A023432 -id:A023432 - cp's OEIS frontend

A212364 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 5).

1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 16, 23, 35, 57, 96, 161, 264, 425, 682, 1106, 1821, 3030, 5055, 8412, 13956, 23145, 38487, 64261, 107673, 180762, 303651, 510187, 857692, 1443597, 2433495, 4108299, 6943862, 11746362, 19883655, 33681015, 57096874, 96874214

Offset: 0

Alois P. Heinz, May 10 2012

nonn

			a(0) = 1: the empty path.
a(1) = 1: UD.
a(5) = 1: UDUDUDUDUD.
a(6) = 2: UDUDUDUDUDUD, UUUUUUDDDDDD.
a(7) = 4: UDUDUDUDUDUDUD, UDUUUUUUDDDDDD, UUUUUUDDDDDDUD, UUUUUUDUDDDDDD.
a(8) = 7: UDUDUDUDUDUDUDUD, UDUDUUUUUUDDDDDD, UDUUUUUUDDDDDDUD, UDUUUUUUDUDDDDDD, UUUUUUDDDDDDUDUD, UUUUUUDUDDDDDDUD, UUUUUUDUDUDDDDDD.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=5 of A212363.

Cf. A023432 (m=3), A023427 (m=4), this sequence (m=5), A212386(m=6).

a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1) +add(a(k)*a(n-5-k), k=1..n-5))
    end:
seq(a(n), n=0..50);
# second Maple program:
a:= n-> coeff(series(RootOf(A=1+A*(x-x^5*(1-A)), A), x, n+1), x, n):
seq(a(n), n=0..50);

CoefficientList[Series[(1-x+x^5-Sqrt[-4*x^5+(1-x+x^5)^2])/(2*x^5),{x,0,20}],x] (* Vaclav Kotesovec, Mar 20 2014 *)

G.f. satisfies: A(x) = 1+A(x)*(x-x^5*(1-A(x))).
a(n) = a(n-1) + Sum_{k=1..n-5} a(k)*a(n-5-k) if n>0; a(0) = 1.
Recurrence: (n+5)*a(n) = (2*n+7)*a(n-1) - (n+2)*a(n-2) + (2*n-5)*a(n-5) + 2*(n-4)*a(n-6) - (n-10)*a(n-10). - Vaclav Kotesovec, Mar 20 2014
a(n) = Sum_{k=0..(n-1)/4} C(n-4*k,k)*C(n-4*k,k+1)/(n-4*k) for n>0, a(0)=1. - Vladimir Kruchinin, Jan 21 2019

A212363 Number A(n,k) of Dyck n-paths all of whose ascents and descents have lengths equal to 1+k*m (m>=0); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

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, 4, 42, 1, 1, 1, 1, 1, 2, 8, 132, 1, 1, 1, 1, 1, 1, 4, 17, 429, 1, 1, 1, 1, 1, 1, 2, 7, 37, 1430, 1, 1, 1, 1, 1, 1, 1, 4, 12, 82, 4862, 1, 1, 1, 1, 1, 1, 1, 2, 7, 22, 185, 16796, 1

Offset: 0

Alois P. Heinz, May 10 2012

			A(3,0) = 1: UDUDUD.
A(3,1) = 5: UDUDUD, UDUUDD, UUDDUD, UUDUDD, UUUDDD.
A(4,2) = 4: UDUDUDUD, UDUUUDDD, UUUDDDUD, UUUDUDDD.
A(5,2) = 8: UDUDUDUDUD, UDUDUUUDDD, UDUUUDDDUD, UDUUUDUDDD, UUUDDDUDUD, UUUDUDDDUD, UUUDUDUDDD, UUUUUDDDDD.
A(5,3) = 4: UDUDUDUDUD, UDUUUUDDDD, UUUUDDDDUD, 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,  4,  2,  1,  1,  1,  1, ...
  1,  42,  8,  4,  2,  1,  1,  1, ...
  1, 132, 17,  7,  4,  2,  1,  1, ...
  1, 429, 37, 12,  7,  4,  2,  1, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000012, A000108, A004148, A023432, A023427, A212364, A212365, A212366, A212367, A212368, A212369.

