A365701 -id:A365701 - cp's OEIS frontend

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

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

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).

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

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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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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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)).

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

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