A364371 -id:A364371 - cp's OEIS frontend

A073157 Number of Schroeder n-paths containing no FFs.

1, 2, 5, 18, 70, 293, 1283, 5808, 26960, 127628, 613814, 2990681, 14730713, 73229291, 366936231, 1851352820, 9397497758, 47957377934, 245903408244, 1266266092112, 6545667052320, 33954266444498, 176689391245146

Offset: 0

Paul D. Hanna, Jul 29 2002

Number of Schroeder n-paths containing no FFs. A Schroeder n-path (A006318) consists of steps U=(1,1),F=(2,0),D=(1,-1) starting at (0,0), ending at (2n,0), and never going below the x-axis. Example: a(2)=5 counts UFD, UUDD, UDF, FUD, UDUD. - David Callan, Aug 23 2011

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 70*x^4 + 293*x^5 + 1283*x^6 + ...

Muniru A Asiru, Table of n, a(n) for n = 0..300

Leftmost column of triangle A073154 (was previous name).

Cf. A000108, A073155, A073156, A073153, A143330, A198953, A215715, A216359, A364335, A364371.

List([0..25],n->Sum([0..n],i->Binomial(2*i+1,i)*Binomial(2*i+1,n-i)/(2*i+1))); # Muniru A Asiru, Oct 11 2018

a:=n->add(binomial(2*i+1,i)*binomial(2*i+1,n-i)/(2*i+1),i=0..n): seq(a(n),n=0..25); # Muniru A Asiru, Oct 11 2018

Table[Sum[Binomial[2*i + 1, i]*Binomial[2*i + 1, n - i]/(2*i + 1), {i, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 11 2018 *)

a(n):=sum((sum((binomial(2*k+2,j-k)*binomial(2*k,k)/(k+1)),k,0,j))*(-1)^(n-j),j,0,n); /* Vladimir Kruchinin, Mar 13 2016 */

{a(n)=local(A=1); for(i=0,n-1,A=(1+x)*(1+x*(A+x*O(x^n))^2));polcoeff(A,n)} /* Paul D. Hanna, Mar 03 2008 */

A073155(n+1) = Sum_{k=0..n} a(k)*a(n-k), that is, convolution yields sequence A073155 minus the 0th term.
G.f.: A(x) = (1 - sqrt(1 - 4*x*(1+x)^2))/(2*x*(1+x)) satisfies A(x) = (1+x)*(1 + x*A(x)^2);
G.f.: A(x) = (1+x)*C(x*(1+x)^2) where C(x) is the Catalan g.f. of A000108. - Paul D. Hanna, Mar 03 2008
a(n) = Sum_{j=0..n}((Sum_{k=0..j}((binomial(2*k+2,j-k)*C(k))))*(-1)^(n-j)), where C(k) = A000108(k). - Vladimir Kruchinin, Mar 13 2016
a(n) = Sum_{i=0..n} C(2*i+1,i)*C(2*i+1,n-i)/(2*i+1). - Vladimir Kruchinin, Oct 11 2018
Recurrence: (n+1)*a(n) = 3*(n-1)*a(n-1) + 6*(2*n - 3)*a(n-2) + 6*(2*n - 5)*a(n-3) + 2*(2*n - 7)*a(n-4). - Vaclav Kotesovec, Oct 11 2018
From Peter Bala, Aug 25 2024: (Start)
(1/x) * series_reversion(x/A(x)) = 1 + 2*x + 9*x^2 + 56*x^3 + 400*x^4 + 3095*x^5 + 25240*x^6 + ... is the g.f. of A198953.
(1/x) * series_reversion(x*A(-x)) = 1 + 2*x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 289*x^6 + ... = G(x) + x, where G(x) = (1 - x^2 - sqrt(1 - 4*x - 2*x^2 + x^4))/(2*x) is the g.f. of A143330. (End)
Define a sequence operator R: {u(n): n >= 0} -> {v(n): n >= 0} by Sum_{n >= 0} v(n)*x^n = (1/x) * series_reversion(x/Sum_{n >= 0} u(n)*x^n). Then R({a(n)}) = A198953, R^2({a(n)}) = A215715 and R^3({a(n)}) = A364335. Cf. A216359. - Peter Bala, Sep 13 2024

More terms from Paul D. Hanna, Mar 03 2008

New name using a comment from David Callan, Peter Luschny, Oct 14 2018

A364374 G.f. satisfies A(x) = (1 + xA(x)) (1 - x*A(x)^2).

