A212383 -id:A212383 - cp's OEIS frontend

A212382 Number A(n,k) of Dyck n-paths all of whose ascents have lengths equal to 1+k*m (m>=0); square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 1, 2, 14, 1, 1, 1, 1, 1, 5, 42, 1, 1, 1, 1, 1, 2, 12, 132, 1, 1, 1, 1, 1, 1, 6, 30, 429, 1, 1, 1, 1, 1, 1, 2, 16, 79, 1430, 1, 1, 1, 1, 1, 1, 1, 7, 37, 213, 4862, 1, 1, 1, 1, 1, 1, 1, 2, 22, 83, 584, 16796, 1

Offset: 0

Alois P. Heinz, May 12 2012

Lengths of descents are unrestricted.

For p>0 is column p asymptotic to a(n) ~ s^2 / (n^(3/2) * r^(n-1/2) * sqrt(2*Pi*p*(s-1)*(1+s/(1+p*(s-1))))), where r and s are real roots (0 < r < 1) of the system of equations r = p*(s-1)^2 / (s*(1-p+p*s)), (r*s)^p = (s-1-r*s)/(s-1). - Vaclav Kotesovec, Jul 16 2014

			A(0,k) = 1: the empty path.
A(3,0) = 1: UDUDUD.
A(3,1) = 5: UDUDUD, UDUUDD, UUDDUD, UUDUDD, UUUDDD.
A(3,2) = 2: UDUDUD, UUUDDD.
A(5,3) = 6: UDUDUDUDUD, UDUUUUDDDD, UUUUDDDDUD, UUUUDDDUDD, UUUUDDUDDD, UUUUDUDDDD.
Square array A(n,k) begins:
  1,   1,  1,  1,  1,  1,  1,  1, ...
  1,   1,  1,  1,  1,  1,  1,  1, ...
  1,   2,  1,  1,  1,  1,  1,  1, ...
  1,   5,  2,  1,  1,  1,  1,  1, ...
  1,  14,  5,  2,  1,  1,  1,  1, ...
  1,  42, 12,  6,  2,  1,  1,  1, ...
  1, 132, 30, 16,  7,  2,  1,  1, ...
  1, 429, 79, 37, 22,  8,  2,  1, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Vaclav Kotesovec, Asymptotic of subsequences of A212382

Columns k=0-10 give: A000012, A000108, A101785, A212383, A212384, A212385, A212386, A212387, A212388, A212389, A212390.

A(2n,n) gives A323229.

b:= proc(x, y, k, u) option remember;
      `if`(x<0 or y `if`(k=0, 1, b(n, n, k, true)):
seq(seq(A(n, d-n), n=0..d), d=0..15);
# second Maple program
A:= (n, k)-> `if`(k=0, 1, coeff(series(RootOf(
               A||k=1+x*A||k/(1-(x*A||k)^k), A||k), x, n+1), x, n)):
seq(seq(A(n, d-n), n=0..d), d=0..15);

b[x_, y_, k_, u_] := b[x, y, k, u] = If[x<0 || yJean-François Alcover, Jan 15 2014, translated from first Maple program *)

G.f. of column k>0 satisfies: A_k(x) = 1+x*A_k(x)/(1-(x*A_k(x))^k), g.f. of column k=0: A_0(x) = 1/(1-x).

G.f. of column k>0 is series_reversion(B(x))/x where B(x) = x/(1 + x + x^(k+1) + x^(2*k+1) + x^(3*k+1) + ... ) = x/(1+x/(1-x^k)); for Dyck paths with allowed ascent lengths {u_1, u_2, ...} use B(x) = x/( 1 + sum(k>=1, x^{u_k} ) ). - Joerg Arndt, Apr 23 2016

A365244 G.f. satisfies A(x) = 1 + xA(x)/(1 - x^2A(x)^3).

Original entry on oeis.org

1, 1, 1, 2, 6, 17, 48, 144, 449, 1422, 4568, 14893, 49139, 163665, 549570, 1858754, 6326343, 21651064, 74462327, 257219221, 892047965, 3104749126, 10841192392, 37967942203, 133333407639, 469405472729, 1656383420850, 5857371543403, 20754268304707

Offset: 0

Seiichi Manyama, Aug 28 2023

nonn

Cf. A212383, A365245.

