parity. Equivalently, the number of RNA secondary structures on n vertices where only vertices of the same parity can be matched. - Alexander Burstein, May 17 2021

			a(5) = 6: fUfFd, fUfDf, fUdUd, fUdFf, fFfUd, fFfFf showing odd-numbered steps in lower case.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Cf. A109081, A106228, A114465, A084075.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, 1, b(x-1, y) +b(x-1, y+1) +
      `if`(irem(x, 2)=1, 0, b(x-1, y-1)) ))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 04 2013

f[n_,x_,y_]:=f[n,x,y] = If[x>n||y<0,0,If[x==n&&y==0,1, If[EvenQ[x],0,f[n,x+1,y+1]] +f[n,x+1,y-1] + f[n,x+1,y]]]; Table[f[n,0,0],{n,0,35}]

{a(n)=polcoeff((1/x)*serreverse(x*(3+2*x+x^2 - sqrt((1+x^2)*(1+4*x+x^2)+x^2*O(x^n)))/(2*(1+x+x^2+x^2*O(x^n)))),n)} \\ Paul D. Hanna, Aug 02 2012

from mpmath import mp
mp.dps = 25; mp.pretty = True
def A215067(n) :
    m = n%2; r = n//2 if n>0 else 1
    return r^(1-m)*mp.hyper([-r,1-r-2*m,1+r+m],[(3-m)/2,(4-m)/2],1/4)
[int(A215067(i)) for i in (0..32)]  # Peter Luschny, Aug 03 2012

a(2*n) = Sum_{k=0..n} binomial(n+k-1,n-k) * binomial(n,k)/(n-k+1);
a(2*n+1) = Sum_{k=0..n} binomial(n+k+1,n-k) * binomial(n,k)/(n-k+1).
G.f.: (1/x)*Series_Reversion( x*(3+2*x+x^2 - sqrt((1+x^2)*(1+4*x+x^2)))/(2*(1+x+x^2)) ). - Paul D. Hanna, Aug 02 2012
G.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) = (3+2*x+x^2 + sqrt((1+x^2)*(1+4*x+x^2)))/4. - Paul D. Hanna, Aug 02 2012
G.f. satisfies: Series_Reversion(x*A(x)) = x - x^2*F(-x) where F(x) = g.f. of A114465. - Paul D. Hanna, Aug 02 2012
a(n) = 3_F_2([-r,1-r-2*m,1+r+m],[(3-m)/2,(4-m)/2],1/4)*r^(1-m) for n>0 where m = n mod 2 and r = floor(n/2). - Peter Luschny, Aug 03 2012

A365134 G.f. satisfies A(x) = (1 + xA(x)/(1 - xA(x))^2)^3.

Original entry on oeis.org

1, 3, 18, 130, 1041, 8889, 79310, 730593, 6895575, 66337179, 648087750, 6412437474, 64125877361, 647102364990, 6581050832082, 67384499298690, 694077333315363, 7186898222178342, 74767377019254450, 781105293655408554, 8191332027277068543

Offset: 0

Seiichi Manyama, Aug 23 2023

nonn

Cf. A109081, A365133.

Cf. A365121.

a(n, s=2, 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).

A366050 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1-x+x^2) ).

Original entry on oeis.org

1, 3, 16, 104, 750, 5769, 46373, 384885, 3273118, 28372354, 249762585, 2226782078, 20065651123, 182457467898, 1672073116401, 15427427247088, 143191280370438, 1336062703751262, 12524930325385008, 117910257665608080, 1114233543986585741

Offset: 0

Seiichi Manyama, Sep 27 2023

nonn

Cf. A005043, A109081, A366049.

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

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

cp's OEIS Frontend

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

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A366049 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1-x+x^2) ).

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A262440 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k-1,n-k).

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A367232 G.f. satisfies A(x) = 1 + x*A(x)^3 / (1 - x*A(x))^2.

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A262442 a(n) = Sum_{k=0..n}(binomial(n-1,n-k)*binomial(n+k-1,n-k)).

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A365133 G.f. satisfies A(x) = (1 + x*A(x)/(1 - x*A(x))^2)^2.

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A366081 Expansion of (1/x) * Series_Reversion( x*(1-x)^2/(1-x-x^2) ).

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A215067 Number of Motzkin n-paths avoiding odd-numbered steps that are up steps.

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A367232 G.f. satisfies A(x) = 1 + xA(x)^3 / (1 - xA(x))^2.

A365133 G.f. satisfies A(x) = (1 + xA(x)/(1 - xA(x))^2)^2.

A365134 G.f. satisfies A(x) = (1 + xA(x)/(1 - xA(x))^2)^3.