Original entry on oeis.org

1, 0, -1, 1, 2, -6, -1, 28, -31, -98, 288, 131, -1730, 1638, 7431, -19583, -15502, 135642, -99523, -664050, 1535896, 1816196, -11902728, 5944326, 64487669, -129346490, -213116764, 1112382523, -277762230, -6572175490, 11287106695, 25078981772, -107983368519, -1826241850

Offset: 0

Seiichi Manyama, Jul 21 2023

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

Cf. A364375, A364376.

Cf. A057078, A007863, A364371.

A364374 := proc(n)
    add( (-1)^k*binomial(n+k+1,k) * binomial(n+k+1,n-k)/(n+k+1),k=0..n) ;
end proc:
seq(A364374(n),n=0..80); # R. J. Mathar, Jul 25 2023

nmax = 33;
A[_] = 1;
Do[A[x_] = (1+x*A[x])*(1-x*A[x]^2) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 21 2023 *)

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

a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1).
D-finite with recurrence 15*n*(n+1)*a(n) +2*n*(13*n-11)*a(n-1) +12*(9*n^2-19*n+9)*a(n-2) +2*(10*n^2-65*n+99)*a(n-3) -4*(n-3)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jul 25 2023
A(x) = (1/x) * series_reversion(x*(1 + x + x^2)/(1 + x)). - Peter Bala, Sep 08 2024

A364372 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^3).

Original entry on oeis.org

1, 0, -1, 3, -6, 6, 15, -107, 349, -672, 39, 5835, -27654, 75765, -95799, -279129, 2297970, -8377854, 17663640, -996624, -177445221, 888491025, -2551959604, 3337931168, 10407149226, -87719805853, 328682535695, -708428979213, 15252552804, 7616368090377, -38693979668535

Offset: 0

Seiichi Manyama, Jul 20 2023

Cf. A364371, A364373, A364376.

Cf. A364336.

A364372 := proc(n)
    add( (-1)^k*binomial(3*k+1,k) * binomial(3*k+1,n-k)/(3*k+1),k=0..n) ;
end proc:
seq(A364372(n),n=0..80); # R. J. Mathar, Jul 25 2023

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

a(n) = Sum_{k=0..n} (-1)^k * binomial(3*k+1,k) * binomial(3*k+1,n-k) / (3*k+1).
D-finite with recurrence 2*n*(2*n+1)*a(n) +(51*n-26)*(n-1)*a(n-1) +(279*n^2 -931*n +766)*a(n-2) +2*(413*n^2 -2127*n +2728)*a(n-3) +6*(75*n-244) *(3*n-11)*a(n-4) +9*(45*n-179) *(3*n-14)*a(n-5) +63*(3*n-14) *(3*n-17)*a(n-6) +12*(3*n-16) *(3*n-20)*a(n-7)=0. - R. J. Mathar, Jul 25 2023
From Peter Bala, Aug 25 2024: (Start)
Fifth-order recurrence: 2*(n-2)*(n-1)*n*(2*n+1)*a(n) + (n-2)*(n-1)*(31*n^2-27*n+6)*a(n-1) + 3*(n-2)*(3*n-5)*(12*n^2-19*n+2)*a(n-2) + 3*(3*n-8)*(18*n^3-69*n^2+63*n-2)*a(n-3) + 3*n*(3*n-11)*(12*n^2-49*n+42)*a(n-4) + 3*n*(n-1)*(3*n-10)*(3*n-14)*a(n-5) = 0 with a(0) = 1, a(1) = 0, a(2) = -1, a(3) = 3 and a(4) = -6.
The g.f. A(x) satisfies (1/x) * series_reversion(x/A(x)) = 1 - x^2 + 3*x^3 - 4*x^4 - 9*x^5 + 73*x^6 - ..., the g.f. of A364376. (End)

A364375 G.f. satisfies A(x) = (1 + xA(x)) (1 - x*A(x)^3).

