A367640 -id:A367640 - cp's OEIS frontend

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

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

A367639 G.f. A(x) satisfies A(x) = (1 + x)^2 + x*A(x)^2 / (1 + x).

Original entry on oeis.org

1, 3, 6, 16, 52, 184, 688, 2672, 10672, 43552, 180800, 761088, 3241088, 13937408, 60435968, 263962880, 1160188672, 5127762432, 22775636992, 101608357888, 455105255424, 2045751037952, 9225923895296, 41731062358016, 189275050729472, 860630181167104

Offset: 0

Seiichi Manyama, Nov 25 2023

Cf. A006318, A025227, A367640, A333090, A367641.

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

G.f.: A(x) = 2*(1+x)^2 / (1+sqrt(1-4*x*(1+x))).
a(n) = Sum_{k=0..n} binomial(k+2,n-k) * binomial(2*k,k)/(k+1).
a(n) ~ 2^(n - 5/4) * (1 + sqrt(2))^(n + 3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 25 2023
D-finite with recurrence (n+1)*a(n) +(-3*n+1)*a(n-1) +2*(-4*n+9)*a(n-2) +4*(-n+4)*a(n-3)=0. - R. J. Mathar, Dec 04 2023
From Peter Bala, May 05 2024: (Start)
A(x) = (1 + x)*S(x/(1 + x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the g.f. of the large Schröder numbers A006318. Cf. A025227.
A333090(n) = [x^n] A(x)^n. (End)

A381860 G.f. A(x) satisfies A(x) = (1 + x)^3 * C(x*A(x)), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 4, 12, 55, 327, 2157, 15141, 110853, 836790, 6465309, 50876776, 406335099, 3285202335, 26835060422, 221128733649, 1835973630276, 15344202894457, 128983332603009, 1089803313492966, 9250137181234430, 78837133437062307, 674408139329393187, 5788618956395607745

Offset: 0

Seiichi Manyama, Mar 08 2025

nonn

Cf. A367640, A381787.

Cf. A000108, A381882.

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

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

D-finite with recurrence -2*n*(2*n+1)*a(n) +3*(n^2+13*n-6)*a(n-1) +3*(69*n^2-221*n+150)*a(n-2) +2*(397*n^2-2431*n+3471)*a(n-3) +6*(225*n^2-1953*n+4079)*a(n-4) +9*(135*n^2-1503*n+4084)*a(n-5) +9*(63*n^2-855*n+2860)*a(n-6) +12*(3*n-22)*(3*n-26)*a(n-7)=0. - R. J. Mathar, Mar 10 2025

A381938 G.f. A(x) satisfies A(x) = (1 + x)^2 * B(x*A(x)), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 3, 9, 52, 380, 3066, 26304, 235314, 2170312, 20487963, 196988392, 1922327792, 18990571724, 189548947601, 1908604524752, 19364096602370, 197761735366804, 2031444188437719, 20974821788118024, 217561484977675026, 2265961977605950416, 23688432825547509283

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A366694, A367640, A381941.

Cf. A001764, A381785.

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

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

a(n) = A381785(n) + A381785(n-1).

A381941 G.f. A(x) satisfies A(x) = (1 + x)^2 * B(x*A(x)), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 3, 10, 71, 644, 6461, 68971, 768054, 8820281, 103694479, 1241799996, 15095075897, 185769856443, 2310006893997, 28978952155943, 366315306556482, 4661272734504606, 59659914501348239, 767539555514812321, 9920124234695256009, 128744011085858468131, 1677087982747514335025

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A366694, A367640, A381938.

Cf. A002293.

a(n) = sum(k=0, n, binomial(5*k+1, k)*binomial(2*k+2, n-k)/(5*k+1));

a(n) = Sum_{k=0..n} binomial(5*k+1,k) * binomial(2*k+2,n-k)/(5*k+1).

cp's OEIS Frontend

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

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A367639 G.f. A(x) satisfies A(x) = (1 + x)^2 + x*A(x)^2 / (1 + x).

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A381860 G.f. A(x) satisfies A(x) = (1 + x)^3 * C(x*A(x)), where C(x) is the g.f. of A000108.

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A381938 G.f. A(x) satisfies A(x) = (1 + x)^2 * B(x*A(x)), where B(x) is the g.f. of A001764.

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A381941 G.f. A(x) satisfies A(x) = (1 + x)^2 * B(x*A(x)), where B(x) is the g.f. of A002293.

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