A367160
E.g.f. satisfies A(x) = 1 - A(x)^3 * log(1 - x*A(x)).
Original entry on oeis.org
1, 1, 9, 161, 4412, 164024, 7721898, 440550102, 29548655696, 2278884869640, 198709176600696, 19331290511231280, 2075887738522499664, 243905626745780976144, 31125204862136659763040, 4287017279890868817295728, 633888142969750426274770944
Offset: 0
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a(n) = sum(k=0, n, (n+3*k)!/(n+2*k+1)!*abs(stirling(n, k, 1)));
A377360
E.g.f. satisfies A(x) = ( 1 - log(1 - x*A(x)) )^2.
Original entry on oeis.org
1, 2, 12, 130, 2082, 44488, 1192964, 38557860, 1459988440, 63414711072, 3108861424032, 169829819311392, 10230860299538400, 673850170929176928, 48176129912775680160, 3715759452364764485280, 307545698210584533055488, 27190399275422185989742080, 2557448587458299889542868480
Offset: 0
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nmax = 20; CoefficientList[1/x * InverseSeries[Series[x/(1 - Log[1 - x])^2, {x, 0, nmax + 1}], x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 27 2025 *)
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a(n) = 2*(2*n+1)!*sum(k=0, n, abs(stirling(n, k, 1))/(2*n-k+2)!);
A377361
E.g.f. satisfies A(x) = ( 1 - log(1 - x*A(x)) )^3.
Original entry on oeis.org
1, 3, 27, 435, 10308, 324942, 12831540, 610024398, 33948639024, 2165995595208, 155913776865216, 12501945620113320, 1105228405532295216, 106806396107364409440, 11201958792185117156640, 1267313834232739887340464, 153842580381390055963315200, 19946923686925035463312117632
Offset: 0
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a(n) = 3*(3*n+2)!*sum(k=0, n, abs(stirling(n, k, 1))/(3*n-k+3)!);
A377789
Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 - 2*log(1-x)) ).
Original entry on oeis.org
1, 2, 10, 88, 1148, 20088, 442896, 11802096, 369132256, 13261156416, 538227938880, 24359100451200, 1216403663398656, 66440221207025664, 3940468338389603328, 252190997066643909120, 17324237625466992906240, 1271459220768570290626560, 99289436336361780797288448
Offset: 0
-
my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1-2*log(1-x)))/x))
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a(n) = n!*sum(k=0, n, 2^k*abs(stirling(n, k, 1))/(n-k+1)!);
A377790
Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 - 3*log(1-x)) ).
Original entry on oeis.org
1, 3, 21, 249, 4302, 98982, 2860686, 99779418, 4081683744, 191696903424, 10168315038360, 601321398385320, 39230551252853424, 2799199551778309872, 216856533870111053520, 18127987493141612555280, 1626479163148212406506240, 155902932291162161594016000
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1-3*log(1-x)))/x))
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a(n) = n!*sum(k=0, n, 3^k*abs(stirling(n, k, 1))/(n-k+1)!);
A177379
E.g.f. satisfies: A(x) = 1/(1-x - x*log(A(x))).
Original entry on oeis.org
1, 1, 4, 27, 260, 3270, 50904, 946134, 20462896, 505137312, 14020517520, 432340670520, 14667108820704, 542979374426736, 21784934875431168, 941691211940974320, 43634507604383543040, 2157698329617806488320
Offset: 0
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 260*x^4/4! +...
Log(A(x)) = G(x) - 1 where G(x) = e.g.f. of A138013 begins:
G(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 146*x^4/4! + 1694*x^5/5! + ...
and satisfies: exp(1 - G(x)) = 1 - x*G(x).
Contribution from _Paul D. