A212385 -id:A212385 - 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

A226974 a(n) = Sum_{k=0..floor(n/3)} binomial(n,3k)binomial(4k,k)/(3k+1).

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 25, 64, 169, 443, 1181, 3224, 8897, 24701, 69161, 195255, 554577, 1583109, 4541461, 13086574, 37856437, 109892403, 320034309, 934774902, 2737689189, 8037746691, 23652564261, 69749727716, 206091735797, 610061655665, 1808962146529

Offset: 0

Karol A. Penson, Jun 25 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A049130, A212385, A226910, A227035.

A226974 := proc(n)
    hypergeom([-n/3,-n/3+2/3,-n/3+1/3,1/4,1/2,3/4],[1/3,2/3,2/3,1,4/3],-256/27) ;
    simplify(%) ;
end proc:
seq(A226974(n),n=0..40) ; # R. J. Mathar, Jan 10 2023

Table[Sum[Binomial[n,3*k]*Binomial[4*k,k]/(3*k+1), {k,0,Floor[n/3]}],{n,0,20}] (* Vaclav Kotesovec, Jun 28 2013 *)

a(n):=if n<0 then 0 else if n=0 then 1 else sum(sum(sum(a(l)*a(i)*a(j)*a(n-i-j-l-3),l,0,n-3-i-j),j,0,n-3-i),i,0,n-3)+1; /* Vladimir Kruchinin, May 17 2020 */

a(n) = sum(k=0, n\3, binomial(n,3*k)*binomial(4*k,k)/(3*k+1)); \\ Michel Marcus, Sep 16 2021

a(n) = Sum_{k=0..floor(n/3)} binomial(n,3*k)*A002293(k).
Representation in terms of special values of hypergeometric function of type 6F5: a(n) = hypergeom([1/4, 1/2, 3/4, -(1/3)*n, -(1/3)*n+2/3, -(1/3)*n+1/3], [1/3, 2/3, 2/3, 1, 4/3],-4^4/3^3), n>=0.
Recurrence: 27*n*(n+1)*(n-1)*a(n) = 162*n*(n-1)^2*a(n-1) - 81*(5*n^2-15*n+12)*(n-1)*a(n-2) + 4*(199*n^3 - 1098*n^2 + 2043*n - 1296)*a(n-3) - (n-3)*(1173*n^2 - 5097*n + 5584)*a(n-4) + 6*(n-4)*(n-3)*(155*n-401)*a(n-5) - 283*(n-5)*(n-4)*(n-3)*a(n-6). - Vaclav Kotesovec, Jun 28 2013
a(n) ~ (3+4^(1+1/3))^(n+3/2)/(8*3^(n+1)*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 28 2013
G.f. satisfies A(x)=1+x^3*A(x)^4+x/(1-x). - Vladimir Kruchinin, May 17 2020
From Peter Bala, Sep 15 2021: (Start)
O.g.f.: A(x) = (1/x)*series reversion( x*(1 - x^3)/(1 + x*(1 - x^3)) ).
The g.f. of the m-th binomial transform of this sequence is equal to (1/x)*series reversion( x*(1 - x^3)/(1 + (m + 1)*x*(1 - x^3)) ). The case m = -1 gives the sequence [1,0,0,1,0,0,4,0,0,22,0,0,140,...] - an aerated version of A002293. (End)

A226910 a(n) = Sum_{k=0..floor(n/5)} binomial(n,5k)binomial(6k,k)/(5k+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 259, 529, 1189, 3004, 8009, 21073, 53233, 129813, 312733, 763573, 1915251, 4914736, 12720841, 32800186, 83869501, 213261712, 542609237, 1388542312, 3579043987, 9273567337, 24075321925, 62475528190, 161969731985, 419914766965

Offset: 0

Karol A. Penson, Jun 22 2013

Alois P. Heinz, Table of n, a(n) for n = 0..750

Cf. A002295, A049130, A212385, A227035.

Table[Sum[Binomial[n,5*k]*Binomial[6*k,k]/(5*k+1),{k,0,Floor[n/5]}],{n,0,20}] (* Vaclav Kotesovec, Jun 28 2013 *)

a(n)=sum(k=0,n\5,binomial(n,5*k)*binomial(6*k,k)/(5*k+1)) \\ Charles R Greathouse IV, Jun 24 2013

