A366272 -id:A366272 - cp's OEIS frontend

A366595 G.f. A(x) satisfies A(x) = 1 + x^4(1+x)^3A(x)^4.

1, 0, 0, 0, 1, 3, 3, 1, 4, 24, 60, 80, 82, 222, 796, 1848, 2912, 4452, 11088, 31592, 70467, 125437, 231105, 551775, 1399069, 3068219, 5942937, 12017739, 27966515, 66675777, 145719483, 298344501, 632955999, 1449806573, 3346606719, 7335193353, 15557399668

Offset: 0

Seiichi Manyama, Oct 14 2023

nonn

Cf. A366272, A366593, A366594.
Cf. A366589, A366592.
Cf. A366558.

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

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

A367641 G.f. A(x) satisfies A(x) = (1 + x)^2 + x*A(x)^4 / (1 + x)^3.

Original entry on oeis.org

1, 3, 10, 64, 504, 4368, 40208, 385728, 3813888, 38590208, 397648384, 4158436864, 44020882944, 470804670464, 5079479547904, 55217003536384, 604200374845440, 6649658071007232, 73560096496779264, 817467602640830464, 9121818467786162176

Offset: 0

Seiichi Manyama, Nov 25 2023

nonn

Cf. A367639, A367640.

Cf. A366267, A366272.

a(n) = sum(k=0, n, binomial(3*k+2, n-k)*binomial(4*k, k)/(3*k+1));

a(n) = Sum_{k=0..n} binomial(3*k+2,n-k) * binomial(4*k,k)/(3*k+1).

D-finite with recurrence 3*n*(5589*n-14914)*(3*n-1)*(3*n+1)*a(n) +(150903*n^4 -5939762*n^3 +21653157*n^2 -22049842*n +6856944)*a(n-1) +6*(-2312427*n^4 +15333754*n^3 -28367401*n^2 +6040114*n +14892656)*a(n-2) +24*(-3942141*n^4 +46541449*n^3 -199851671*n^2 +367766019*n -243569600)*a(n-3) -32*(n-5)*(8043984*n^3 -85808428*n^2 +305023231*n -361082892)*a(n-4) -384*(n-5)*(n-6)*(885234*n^2 -6808468*n +12951185)*a(n-5) -1536*(n-6)*(n-7)*(144699*n^2 -1203919*n +2337211)*a(n-6) -2048*(n-6)*(n-7)*(n-8)*(27819*n-74186)*a(n-7)=0. - R. J. Mathar, Dec 04 2023

A366593 G.f. A(x) satisfies A(x) = 1 + x^2(1+x)^3A(x)^4.

Original entry on oeis.org

1, 0, 1, 3, 7, 25, 82, 278, 992, 3552, 12985, 48107, 179977, 680079, 2589915, 9931573, 38319117, 148640195, 579349123, 2267818509, 8911575579, 35141656433, 139018921717, 551557089103, 2194155973751, 8750097458849, 34973989188202, 140085055366350

Offset: 0

Seiichi Manyama, Oct 14 2023

nonn

Cf. A366272, A366594, A366595.

Cf. A137954.

a(n) = sum(k=0, n\2, binomial(3*k, n-2*k)*binomial(4*k, k)/(3*k+1));

a(n) = Sum_{k=0..floor(n/2)} binomial(3*k,n-2*k) * binomial(4*k,k)/(3*k+1).

A378786 G.f. A(x) satisfies A(x) = 1 + x * (1+x)^2 * A(x)^4.

Original entry on oeis.org

1, 1, 6, 39, 296, 2435, 21138, 190603, 1767968, 16761424, 161697576, 1582171216, 15664531716, 156637712953, 1579664567130, 16048129755157, 164085811289360, 1687224436103842, 17436287104620980, 181001686332329224, 1886522317836670988, 19734386503541838083

Offset: 0

Seiichi Manyama, Dec 07 2024

nonn

Cf. A365178, A366216, A366272.
Cf. A002478, A073155, A366221.
Cf. A002293.

a(n, r=1, s=2, t=4, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(2*k,n-k)/(4*k+1) = Sum_{k=0..n} binomial(2*k,n-k) * A002293(k).

A382892 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1+x)^3 * A(x) )^3.

Original entry on oeis.org

1, 3, 24, 190, 1659, 15309, 146986, 1453536, 14704917, 151479031, 1583533308, 16756882194, 179149227231, 1932144798513, 20996553430206, 229678298803028, 2527034248221849, 27947027713469307, 310494250880357488, 3463870813896354726, 38787008808135775299

Offset: 0

Seiichi Manyama, Apr 08 2025

nonn

Cf. A360076, A382894.

Cf. A366272, A382614.

a(n, r=3, s=3, t=4, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

G.f. A(x) satisfies A(x) = ( 1 + x * (1+x)^3 * A(x)^(4/3) )^3.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r).
G.f.: B(x)^3, where B(x) is the g.f. of A366272.

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A366595 G.f. A(x) satisfies A(x) = 1 + x^4*(1+x)^3*A(x)^4.

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A367641 G.f. A(x) satisfies A(x) = (1 + x)^2 + x*A(x)^4 / (1 + x)^3.

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A366593 G.f. A(x) satisfies A(x) = 1 + x^2*(1+x)^3*A(x)^4.

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A378786 G.f. A(x) satisfies A(x) = 1 + x * (1+x)^2 * A(x)^4.

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A382892 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1+x)^3 * A(x) )^3.

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A366595 G.f. A(x) satisfies A(x) = 1 + x^4(1+x)^3A(x)^4.

A366593 G.f. A(x) satisfies A(x) = 1 + x^2(1+x)^3A(x)^4.