A235135
Expansion of e.g.f. 1/(1 - sinh(3*x))^(1/3).
Original entry on oeis.org
1, 1, 4, 37, 424, 6241, 113824, 2460277, 61504384, 1746727201, 55545439744, 1955176596517, 75470959673344, 3169939381277761, 143927870364811264, 7024566555751464757, 366742587098140770304, 20394984041632355113921, 1203587891190987380752384, 75124090160952970927512997
Offset: 0
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CoefficientList[Series[1/(1-Sinh[3*x])^(1/3), {x, 0, 20}], x] * Range[0, 20]!
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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)*3^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025
A385281
Expansion of e.g.f. 1/(1 - 2 * x * cosh(2*x))^(1/2).
Original entry on oeis.org
1, 1, 3, 27, 249, 2825, 41355, 708883, 13888497, 309267729, 7698772755, 211585744139, 6367841422569, 208299923870233, 7357493992966299, 279095125351544835, 11316313498670411745, 488403056864943302177, 22355228989851909617187, 1081663315375339026249211
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)*2^(n-k)*a185951(n, k));
A385304
Expansion of e.g.f. 1/(1 - 2 * sinh(x))^(1/2).
Original entry on oeis.org
1, 1, 3, 16, 117, 1096, 12543, 169576, 2644617, 46735936, 922993083, 20145579136, 481555537917, 12511452674176, 351058439096823, 10579734482269696, 340820224678288017, 11687491783287586816, 425075150516293691763, 16343274366458168160256, 662325275389743380902917
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));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*a136630(n, k));
A385283
Expansion of e.g.f. 1/(1 - 2 * x * cos(2*x))^(1/2).
Original entry on oeis.org
1, 1, 3, 3, -39, -775, -9045, -85813, -426447, 7321329, 325555155, 7786757011, 137053423881, 1388713844713, -21121997539461, -1827406866674085, -69034283067822495, -1852635543265039903, -30574875232261547613, 308376017794648053539, 54871741689019890859065
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)*(2*I)^(n-k)*a185951(n, k));
Showing 1-4 of 4 results.
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