A:= proc(n, k) option remember;
      `if`(k=0, 1, `if`(n=0, 1, A(n-1, k)
                   +add(A(j, k)*A(n-k-j, k), j=1..n-k)))
    end:
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+A||k*(x-x^k*(1-A||k)), A||k), x, n+1), x, n)):
seq(seq(A(n, d-n), n=0..d), d=0..15);

A[n_, k_] := A[n, k] = If[k == 0, 1, If[n == 0, 1, A[n-1, k] + Sum[A[j, k]*A[n-k-j, k], {j, 1, n-k}]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from first Maple program *)

G.f. of column k>0 satisfies: A_k(x) = 1+A_k(x)*(x-x^k*(1-A_k(x))), g.f. of column k=0: A_0(x) = 1/(1-x).
A(n,k) = A(n-1,k) + Sum_{j=1..n-k} A(j,k)*A(n-k-j,k) for n,k>0; A(n,0) = A(0,k) = 1.
G.f. of column k > 0: (1 - x + x^k - sqrt((1 - x + x^k)^2 - 4*x^k)) / (2*x^k). - Vaclav Kotesovec, Sep 02 2014

A365695 G.f. satisfies A(x) = 1 + x^3A(x)^5 / (1 - xA(x)).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 6, 12, 19, 62, 156, 318, 852, 2254, 5262, 13441, 35543, 88772, 226880, 596937, 1539188, 3980364, 10468270, 27410289, 71702956, 189169352, 499529048, 1318355542, 3493861461, 9278408639, 24647900618, 65620808508, 175037591303, 467277998136

Offset: 0

Seiichi Manyama, Sep 16 2023

nonn

Cf. A023426, A023432, A054514, A114997, A365694.

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

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

A365694 G.f. satisfies A(x) = 1 + x^3A(x)^2 / (1 - xA(x)).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 42, 84, 170, 354, 740, 1549, 3269, 6945, 14811, 31711, 68177, 147091, 318313, 690837, 1503351, 3279445, 7169907, 15708485, 34482475, 75830981, 167042763, 368548926, 814341362, 1801867812, 3992172298, 8855912464, 19668236110

Offset: 0

Seiichi Manyama, Sep 16 2023

nonn

Cf. A023426, A023432, A054514, A114997, A365695.

Cf. A365243.

CoefficientList[Series[2/(1 + x + Sqrt[1 + x*(-2 + x - 4*x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 16 2023 *)

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

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

G.f.: A(x) = 2/(1 + x + sqrt(1 + x*(-2 + x - 4*x^2))). - Vaclav Kotesovec, Sep 16 2023

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).

A216116 G.f. satisfies: A(x) = (1 + xA(x)) (1 + x^4*A(x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 7, 11, 17, 28, 49, 87, 152, 262, 453, 794, 1408, 2507, 4462, 7943, 14179, 25415, 45713, 82398, 148731, 268859, 486890, 883411, 1605582, 2922259, 5325377, 9716564, 17750332, 32464980, 59443403, 108951953, 199886003, 367052947, 674620772, 1240963218

Offset: 0

Paul D. Hanna, Oct 29 2012

nonn

			A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 11*x^7 + 17*x^8 + 28*x^9 +...
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x^3)*x +
(1 + 2^2*x^3 + x^6)*x^2/2 +
(1 + 3^2*x^3 + 3^2*x^6 + x^9)*x^3/3 +
(1 + 4^2*x^3 + 6^2*x^6 + 4^2*x^9 + x^12)*x^4/4 +
(1 + 5^2*x^3 + 10^2*x^6 + 10^2*x^9 + 5^2*x^12 + x^15)*x^5/5 +
(1 + 6^2*x^3 + 15^2*x^6 + 20^2*x^9 + 15^2*x^12 + 6^2*x^15 + x^18)*x^6/6 +...

Cf. A023427, A023432, A004148.

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1+x*A)*(1+x^4*A)+x*O(x^n)); polcoeff(A, n)}

{a(n)=polcoeff(exp(sum(m=1, n+1, sum(j=0, m, binomial(m, j)^2*(x+x*O(x^n))^(3*j))*x^m/m)), n)}

{a(n)=polcoeff(((1-x-x^4) - sqrt((1-x-x^4)^2 - 4*x^5 +x^6*O(x^n)))/(2*x^5), n)}
for(n=0,45,print1(a(n),", "))

G.f.: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * x^(3*k) ).
G.f.: A(x) = ((1-x-x^4) - sqrt((1-x-x^4)^2 - 4*x^5))/(2*x^5).
a(n) = A023427(n+1) for n>=0.

