A349719
E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x))/2 ).
Original entry on oeis.org
1, 1, 0, 1, -4, 26, -212, 2108, -24720, 334072, -5112544, 87396728, -1650607040, 34132685120, -767025716736, 18612106195456, -485013257865472, 13509071081429888, -400505695457942528, 12592502771190979712, -418524228123134068224
Offset: 0
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a[n_] := (1/2^n) * Sum[If[k == n == 1, 1, (-k + 1)^(n - 1)] * Binomial[n, k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Nov 27 2021 *)
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a(n) = sum(k=0, n, (-k+1)^(n-1)*binomial(n, k))/2^n;
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my(N=40, x='x+O('x^N)); Vec(serlaplace((x/2)/lambertw(x/2*exp(-x/2))))
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my(N=40, x='x+O('x^N)); Vec(2*sum(k=0, N, (-k+1)^(k-1)*x^k/(2-(-k+1)*x)^(k+1)))
A349714
E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^3)/2 ).
Original entry on oeis.org
1, 1, 4, 37, 532, 10426, 259300, 7823908, 277713904, 11339452792, 523621438336, 26982030104536, 1534947906550528, 95550736737542464, 6460746383585984512, 471533064029919744256, 36946948091091750496000, 3093472887944746070621056
Offset: 0
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a[n_] := (1/2^n) * Sum[(3*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)
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a(n) = sum(k=0, n, (3*k+1)^(n-1)*binomial(n, k))/2^n;
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my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-3*x/2*exp(3*x/2))/(3*x/2))^(1/3)))
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my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (3*k+1)^(k-1)*x^k/(2-(3*k+1)*x)^(k+1)))
A349716
E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^5)/2 ).
Original entry on oeis.org
1, 1, 6, 91, 2156, 69926, 2884576, 144555356, 8529135216, 579220982056, 44503081624976, 3816776859516776, 361462121953291456, 37464997600663289216, 4218485281787859411456, 512762346462142021355776, 66919363061333997572830976, 9332997074366800051673277056
Offset: 0
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a[n_] := (1/2^n) * Sum[(5*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)
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a(n) = sum(k=0, n, (5*k+1)^(n-1)*binomial(n, k))/2^n;
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my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-5*x/2*exp(5*x/2))/(5*x/2))^(1/5)))
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my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (5*k+1)^(k-1)*x^k/(2-(5*k+1)*x)^(k+1)))
A349720
E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x)^2)/2 ).
Original entry on oeis.org
1, 1, -1, 7, -63, 801, -13025, 258343, -6048511, 163276417, -4992740289, 170571634311, -6439161507647, 266180947507489, -11958385377911713, 580151397382158631, -30227616424300542975, 1683438461080186841601, -99796591057813372007297
Offset: 0
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a[n_] := (1/2^n) * Sum[(-2*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 27 2021 *)
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a(n) = sum(k=0, n, (-2*k+1)^(n-1)*binomial(n, k))/2^n;
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my(N=20, x='x+O('x^N)); Vec(serlaplace((x/lambertw(x*exp(-x)))^(1/2)))
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my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (-2*k+1)^(k-1)*x^k/(2-(-2*k+1)*x)^(k+1)))
A349721
E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x)^3)/2 ).
Original entry on oeis.org
1, 1, -2, 19, -260, 4966, -121328, 3613996, -127035920, 5147600680, -236245559984, 12112405259560, -686148484748480, 42560312499982720, -2868921992458611200, 208828244778853125376, -16324500711130356582656, 1363986660232205656646272
Offset: 0
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a[n_] := (1/2^n) * Sum[(-3*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)
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a(n) = sum(k=0, n, (-3*k+1)^(n-1)*binomial(n, k))/2^n;
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my(N=20, x='x+O('x^N)); Vec(serlaplace(((3*x/2)/lambertw(3*x/2*exp(-3*x/2)))^(1/3)))
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my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (-3*k+1)^(k-1)*x^k/(2-(-3*k+1)*x)^(k+1)))
Showing 1-5 of 5 results.