A349598
E.g.f. satisfies: log(A(x)) = exp(x*A(x)^2) - 1.
Original entry on oeis.org
1, 1, 6, 71, 1279, 31142, 958127, 35674921, 1560207964, 78410153193, 4453247964775, 282086867840252, 19718661737739301, 1507855981764016549, 125211854842018500134, 11220898483255456505555, 1079389691811367897870339, 110936313685240067472613726
Offset: 0
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a[n_] := Sum[(2*n + 1)^(k - 1)*StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 23 2021 *)
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a(n) = sum(k=0, n, (2*n+1)^(k-1)*stirling(n, k, 2));
A367180
E.g.f. satisfies A(x) = 1 + (exp(x*A(x)^2) - 1)/A(x).
Original entry on oeis.org
1, 1, 3, 19, 187, 2491, 41951, 855387, 20491395, 564179371, 17555839639, 609337562923, 23340215770235, 978038556122811, 44506423393073487, 2185725954288076987, 115224508775345033779, 6490005347933921581195, 388973650645651854960455
Offset: 0
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a(n) = sum(k=0, n, (2*n-k)!/(2*n-2*k+1)!*stirling(n, k, 2));
A349602
E.g.f. satisfies: A(x) * log(A(x)) = 1 - exp(-x*A(x)^2).
Original entry on oeis.org
1, 1, 2, 2, -43, -668, -5908, -1209, 1399400, 37121106, 508366819, -3012861630, -444910083132, -15407930598279, -249403814792546, 5359691081465462, 589889204153846141, 23861630070579997032, 379819221897309026072, -21971010821241361939769
Offset: 0
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a[n_] := Sum[(-1)^(n - k)*(2*n - k + 1)^(k - 1)*StirlingS2[n, k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Nov 23 2021 *)
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a(n) = sum(k=0, n, (-1)^(n-k)*(2*n-k+1)^(k-1)*stirling(n, k, 2));
A349600
E.g.f. satisfies: A(x)^A(x) = 1 + x*A(x)^2.
Original entry on oeis.org
1, 1, 2, 3, -20, -320, -2274, 5474, 487432, 8358480, 37944240, -2286868848, -81319780200, -1139790073968, 18382692073032, 1570867988794680, 42704382709868736, 55662087673489920, -49662902468183117760, -2360239974764654675904, -38098700311039336972800
Offset: 0
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a(n) = sum(k=0, n, (2*n-k+1)^(k-1)*stirling(n, k, 1));
A356972
E.g.f. satisfies log(A(x)) = (exp(x * A(x)^2) - 1) * A(x).
Original entry on oeis.org
1, 1, 8, 128, 3139, 104382, 4393590, 224045271, 13428576766, 925335827928, 72082558060889, 6264277731652096, 600873473776204782, 63059026039778220285, 7187299097301622432156, 884141943373486896560252, 116756337165196381259759707, 16474480747756013055963484442
Offset: 0
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a(n) = sum(k=0, n, (2*n+k+1)^(k-1)*stirling(n, k, 2));
A357011
E.g.f. satisfies A(x) * log(A(x)) = exp(x * A(x)^3) - 1.
Original entry on oeis.org
1, 1, 6, 74, 1407, 36357, 1190476, 47254783, 2205546706, 118378505742, 7184030384361, 486440226752911, 36358328607088010, 2973464028723984551, 264119772408892921774, 25321946948812001539166, 2606224408648404660237647, 286624141573198517220290837
Offset: 0
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a(n) = sum(k=0, n, (3*n-k+1)^(k-1)*stirling(n, k, 2));
Showing 1-6 of 6 results.