A275448 The number of weakly alternating bargraphs of semiperimeter n. A bargraph is said to be weakly alternating if its ascents and descents alternate. An ascent (descent) is a maximal sequence of consecutive U (D) steps.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 28, 65, 146, 327, 749, 1756, 4165, 9913, 23652, 56687, 136627, 330969, 804915, 1963830, 4805523, 11793046, 29019930, 71589861, 177006752, 438561959, 1088714711, 2707615555, 6745272783, 16830750107, 42058592797, 105248042792

Offset: 2

Emeric Deutsch, Sergi Elizalde, Aug 26 2016

nonn

			a(4)=3 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and the corresponding drawings show that only [1,1,1],[2,2], and [3] lead to weakly alternating bargraphs.

M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Cf. A082582, A023432.

g := ((1-3*z+3*z^2-sqrt((1-3*z+z^2)*(1-3*z+5*z^2-4*z^3)))*(1/2))/(z*(1-z)): gser:= series(g,z=0,43): seq(coeff(gser,z,n), n=2..40);

terms = 32;
g[z_] = ((1 - 3z + 3z^2 - Sqrt[(1 - 3z + z^2)(1 - 3z + 5z^2 - 4z^3)])*(1/2) )/(z(1-z));
Drop[CoefficientList[g[z] + O[z]^(terms+2), z], 2] (* Jean-François Alcover, Aug 07 2018 *)

G.f.: g(z) = (1-3z+3z^2 - Q)/(2z(1-z)), where Q = sqrt((1-3z+z^2)(1-3z+5z^2-4z^3)).

D-finite with recurrence (n+1)*a(n) +(-7*n+2)*a(n-1) +3*(7*n-11)*a(n-2) +(-37*n+107)*a(n-3) +3*(13*n-54)*a(n-4) +3*(-7*n+37)*a(n-5) +2*(2*n-13)*a(n-6)=0. - R. J. Mathar, Jul 22 2022

A296201 Expansion of 1/(1 - x/(1 - x/(1 - x^2/(1 - x/(1 - x^3/(1 - x/(1 - x^4/(1 - ...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 50, 120, 290, 704, 1714, 4181, 10212, 24965, 61070, 149458, 365888, 895932, 2194178, 5374262, 13164426, 32248616, 79002180, 193544446, 474168003, 1161691893, 2846131055, 6973047572, 17084140245, 41856763371, 102550935614, 251254982356, 615588531011, 1508227753087, 3695249380509

Offset: 0

Ilya Gutkovskiy, Dec 07 2017

nonn

Cf. A004148, A005169, A023432, A088354, A088355, A296202.

nmax = 34; CoefficientList[Series[1/(1 + ContinuedFractionK[-x^(1 + k (1 + (-1)^k)/4), 1, {k, 0, nmax}]), {x, 0, nmax}], x]

a(n) ~ c * d^n, where d = 2.450066970712861209761227155593662591019701927336233634485900133440192... and c = 0.21656595617747023258115906735909123622190252865232858964820650877171... - Vaclav Kotesovec, Sep 18 2021

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

A212364 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 5).

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A212363 Number A(n,k) of Dyck n-paths all of whose ascents and descents 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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A365695 G.f. satisfies A(x) = 1 + x^3*A(x)^5 / (1 - x*A(x)).

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A365694 G.f. satisfies A(x) = 1 + x^3*A(x)^2 / (1 - x*A(x)).

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

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A216116 G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x^4*A(x)).

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A275448 The number of weakly alternating bargraphs of semiperimeter n. A bargraph is said to be weakly alternating if its ascents and descents alternate. An ascent (descent) is a maximal sequence of consecutive U (D) steps.

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Mathematica

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A296201 Expansion of 1/(1 - x/(1 - x/(1 - x^2/(1 - x/(1 - x^3/(1 - x/(1 - x^4/(1 - ...)))))))), a continued fraction.

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

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A365695 G.f. satisfies A(x) = 1 + x^3A(x)^5 / (1 - xA(x)).

A365694 G.f. satisfies A(x) = 1 + x^3A(x)^2 / (1 - xA(x)).

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

A216116 G.f. satisfies: A(x) = (1 + xA(x)) (1 + x^4*A(x)).

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