A385311
Expansion of e.g.f. 1/(1 - 3 * x * cos(x))^(1/3).
Original entry on oeis.org
1, 1, 4, 25, 232, 2805, 41920, 744933, 15340416, 359136073, 9419223040, 273558859409, 8714789788672, 302151400126589, 11326084055150592, 456421403198919325, 19677025400034590720, 903660903945306053137, 44042354270955276599296, 2270411632567521580120713
Offset: 0
-
a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*I^(n-k)*a185951(n, k));
A385305
Expansion of e.g.f. 1/(1 - 3 * sinh(x))^(1/3).
Original entry on oeis.org
1, 1, 4, 29, 296, 3921, 63904, 1236509, 27700096, 705098241, 20100847104, 634406699389, 21959759364096, 827184049670161, 33684401687855104, 1474548883501060669, 69051807696652599296, 3444499079760040247681, 182339939994632235515904, 10209271857672376613472349
Offset: 0
-
a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*a136630(n, k));
A385306
Expansion of e.g.f. 1/(1 - 2 * sin(x))^(1/2).
Original entry on oeis.org
1, 1, 3, 14, 93, 796, 8343, 103424, 1479993, 24008656, 435364683, 8726775584, 191601310293, 4572794295616, 117871476051423, 3263515787807744, 96591500816346993, 3043368045293138176, 101702692426476460563, 3592948632452749243904, 133794496537591022166093
Offset: 0
-
With[{nn=20},CoefficientList[Series[1/Sqrt[1-2Sin[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 09 2025 *)
-
a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*I^(n-k)*a136630(n, k));
A385309
Expansion of e.g.f. 1/(1 - 3 * x * cosh(x))^(1/3).
Original entry on oeis.org
1, 1, 4, 31, 328, 4485, 75520, 1509347, 34916224, 917703145, 27011107840, 880133628231, 31451749424128, 1223047891889837, 51414400611438592, 2323391075748100555, 112315439676217262080, 5783449255108473820497, 316034972288791445241856, 18265740423344520141491951
Offset: 0
-
a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*a185951(n, k));
Showing 1-4 of 4 results.