A365724 -id:A365724 - 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).

A365725 G.f. satisfies A(x) = 1 + x^3A(x)^4(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 4, 9, 5, 22, 78, 91, 175, 680, 1224, 1938, 6270, 14630, 24794, 63756, 166980, 322920, 720720, 1900080, 4125888, 8803008, 22151360, 51778804, 111882100, 267682272, 645736432, 1442390092, 3346519020, 8094247798, 18657762006, 42890295734

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Index entries for reversions of series

Cf. A308616, A365723, A365724, A365726.
Cf. A001002, A217358, A365731.
Cf. A054514.

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

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

G.f.: (1/x) * Series_Reversion( x*(1 - x^3*(1 + x)) ). - Seiichi Manyama, Sep 24 2024

A365723 G.f. satisfies A(x) = 1 + x^3A(x)^2(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 5, 3, 5, 21, 28, 26, 84, 180, 207, 385, 990, 1562, 2288, 5278, 10439, 15925, 30212, 64738, 112268, 192780, 396338, 761634, 1317840, 2512940, 5015867, 9146049, 16729462, 32927950, 62851525, 115101637, 220051350, 427898900, 800956600, 1505524800

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Cf. A308616, A365724, A365725, A365726.

Cf. A365694.

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

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

A365726 G.f. satisfies A(x) = 1 + x^3A(x)^5(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 5, 11, 6, 35, 120, 136, 336, 1330, 2310, 4301, 15456, 35100, 64701, 193662, 508921, 1023000, 2643432, 7298984, 16196682, 38795055, 105939288, 254015541, 596987183, 1575487320, 3959803694, 9418896773, 24081344034, 61781452530, 150293865540

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Cf. A308616, A365723, A365724, A365725.

Cf. A365695.

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

a(n) = Sum_{k=0..floor(n/3)} binomial(k,n-3*k) * binomial(n+2*k+1,k) / (n+2*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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A365725 G.f. satisfies A(x) = 1 + x^3A(x)^4(1 + x*A(x)).

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A365723 G.f. satisfies A(x) = 1 + x^3A(x)^2(1 + x*A(x)).

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A365726 G.f. satisfies A(x) = 1 + x^3A(x)^5(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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A365725 G.f. satisfies A(x) = 1 + x^3*A(x)^4*(1 + x*A(x)).

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A365723 G.f. satisfies A(x) = 1 + x^3*A(x)^2*(1 + x*A(x)).

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A365726 G.f. satisfies A(x) = 1 + x^3*A(x)^5*(1 + x*A(x)).

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

A365725 G.f. satisfies A(x) = 1 + x^3A(x)^4(1 + x*A(x)).

A365723 G.f. satisfies A(x) = 1 + x^3A(x)^2(1 + x*A(x)).

A365726 G.f. satisfies A(x) = 1 + x^3A(x)^5(1 + x*A(x)).