A381787 -id:A381787 - cp's OEIS frontend

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

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

A381937 G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 2, 6, 35, 240, 1805, 14386, 119365, 1020136, 8918423, 79380514, 716911887, 6553219720, 60513355786, 563648995020, 5289485238552, 49963186247220, 474655663418546, 4532279676629700, 43473774550929628, 418706702628897708, 4047555977981218963

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A025227, A381787, A381940.

Cf. A001764, A346646, A365178.

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

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

a(n) = A365178(n) + A365178(n-1).

A381940 G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 2, 7, 51, 440, 4170, 41921, 438972, 4736281, 52286520, 587774685, 6705201456, 77426676892, 903251324476, 10629495065550, 126032922655030, 1504194199010435, 18056321542477095, 217859030049153565, 2640609137351540510, 32137554969392230950, 392580762083089376630

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A025227, A381787, A381937.

Cf. A002293, A346647, A365184.

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

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

a(n) = A365184(n) + A365184(n-1).

cp's OEIS Frontend

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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A381937 G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A001764.

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A381940 G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A002293.

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