A371117
E.g.f. satisfies A(x) = 1 - x*log(1 - x*A(x)).
Original entry on oeis.org
1, 0, 2, 3, 32, 210, 2184, 26460, 373344, 6150816, 113958720, 2362345920, 54094694400, 1355708296800, 36926213869440, 1085886303989760, 34291129916574720, 1157362522046277120, 41576054625791078400, 1583864892141097098240, 63779322541075124428800
Offset: 0
-
nmax = 20; A[] = 0; Do[A[x] = 1 - x*Log[1 - x*A[x]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0,nmax]! (* Vaclav Kotesovec, Mar 11 2024 *)
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a(n) = n!*sum(k=0, n\2, abs(stirling(n-k, k, 1))/(n-2*k+1)!);
A371119
E.g.f. satisfies A(x) = 1 + x*A(x)*(exp(x*A(x)) - 1).
Original entry on oeis.org
1, 0, 2, 3, 52, 305, 4866, 57337, 1048776, 18547713, 407900710, 9436057961, 248501026236, 7021087254337, 217488458525898, 7223642070331065, 258233053457437456, 9841074705853124609, 399304906991091898830, 17163110041947804495817, 779646387683354742170820
Offset: 0
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a(n) = n!^2*sum(k=0, n\2, stirling(n-k, k, 2)/((n-k)!*(n-k+1)!));
A371116
E.g.f. satisfies A(x) = 1 + x^2*(exp(x*A(x)) - 1).
Original entry on oeis.org
1, 0, 0, 6, 12, 20, 750, 7602, 47096, 589752, 11823930, 169812830, 2287327812, 46793930196, 1061518458182, 21163158296490, 458344052450160, 12165772611938672, 329982890581149426, 8764089834124752822, 255655700917556204540, 8220667673623130347020
Offset: 0
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nmax = 20; CoefficientList[Series[1 - x^2 - ProductLog[-E^(x*(1 - x^2))*x^3]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 11 2024 *)
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a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 2)/(n-3*k+1)!);
A371120
E.g.f. satisfies A(x) = 1 + x*A(x)^3*(exp(x*A(x)) - 1).
Original entry on oeis.org
1, 0, 2, 3, 100, 545, 17946, 203497, 7194440, 132963777, 5172409630, 135827977241, 5868623306844, 200952952956769, 9665278822378466, 407661518051710665, 21789972653746494736, 1088515671895571005313, 64406426353877958253254
Offset: 0
-
a(n) = n!*sum(k=0, n\2, (n+2*k)!*stirling(n-k, k, 2)/((n-k)!*(n+k+1)!));
A377688
E.g.f. satisfies A(x) = (1 + x * (exp(x*A(x)) - 1))^2.
Original entry on oeis.org
1, 0, 4, 6, 128, 850, 13872, 195314, 3586592, 74163618, 1694735840, 44196946882, 1244904944208, 38788984768274, 1302631536943856, 47297768099973330, 1840951270666885952, 76501162074673415746, 3382517582789739956928, 158445187728836733069986
Offset: 0
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a(n) = 2*n!*sum(k=0, n\2, (2*n-2*k+1)!*stirling(n-k, k, 2)/((n-k)!*(2*n-3*k+2)!));
A377689
E.g.f. satisfies A(x) = (1 + x * (exp(x*A(x)) - 1))^3.
Original entry on oeis.org
1, 0, 6, 9, 300, 1995, 48438, 720111, 17965944, 422161011, 12234150930, 380328463383, 13151800946628, 497667965729259, 20320277028840558, 899482574279597535, 42525760204244934768, 2153233176660303831267, 115738033009558749725610, 6600044862098481204272487
Offset: 0
-
a(n) = 3*n!*sum(k=0, n\2, (3*n-3*k+2)!*stirling(n-k, k, 2)/((n-k)!*(3*n-4*k+3)!));
Showing 1-6 of 6 results.