A307874
E.g.f. A(x) satisfies: d/dx A(x) = 1 + A(log(1+x)).
Original entry on oeis.org
1, 1, 0, -1, 4, -12, -3, 640, -9721, 107849, -766116, -5716810, 438016259, -13557651987, 318299775147, -5284369281919, -5483686862123, 6119663470743306, -388801742002632589, 17841761552418336070, -645131407697518621805, 14383670984970068901209, 384858376828629625293001
Offset: 1
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terms = 23; A[] = 0; Do[A[x] = Normal[Integrate[1 + A[Log[1 + x] + O[x]^(terms + 1)], x] + O[x]^(terms + 1)], terms]; Rest[CoefficientList[A[x], x] Range[0, terms]!]
a[n_] := a[n] = Sum[StirlingS1[n - 1, k] a[k], {k, 1, n - 1}]; a[1] = 1; Table[a[n], {n, 1, 23}]
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a_vector(n) = my(v=vector(n)); v[1]=1; for(i=1, n-1, v[i+1]=sum(j=1, i, stirling(i, j, 1)*v[j])); v; \\ Seiichi Manyama, Jun 24 2022
A355203
E.g.f. A(x) satisfies A'(x) = 1 + A(1 - exp(-x)).
Original entry on oeis.org
1, 1, 0, -2, 4, 10, -150, 838, 222, -82616, 1408364, -13862308, -18747672, 5307622274, -170657860276, 3561218897884, -33756455501714, -1481233045213718, 116803294574962288, -5108843717328225572, 157037998518149186728, -1976107915155933805542
Offset: 1
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a_vector(n) = my(v=vector(n)); v[1]=1; for(i=1, n-1, v[i+1]=sum(j=1, i, (-1)^(i-j)*stirling(i, j, 2)*v[j])); v;
A355205
E.g.f. A(x) satisfies A'(x) = 1 + 2 * A(-log(1-x)).
Original entry on oeis.org
1, 2, 6, 28, 184, 1596, 17508, 235592, 3799736, 72125344, 1587567768, 40027332256, 1144113365576, 36747710168568, 1316192996129064, 52219780699310176, 2281487895137577232, 109193200290592216368, 5698144666408068511472
Offset: 1
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a_vector(n) = my(v=vector(n)); v[1]=1; for(i=1, n-1, v[i+1]=2*sum(j=1, i, abs(stirling(i, j, 1))*v[j])); v;
A355209
E.g.f. A(x) satisfies A'(x) = 1 + A(-2 * log(1-x)).
Original entry on oeis.org
1, 2, 10, 108, 2308, 94384, 7315728, 1077605632, 304189296192, 166216599473344, 177463576125821632, 373017466526422396288, 1552199775052648327045760, 12835792253795957289436533760, 211464475635678910995043533156352
Offset: 1
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a_vector(n) = my(v=vector(n)); v[1]=1; for(i=1, n-1, v[i+1]=sum(j=1, i, 2^j*abs(stirling(i, j, 1))*v[j])); v;
Original entry on oeis.org
1, 1, 1, 2, 3, 2, 6, 11, 12, 7, 24, 50, 70, 70, 36, 120, 274, 450, 595, 540, 250, 720, 1764, 3248, 5145, 6300, 5250, 2229
Offset: 0
First few rows of the triangle:
1;
1, 1;
2, 3, 2;
6, 11, 12, 7;
24, 50, 70, 70, 36;
120, 274, 450, 595, 540, 250;
720, 1764, 3248, 5145, 6300, 5250, 2229;
...
A317275
a(1) = 1; a(n) = Sum_{k=1..n-1} |Stirling1(n-1,k)|*a(k)*a(n-k).
Original entry on oeis.org
1, 1, 2, 9, 97, 3105, 409318, 301069244, 1523141657289, 61447697339843710, 22299766257043761657829, 80922067241038150103930448880, 3230152742688615187688660954252643194, 1547248455508510864175770056662224501358437847
Offset: 1
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a[n_] := a[n] = Sum[Abs[StirlingS1[n - 1, k]] a[k] a[n - k], {k, n - 1}]; a[1] = 1; Table[a[n], {n, 14}]
Showing 1-6 of 6 results.
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