A264037
Stirling transform of A077957 (aerated powers of 2) with 0 prepended [0, 1, 0, 2, 0, 4, 0, 8, ...].
Original entry on oeis.org
0, 1, 1, 3, 13, 55, 241, 1171, 6357, 37567, 236521, 1574331, 11068333, 82110535, 640794337, 5239439011, 44723250501, 397481121295, 3671081354137, 35176098791115, 349120380267421, 3583273413146647, 37975511840454673, 415004245048757299, 4670891190907818165
Offset: 0
G.f. = x + x^2 + 3*x^3 + 13*x^4 + 55*x^5 + 241*x^7 + 1171*x^8 + 6357*x^9 + ...
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Table[(BellB[n, Sqrt[2]] - BellB[n, -Sqrt[2]])/(2 Sqrt[2]), {n, 0, 24}]
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vector(100, n, n--; sum(k=0, n\2, 2^k*stirling(n, 2*k+1, 2))) \\ Altug Alkan, Nov 01 2015
A357598
Expansion of e.g.f. sinh(2 * (exp(x)-1)) / 2.
Original entry on oeis.org
0, 1, 1, 5, 25, 117, 601, 3509, 22457, 153141, 1105561, 8453557, 68339833, 581495605, 5184047961, 48259748533, 468040609593, 4719817792565, 49396003390489, 535526127566773, 6004124908829177, 69509047405180213, 829801009239621849, 10202835010223731893
Offset: 0
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my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh(2*(exp(x)-1))/2)))
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a(n) = sum(k=0, (n-1)\2, 4^k*stirling(n, 2*k+1, 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, 2)-Bell_poly(n, -2)))/4;
A357615
Expansion of e.g.f. cosh(sqrt(3) * (exp(x) - 1)).
Original entry on oeis.org
1, 0, 3, 9, 30, 135, 705, 3906, 22953, 145053, 985800, 7136613, 54544485, 437961888, 3685605735, 32441696325, 297977767662, 2848636972971, 28278241848309, 290931124989546, 3097051613077269, 34064462020306473, 386600759467746528, 4521440483724439521
Offset: 0
G.f. = 1 + 3*x^2 + 9*x^3 + 30*x^4 + 135*x^5 + 705*x^6 + ... - _Michael Somos_, Oct 06 2022
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a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Cosh[Sqrt[3] * (Exp@x - 1)], {x, 0, n}]]; (* Michael Somos, Oct 06 2022 *)
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a(n) = sum(k=0, n\2, 3^k*stirling(n, 2*k, 2));
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my(x='x+O('x^30)); apply(round, Vec(serlaplace(cosh(sqrt(3) * (exp(x) - 1))))) \\ Michel Marcus, Oct 06 2022
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{a(n) = if(n<0, 0, n!*simplify(polcoeff( cosh(quadgen(12) * (exp(x + x*O(x^n)) - 1)), n)))}; /* Michael Somos, Oct 06 2022 */
A357665
Expansion of e.g.f. sinh( (exp(3*x) - 1)/sqrt(3) )/sqrt(3).
Original entry on oeis.org
0, 1, 3, 12, 81, 765, 7938, 85239, 963819, 11801862, 158533443, 2320621569, 36425289816, 604576791405, 10532817901791, 192197187209484, 3673078679995677, 73486862051182425, 1536507360834633666, 33482575797899354235, 758209049155176114807
Offset: 0
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my(N=30, x='x+O('x^N)); concat(0, apply(round, Vec(serlaplace(sinh((exp(3*x)-1)/sqrt(3))/sqrt(3)))))
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a(n) = sum(k=0, (n-1)\2, 3^(n-1-k)*stirling(n, 2*k+1, 2));
A357668
Expansion of e.g.f. sinh( 3 * (exp(x) - 1) )/3.
Original entry on oeis.org
0, 1, 1, 10, 55, 307, 2026, 14779, 114157, 933616, 8110261, 74525167, 719925328, 7279859485, 76855303981, 845280487018, 9663800287483, 114601481983855, 1407040763488354, 17856103120048783, 233883061849700137, 3157648445216335528, 43887908697233605489
Offset: 0
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my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh(3*(exp(x)-1))/3)))
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a(n) = sum(k=0, (n-1)\2, 9^k*stirling(n, 2*k+1, 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)))/6;
A357737
Expansion of e.g.f. sin( sqrt(3) * (exp(x) - 1) )/sqrt(3).
Original entry on oeis.org
0, 1, 1, -2, -17, -65, -134, 331, 5797, 41092, 199621, 500731, -2996432, -58995155, -573624323, -4065029714, -19194210269, 7657775035, 1581081323122, 24363365708815, 260409006907921, 2127851409822892, 11143555796154673, -27234657667343081
Offset: 0
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my(N=30, x='x+O('x^N)); concat(0, apply(round, Vec(serlaplace(sin(sqrt(3)*(exp(x)-1))/sqrt(3)))))
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a(n) = sum(k=0, (n-1)\2, (-3)^k*stirling(n, 2*k+1, 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, sqrt(3)*I)-Bell_poly(n, -sqrt(3)*I))/(2*sqrt(3)*I));
Showing 1-6 of 6 results.
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