A212364 -id:A212364 - cp's OEIS frontend

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.

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

A365699 G.f. satisfies A(x) = 1 + x^5A(x)^2 / (1 - xA(x)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 3, 6, 10, 15, 21, 33, 57, 101, 175, 291, 477, 791, 1341, 2310, 3986, 6839, 11681, 19966, 34300, 59245, 102647, 177963, 308483, 534973, 929147, 1616981, 2818967, 4920299, 8594665, 15023561, 26283971, 46030771, 80695333, 141593087

Offset: 0

Seiichi Manyama, Sep 16 2023

nonn

Cf. A212364, A365698, A365700, A365701, A365702.

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

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

A365700 G.f. satisfies A(x) = 1 + x^5A(x)^3 / (1 - xA(x)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 4, 8, 13, 19, 26, 46, 88, 163, 284, 466, 781, 1369, 2468, 4449, 7856, 13724, 24084, 42788, 76759, 137785, 246418, 439757, 786132, 1411148, 2541368, 4581906, 8259500, 14889781, 26871106, 48573823, 87934175, 159333544, 288857216

Offset: 0

Seiichi Manyama, Sep 16 2023

nonn

Cf. A212364, A365698, A365699, A365701, A365702.

terms = 43; A[] = 0; Do[A[x] = 1 + x^5*A[x]^3 / (1 - x*A[x])+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, May 29 2025 *)

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

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

a(n) ~ s*sqrt((1 - r*s)*(5 - 4*r*s)/(Pi*(3 - r*s*(3 - r*s)))) / (2*n^(3/2)*r^n), where r = 0.53247307479161512230023149440436598140650951738583 and s = 1.2504652351088857309836364363044636883260447207988... are roots of the system of equations r^5*s^3 = (s-1)*(1 - r*s), (s-1)*(3 - 2*r*s) = s*(1 - r*s). - Vaclav Kotesovec, May 29 2025

A365701 G.f. satisfies A(x) = 1 + x^5A(x)^4 / (1 - xA(x)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 5, 10, 16, 23, 31, 62, 128, 243, 423, 686, 1192, 2223, 4223, 7843, 13991, 24856, 45108, 83673, 156223, 288535, 527971, 966803, 1784663, 3319988, 6183424, 11483613, 21284475, 39499855, 73558147, 137347615, 256616567, 479231240

Offset: 0

Seiichi Manyama, Sep 16 2023

nonn

Cf. A212364, A365698, A365699, A365700, A365702.

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

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

A365702 G.f. satisfies A(x) = 1 + x^5A(x)^5 / (1 - xA(x)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 6, 12, 19, 27, 36, 81, 177, 341, 592, 951, 1726, 3417, 6766, 12812, 22951, 41531, 78222, 151291, 291957, 550015, 1024683, 1924543, 3671017, 7063893, 13532120, 25730347, 48840523, 93154161, 178806493, 343926597, 660308308, 1265195467

Offset: 0

Seiichi Manyama, Sep 16 2023

nonn

Cf. A212364, A365698, A365699, A365700, A365701.

Cf. A000108, A005043, A114997, A215341.

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

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

A365698 G.f. satisfies A(x) = 1 + x^5 / (1 - x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 16, 22, 31, 47, 76, 126, 207, 331, 517, 801, 1251, 1987, 3206, 5212, 8465, 13677, 21997, 35341, 56937, 92169, 149860, 244274, 398383, 649379, 1058055, 1724575, 2814475, 4600923, 7533150, 12347908, 20252837, 33230545

Offset: 0

Seiichi Manyama, Sep 16 2023

nonn

Cf. A212364, A365699, A365700, A365701, A365702.

Cf. A001006, A023426, A025246, A357308.

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

G.f.: A(x) = 2*(1+x^5) / (1+x+sqrt( (1+x)^2 - 4*x*(1+x^5) )).

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

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

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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A365699 G.f. satisfies A(x) = 1 + x^5*A(x)^2 / (1 - x*A(x)).

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A365700 G.f. satisfies A(x) = 1 + x^5*A(x)^3 / (1 - x*A(x)).

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A365701 G.f. satisfies A(x) = 1 + x^5*A(x)^4 / (1 - x*A(x)).

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A365702 G.f. satisfies A(x) = 1 + x^5*A(x)^5 / (1 - x*A(x)).

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A365698 G.f. satisfies A(x) = 1 + x^5 / (1 - x*A(x)).

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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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A365699 G.f. satisfies A(x) = 1 + x^5A(x)^2 / (1 - xA(x)).

A365700 G.f. satisfies A(x) = 1 + x^5A(x)^3 / (1 - xA(x)).

A365701 G.f. satisfies A(x) = 1 + x^5A(x)^4 / (1 - xA(x)).

A365702 G.f. satisfies A(x) = 1 + x^5A(x)^5 / (1 - xA(x)).

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).