A365702 -id:A365702 - 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

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).

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).

A366057 Expansion of (1/x) * Series_Reversion( x/(1-x+x^5) ).

Original entry on oeis.org

1, -1, 1, -1, 1, 0, -5, 20, -55, 125, -246, 406, -461, -144, 3004, -11978, 35113, -86293, 181663, -314603, 365922, 150023, -2696308, 10969573, -32970453, 82976409, -178372934, 314133884, -367436684, -179661091, 2923282216, -11972239216, 36369188841, -92517132841

Offset: 0

Seiichi Manyama, Sep 27 2023

sign

Cf. A364522, A365702, A366051.

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

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

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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A365701 G.f. satisfies A(x) = 1 + x^5*A(x)^4 / (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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A366057 Expansion of (1/x) * Series_Reversion( x/(1-x+x^5) ).

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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)).