A385310
Expansion of e.g.f. 1/(1 - 2 * x * cos(x))^(1/2).
Original entry on oeis.org
1, 1, 3, 12, 69, 500, 4455, 46928, 571977, 7914384, 122585355, 2100940864, 39470867469, 806555184448, 17808628411119, 422498774818560, 10717948285126545, 289501146405400832, 8295124400250875667, 251300745071590317056, 8025654235707259740885, 269482309052945201181696
Offset: 0
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a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(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
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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 *)
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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));
A385308
Expansion of e.g.f. 1/(1 - 2 * x * cosh(x))^(1/2).
Original entry on oeis.org
1, 1, 3, 18, 141, 1400, 17055, 245392, 4070073, 76483584, 1606033755, 37267953536, 947051118981, 26156846230528, 780174007426359, 24992424003517440, 855795857724702705, 31193844533488074752, 1205893835653392258867, 49280187764171870470144, 2122704756621224015194365
Offset: 0
-
a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*a185951(n, k));
Showing 1-4 of 4 results.