A127902 -id:A127902 - cp's OEIS frontend

A379463 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -3), and staying on or above y = -1.

1, 1, 1, 1, 3, 11, 31, 71, 150, 334, 826, 2146, 5498, 13690, 33762, 84306, 214451, 551107, 1417291, 3637627, 9343555, 24096675, 62439587, 162331747, 422773098, 1102422546, 2879207046, 7534606366, 19756893196, 51894005428, 136496647696, 359478351816, 947912008073

Offset: 0

Emely Hanna Li Lobnig, Dec 23 2024

nonn

			For n = 4, the a(4)=3 paths are HHHH, UUDU, UUUD, where U=(1,1), D=(1,-3) and H=(1,0).

Cf. A069271, A127902, A379464.

A379463 := proc(n)
    add(2*binomial(n, k*4)*binomial(4*k+1, k)/(3*k+2),k=0..floor(n/4)) ;
end proc:
seq(A379463(n),n=0..50) ; # R. J. Mathar, Jan 29 2025

a(n) = sum(k=0, floor(n/4), 2*binomial(n, k*4)*binomial(4*k+1, k)/(3*k+2)) \\ Thomas Scheuerle, Jan 07 2025

a(n) ~ 2^(3/2) * (1 + 4/3^(3/4))^(n + 3/2) / (3^(11/8) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 15 2025
Conjecture D-finite with recurrence 3*n*(3*n+4)*(n-3)*(3*n+8)*a(n) +3*(-45*n^4+54*n^3+192*n^2-27*n-20)*a(n-1)
+9*(n-1)*(30*n^3-72*n^2-7*n+20)*a(n-2) -3*(n-1)*(n-2)*(90*n^2-234*n+95)*a(n-3) -(n-1)*(n-2)*(n-3)*(121*n+499)*a(n-4) +229*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 29 2025

More terms from Jinyuan Wang, Jan 07 2025

A379464 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -3), and staying on or above y = -2.

Original entry on oeis.org

1, 1, 1, 1, 4, 16, 46, 106, 226, 514, 1306, 3466, 9002, 22634, 56330, 142026, 364743, 945303, 2448511, 6323695, 16336885, 42363693, 110340297, 288229377, 753920796, 1973799396, 5174280216, 13588243696, 35748326836, 94188788164, 248464963876, 656148369796

Offset: 0

Emely Hanna Li Lobnig, Dec 23 2024

nonn

			For n = 4, the a(4)=4 paths are HHHH, UDUU, UUDU, UUUD, where U=(1,1), D=(1,-3) and H=(1,0).

Robert Israel, Table of n, a(n) for n = 0..2271

Cf. A127902, A379463.

f:= proc(n,y) option remember;
    if n = 0 then if y = 0 then return 1 else return 0 fi fi;
    if y > n then return 0 fi;
    if y >= -1 then procname(n-1,y-1) + procname(n-1,y) + procname(n-1,y+3)
    else procname(n-1,y) + procname(n-1,y+3)
    fi;
end proc:
map(f, [$0..40],0); # Robert Israel, Jan 23 2025

a(n) = sum(k=0, floor(n/4), 3*binomial(n, k*4)*binomial(4*k+3, k)/(4*k+3)) \\ Thomas Scheuerle, Jan 07 2025

a(n) ~ 2^(5/2) * (1 + 4/3^(3/4))^(n + 3/2) / (3^(11/8) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 15 2025

Conjecture D-finite with recurrence +3*(n+4)*(3*n+4)*(3*n+8)*a(n) +3*(-63*n^3-297*n^2-349*n-60)*a(n-1) +3*(189*n^3+270*n^2-229*n-140)*a(n-2) +15*(-63*n^3+117*n^2+44*n-64)*a(n-3) +(689*n^3-5372*n^2+6946*n-1288)*a(n-4) +(n-4)*(201*n^2+2767*n-3011)*a(n-5) -(n-5)*(579*n+257)*(n-4)*a(n-6) +229*(n-5)*(n-6)*(n-4)*a(n-7)=0. - R. J. Mathar, Jan 29 2025

More terms from Jinyuan Wang, Jan 07 2025

A365079 G.f. satisfies A(x) = 1 + xA(x)(1 + x^4*A(x)^3).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 6, 16, 36, 71, 131, 247, 511, 1156, 2696, 6172, 13664, 29563, 63871, 140341, 315185, 717962, 1639822, 3728276, 8432696, 19047924, 43166420, 98378502, 225355290, 517683270, 1190034046, 2735049866, 6287002806, 14467864356, 33355524916

Offset: 0

Seiichi Manyama, Aug 29 2023

nonn

Cf. A127902, A186996, A215340.

a(n) = sum(k=0, n\5, binomial(n-4*k, k)*binomial(n-k+1, n-4*k)/(n-k+1));

a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k,k) * binomial(n-k+1,n-4*k)/(n-k+1).

A367114 G.f. satisfies A(x) = 1 + 2xA(x) + x^4*A(x)^4.

Original entry on oeis.org

1, 2, 4, 8, 17, 42, 124, 408, 1380, 4616, 15184, 49568, 162518, 539580, 1818184, 6203088, 21339916, 73776024, 255853744, 889678688, 3102779785, 10856555130, 38115293308, 134243564056, 474159194316, 1678926445272, 5957812156144, 21183679310048

Offset: 0

Seiichi Manyama, Nov 05 2023

Cf. A002293, A127902, A190590, A367115.

a(n) = sum(k=0, n\4, 2^(n-4*k)*binomial(n, 4*k)*binomial(4*k, k)/(3*k+1));

a(n) = Sum_{k=0..floor(n/4)} 2^(n-4*k) * binomial(n,4*k) * A002293(k).

cp's OEIS Frontend

A379463 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -3), and staying on or above y = -1.

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A379464 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -3), and staying on or above y = -2.

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Maple

PARI

Formula

Extensions

A365079 G.f. satisfies A(x) = 1 + xA(x)(1 + x^4*A(x)^3).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A367114 G.f. satisfies A(x) = 1 + 2xA(x) + x^4*A(x)^4.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

A379463 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -3), and staying on or above y = -1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A379464 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -3), and staying on or above y = -2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A365079 G.f. satisfies A(x) = 1 + x*A(x)*(1 + x^4*A(x)^3).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A367114 G.f. satisfies A(x) = 1 + 2*x*A(x) + x^4*A(x)^4.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A365079 G.f. satisfies A(x) = 1 + xA(x)(1 + x^4*A(x)^3).

A367114 G.f. satisfies A(x) = 1 + 2xA(x) + x^4*A(x)^4.