A365244 := proc(n)
    add( binomial(n-k-1,k)*binomial(n+k+1,n-2*k)/(n+k+1),k=0..floor(n/2)) ;
end proc:
seq(A365244(n),n=0..80); # R. J. Mathar, Aug 29 2023

nmax = 28; A[_] = 1;
Do[A[x_] = 1 + x*A[x]/(1 - x^2*A[x]^3) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 25 2023 *)

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

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

D-finite with recurrence -9*n*(3*n-5) *(3*n+2) *(15657757169*n -38967750523)*a(n) +3*(1246945698477*n^4 -4744568003544*n^3 +3294337649527*n^2 +2214578323972*n -1078893934272) *a(n-1) +6*(98125454565*n^4 -4049050969593*n^3 +21710764341344*n^2 -39026642938410*n +22772957131188) *a(n-2) +6*(1426531749264*n^4 -6603349282173*n^3 -4098111856085*n^2 +51689999346882*n -56245738276010) *a(n-3) +6*(2322713957130*n^4 -32736762801117*n^3 +166244031312630*n^2 -356896536324983*n +268070043432100) *a(n-4) -6*(n-5) *(2*n-9) *(613164767527*n^2 -4657829502565*n +8148618486058) *a(n-5) +2*(n-6) *(2*n-11) *(271184324539*n^2 -2272760427224*n +4256723647917) *a(n-6) -4*(6162243349*n -17166617798) *(2*n-13)*(n-6) *(n-7)*a(n-7)=0. - R. J. Mathar, Aug 29 2023

A365245 G.f. satisfies A(x) = 1 + xA(x)/(1 - x^4A(x)^3).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 6, 16, 36, 72, 139, 283, 631, 1487, 3510, 8086, 18240, 41004, 93364, 216370, 507353, 1193113, 2799681, 6556243, 15368798, 36163695, 85483537, 202768647, 481870474, 1146143965, 2728316757, 6502751833, 15525113876, 37131739582

Offset: 0

Seiichi Manyama, Aug 28 2023

nonn

Cf. A212383, A365244.

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

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

A365757 G.f. satisfies A(x) = 1 + xA(x) / (1 - x^3A(x)^5).

Original entry on oeis.org

1, 1, 1, 1, 2, 8, 29, 86, 229, 619, 1836, 5846, 18802, 59356, 185187, 581476, 1855412, 5997965, 19491730, 63395718, 206433172, 674452128, 2213463944, 7293253791, 24098638133, 79791002807, 264698873350, 879945619711, 2931486913728, 9785457123420, 32721317536787

Offset: 0

Seiichi Manyama, Sep 18 2023

nonn

Cf. A023432, A212383, A365243, A365756.

Cf. A365695, A365759.

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

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

A365756 G.f. satisfies A(x) = 1 + xA(x) / (1 - x^3A(x)^4).

Original entry on oeis.org

1, 1, 1, 1, 2, 7, 22, 58, 142, 363, 1014, 2966, 8645, 24824, 71189, 206742, 609159, 1809493, 5388804, 16073002, 48092377, 144532884, 436168716, 1320372837, 4006489208, 12183544414, 37132838866, 113426618425, 347191793705, 1064688271730, 3270387354434

Offset: 0

Seiichi Manyama, Sep 18 2023

nonn

Cf. A023432, A212383, A365243, A365757.

Cf. A063019.

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

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

cp's OEIS Frontend

A212382 Number A(n,k) of Dyck n-paths all of whose ascents have lengths equal to 1+k*m (m>=0); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A365244 G.f. satisfies A(x) = 1 + x*A(x)/(1 - x^2*A(x)^3).

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A365245 G.f. satisfies A(x) = 1 + x*A(x)/(1 - x^4*A(x)^3).

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A365757 G.f. satisfies A(x) = 1 + x*A(x) / (1 - x^3*A(x)^5).

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A365756 G.f. satisfies A(x) = 1 + x*A(x) / (1 - x^3*A(x)^4).

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A365244 G.f. satisfies A(x) = 1 + xA(x)/(1 - x^2A(x)^3).

A365245 G.f. satisfies A(x) = 1 + xA(x)/(1 - x^4A(x)^3).

A365757 G.f. satisfies A(x) = 1 + xA(x) / (1 - x^3A(x)^5).

A365756 G.f. satisfies A(x) = 1 + xA(x) / (1 - x^3A(x)^4).