Representation in terms of special values of generalized hypergeometric function of type 10F9: a(n) = hypergeom([1/6, 1/3, 1/2, 2/3, 5/6, -(1/5)*n, -(1/5)*n+4/5, -(1/5)*n+3/5, -(1/5)*n+2/5, 1/5-(1/5)*n], [1/5, 2/5, 2/5, 3/5, 3/5, 4/5, 4/5, 1, 6/5], -6^6/5^5), n>=0.
Recurrence: -49781*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*a(n-10) + 10*(n-8)*(n-7)*(n-6)*(n-5)*(26453*n - 123726)*a(n-9) - 15*(n-7)*(n-6)*(n-5)*(40479*n^2 - 351957*n + 782140)*a(n-8) + 120*(n-6)*(n-5)*(7013*n^3 - 87699*n^2 + 378278*n - 565577)*a(n-7) - 6*(n-5)*(148255*n^4 - 2435310*n^3 + 15491085*n^2 - 45173430*n + 50791476)*a(n-6) + 12*(69513*n^5 - 1361100*n^4 + 10838875*n^3 - 43818750*n^2 + 89776250*n - 74437500)*a(n-5) - 93750*(7*n^4 - 98*n^3 + 525*n^2 - 1274*n + 1180)*(n-3)*a(n-4) + 375000*(n-2)*(n^2-6*n+10)*(n-3)^2*a(n-3) - 46875*(n-2)*(n-1)*(3*n^2-15*n+20)*(n-3)*a(n-2) + 31250*(n-2)^2*(n-1)*n*(n-3)*a(n-1) - 3125*(n-2)*(n-1)*n*(n+1)*(n-3)*a(n) = 0. - Vaclav Kotesovec, Jun 28 2013
a(n) ~ (5+6^(1+1/5))^(n+3/2)/(5^(n+1)*6^(1+3/10)*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 28 2013
G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^5 * A(x)^6. - Ilya Gutkovskiy, Jul 25 2021
From Peter Bala, Sep 15 2021: (Start)
O.g.f.: A(x) = (1/x)*series reversion ( x*(1 - x^5)/(1 + x*(1 - x^5)) ).
The g.f. of the m-th binomial transform of this sequence is equal to (1/x)*series reversion ( x*(1 - x^5)/(1 + (m + 1)*x*(1 - x^5)) ). The case m = -1 gives the sequence [1, 0, 0, 0, 0, 1, 0, 0,0, 0, 6, 0, 0, 0, 0, 51, 0, 0, 0, 0, 506, ...] - an aerated version of A002295. (End)

A365734 G.f. satisfies A(x) = 1 + xA(x) / (1 - x^5A(x)^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 5, 11, 21, 36, 58, 94, 163, 306, 599, 1170, 2229, 4140, 7596, 14002, 26228, 49979, 96212, 185491, 356255, 681247, 1300680, 2488500, 4782037, 9231306, 17875306, 34656389, 67194497, 130263382, 252631688, 490513867, 953923030, 1858136173, 3624102244

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Cf. A212364, A212385, A365735, A365736.

Cf. A365732, A365699.

a(n) = sum(k=0, n\5, binomial(n-4*k-1, k)*binomial(n-3*k+1, n-5*k)/(n-3*k+1));

a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k-1,k) * binomial(n-3*k+1,n-5*k) / (n-3*k+1).

A365735 G.f. satisfies A(x) = 1 + xA(x) / (1 - x^5A(x)^3).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 6, 16, 36, 71, 128, 223, 403, 796, 1706, 3775, 8252, 17485, 35986, 72988, 148594, 307833, 650947, 1395846, 3004732, 6443836, 13732127, 29134320, 61792707, 131525272, 281463507, 605273669, 1305373379, 2817407854, 6077804871, 13103021422

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Cf. A212364, A212385, A365734, A365736.

Cf. A365733, A365700.

a(n) = sum(k=0, n\5, binomial(n-4*k-1, k)*binomial(n-2*k+1, n-5*k)/(n-2*k+1));

a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k-1,k) * binomial(n-2*k+1,n-5*k) / (n-2*k+1).

A365736 G.f. satisfies A(x) = 1 + xA(x) / (1 - x^5A(x)^4).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 254, 478, 903, 1838, 4148, 10012, 24417, 58019, 132919, 295699, 649742, 1437719, 3247500, 7504925, 17607055, 41465646, 97197400, 226053017, 522505492, 1205674911, 2790322418, 6495170018, 15209566913, 35761582618

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Cf. A212364, A212385, A365734, A365735.

Cf. A365701.

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

a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k-1,k) * binomial(n-k+1,n-5*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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A226974 a(n) = Sum_{k=0..floor(n/3)} binomial(n,3*k)*binomial(4*k,k)/(3*k+1).

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A226910 a(n) = Sum_{k=0..floor(n/5)} binomial(n,5*k)*binomial(6*k,k)/(5*k+1).

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A365734 G.f. satisfies A(x) = 1 + x*A(x) / (1 - x^5*A(x)^2).

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A365735 G.f. satisfies A(x) = 1 + x*A(x) / (1 - x^5*A(x)^3).

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A365736 G.f. satisfies A(x) = 1 + x*A(x) / (1 - x^5*A(x)^4).

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A226974 a(n) = Sum_{k=0..floor(n/3)} binomial(n,3k)binomial(4k,k)/(3k+1).

A226910 a(n) = Sum_{k=0..floor(n/5)} binomial(n,5k)binomial(6k,k)/(5k+1).

A365734 G.f. satisfies A(x) = 1 + xA(x) / (1 - x^5A(x)^2).

A365735 G.f. satisfies A(x) = 1 + xA(x) / (1 - x^5A(x)^3).

A365736 G.f. satisfies A(x) = 1 + xA(x) / (1 - x^5A(x)^4).