A365728 -id:A365728 - cp's OEIS frontend

A365730 G.f. satisfies A(x) = 1 + x^4A(x)^4(1 + x*A(x)).

1, 0, 0, 0, 1, 1, 0, 0, 4, 9, 5, 0, 22, 78, 91, 35, 140, 680, 1224, 969, 1254, 5985, 14630, 17710, 17710, 55660, 164450, 269100, 299520, 593775, 1805076, 3681405, 4951692, 7594752, 20173560, 47303520, 76404460, 110676324, 239784864, 589602585, 1106339923

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Cf. A365727, A365728, A365729, A365731.
Cf. A001005, A001006, A365724.
Cf. A215341.

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

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

A365731 G.f. satisfies A(x) = 1 + x^4A(x)^5(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 5, 11, 6, 0, 35, 120, 136, 51, 285, 1330, 2310, 1771, 3036, 14950, 35100, 40950, 47502, 175392, 503440, 791120, 927520, 2272424, 7037184, 13803405, 18643560, 33997080, 98920536, 226318196, 359255325, 578590155, 1445166360, 3584815443, 6573439928

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Index entries for reversions of series

Cf. A365727, A365728, A365729, A365730.
Cf. A001002, A217358, A365725.
Cf. A215342.

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

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

G.f.: (1/x) * Series_Reversion( x*(1 - x^4*(1 + x)) ). - Seiichi Manyama, Sep 24 2024

A365727 G.f. satisfies A(x) = 1 + x^4A(x)(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 1, 3, 2, 0, 1, 6, 10, 5, 1, 10, 30, 35, 15, 15, 70, 140, 127, 63, 140, 420, 631, 490, 384, 1050, 2311, 2808, 2136, 2739, 6931, 12057, 12672, 11055, 19449, 42097, 61050, 60060, 66353, 131054, 241670, 306735, 308881, 428792, 835614, 1337765

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Cf. A365728, A365729, A365730, A365731.

Cf. A023427.

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

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

A365729 G.f. satisfies A(x) = 1 + x^4A(x)^3(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 3, 7, 4, 0, 12, 45, 55, 22, 55, 286, 546, 455, 413, 1820, 4760, 6120, 5304, 12597, 38760, 67830, 73587, 108262, 309925, 672980, 928763, 1171390, 2598310, 6270030, 10607025, 14033565, 24293880, 57256875, 112293597, 167416275, 255805056

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Cf. A365727, A365728, A365730, A365731.

Cf. A365697.

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

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

cp's OEIS Frontend

A365730 G.f. satisfies A(x) = 1 + x^4A(x)^4(1 + x*A(x)).

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A365731 G.f. satisfies A(x) = 1 + x^4A(x)^5(1 + x*A(x)).

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A365727 G.f. satisfies A(x) = 1 + x^4A(x)(1 + x*A(x)).

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A365729 G.f. satisfies A(x) = 1 + x^4A(x)^3(1 + x*A(x)).

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cp's OEIS Frontend

A365730 G.f. satisfies A(x) = 1 + x^4*A(x)^4*(1 + x*A(x)).

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A365731 G.f. satisfies A(x) = 1 + x^4*A(x)^5*(1 + x*A(x)).

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A365727 G.f. satisfies A(x) = 1 + x^4*A(x)*(1 + x*A(x)).

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A365729 G.f. satisfies A(x) = 1 + x^4*A(x)^3*(1 + x*A(x)).

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A365730 G.f. satisfies A(x) = 1 + x^4A(x)^4(1 + x*A(x)).

A365731 G.f. satisfies A(x) = 1 + x^4A(x)^5(1 + x*A(x)).

A365727 G.f. satisfies A(x) = 1 + x^4A(x)(1 + x*A(x)).

A365729 G.f. satisfies A(x) = 1 + x^4A(x)^3(1 + x*A(x)).