A356572
Expansion of e.g.f. sinh( (exp(3*x) - 1)/3 ).
Original entry on oeis.org
0, 1, 3, 10, 45, 307, 2718, 26371, 265359, 2778976, 30916863, 372113623, 4873075056, 68908186765, 1037694932823, 16438615126282, 271972422548361, 4687666317874495, 84181305836224422, 1576083180118379527, 30757003280682603699, 624671260245315540568
Offset: 0
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With[{m = 20}, Range[0, m]! * CoefficientList[Series[Sinh[(Exp[3*x] - 1)/3], {x, 0, m}], x]] (* Amiram Eldar, Oct 07 2022 *)
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my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh((exp(3*x)-1)/3))))
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a(n) = sum(k=0, (n-1)\2, 3^(n-1-2*k)*stirling(n, 2*k+1, 2));
A357650
Expansion of e.g.f. cosh( (exp(4*x) - 1)/4 ).
Original entry on oeis.org
1, 0, 1, 12, 113, 1000, 8977, 86996, 959905, 12303888, 179038689, 2840696540, 47684181393, 835731314808, 15277172343409, 292597596283684, 5900038421042753, 125488177929542944, 2809541905807203009, 65903118624174027436, 1610968753088423886257
Offset: 0
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With[{m = 20}, Range[0, m]! * CoefficientList[Series[Cosh[(Exp[4*x] - 1)/4], {x, 0, m}], x]] (* Amiram Eldar, Oct 07 2022 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(cosh((exp(4*x)-1)/4)))
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a(n) = sum(k=0, n\2, 4^(n-2*k)*stirling(n, 2*k, 2));
A357662
Expansion of e.g.f. cosh( (exp(3*x) - 1)/sqrt(3) ).
Original entry on oeis.org
1, 0, 3, 27, 198, 1485, 12825, 132678, 1582497, 20603727, 284290560, 4132840239, 63571690485, 1038868740000, 18022911716439, 330305863479615, 6355242571945878, 127721845479277737, 2672729031195365949, 58142565625982730462, 1313557910179640120061
Offset: 0
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my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cosh((exp(3*x)-1)/sqrt(3)))))
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a(n) = sum(k=0, n\2, 3^(n-k)*stirling(n, 2*k, 2));
A357667
Expansion of e.g.f. cosh( 3 * (exp(x) - 1) ).
Original entry on oeis.org
1, 0, 9, 27, 144, 945, 6273, 44226, 339399, 2796795, 24387786, 223853355, 2159078445, 21827316888, 230536050165, 2536213188519, 28994911890048, 343806474384045, 4220933769308205, 53566838971016418, 701650841036287275, 9473067208871584407
Offset: 0
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my(x='x+O('x^30)); Vec(serlaplace(cosh(3*(exp(x)-1))))
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a(n) = sum(k=0, n\2, 9^k*stirling(n, 2*k, 2));
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Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, 3)+Bell_poly(n, -3)))/2;
Showing 1-4 of 4 results.