Original entry on oeis.org

1, 0, -1, 2, 0, -11, 28, 1, -206, 564, 38, -4711, 13329, 1273, -119762, 344707, 41884, -3251250, 9445976, 1381154, -92305098, 269504686, 45848871, -2707126108, 7921304973, 1532928960, -81375728566, 238196143730, 51591751698, -2493907008116, 7293147604136

Offset: 0

Seiichi Manyama, Jul 21 2023

Cf. A364371, A364372, A364374, A364376.

Cf. A198953.

A364375 := proc(n)
    add( (-1)^k*binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k)/(n+2*k+1),k=0..n) ;
end proc:
seq(A364375(n),n=0..80); # R. J. Mathar, Jul 25 2023

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

a(n) = Sum_{k=0..n} (-1)^k * binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k) / (n+2*k+1).
D-finite with recurrence +2*n*(191553133*n -462036810)*(2*n+1) *(n+1)*a(n) +2*n*(6735679202*n^3 -31340869996*n^2 +39568451245*n -13340358389)*a(n-1) +6*(13937077342*n^4 -106287464449*n^3 +278022830194*n^2 -296712736455*n +108876423952)*a(n-2) +6*(42118990776*n^4 -422141236704*n^3 +1546534911485*n^2 -2448212978721*n +1409411956166)*a(n-3) +6*(72631772298*n^4 -948761263665*n^3 +4512788370945*n^2 -9254886913710*n +6888712179986)*a(n-4) +3*(10147840245*n^4 -513806508936*n^3 +5519825354705*n^2 -22028093493130*n +30003008863784)*a(n-5) +6*(-9503341830*n^4 +235269814455*n^3 -2064338754902*n^2 +7709425316943*n -10409244067330)*a(n-6) +18*(3*n-20)*(n-6) *(156488131*n-746235854) *(3*n-13)*a(n-7)=0. - R. J. Mathar, Jul 25 2023
From Peter Bala, Aug 24 2024: (Start)
G.f. A(x) satisfies (1/x) * series_reversion(x*A(x)) = 1/G(x), where G(x) is the g.f. of A364371.
P-recursive: fifth-order recurrence:      (2*n+1)*(2*n+2)*(3045*n^5-26680*n^4+84901*n^3-123566*n^2+86300*n-25368)*n*a(n) + 6*(18270*n^7-160080*n^6+500851*n^5-666969*n^4+307749*n^3+70849*n^2-76222*n+8288)*n*a(n-1) +  6*(54810*n^8-562455*n^7+2191158*n^6-3956204*n^5+2960986*n^4+88959*n^3-1045774*n^2+187688*n+69888)*a(n-2) + 6*(109620*n^8-1289340*n^7+5897421*n^6-13016841*n^5+13725877*n^4-5967199*n^3+2484230*n^2-3359528*n+1002624)*a(n-3) - 6*(54810*n^8-726885*n^7+3719313*n^6-9080919*n^5+10367473*n^4-4378276*n^3+1152956*n^2-2297912*n+768096)*a(n-4) + (3*n-7)*(3*n-12)*(3*n-14)*(3045*n^5-11455*n^4+8631*n^3+1507*n^2+2376*n-1368)*a(n-5) = 0 with a(0) = 1, a(1) = 0, a(2) = -1, a(3) = 2 and a(4) = 0. (End)

A364373 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^4).

Original entry on oeis.org

1, 0, -1, 4, -12, 26, -14, -236, 1604, -6577, 17827, -14064, -186496, 1437856, -6416576, 18733256, -17358808, -201270728, 1652571996, -7692333934, 23375782030, -23913813710, -250917362258, 2147925544190, -10270145045142, 32053993413694, -35259817590134

Offset: 0

Seiichi Manyama, Jul 20 2023

sign

Cf. A364371, A364372.

Cf. A364337.

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

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

cp's OEIS Frontend

A073157 Number of Schroeder n-paths containing no FFs.

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A364374 G.f. satisfies A(x) = (1 + x*A(x)) * (1 - x*A(x)^2).

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A364372 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^3).

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A364375 G.f. satisfies A(x) = (1 + x*A(x)) * (1 - x*A(x)^3).

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A364373 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^4).

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A364374 G.f. satisfies A(x) = (1 + xA(x)) (1 - x*A(x)^2).

A364375 G.f. satisfies A(x) = (1 + xA(x)) (1 - x*A(x)^3).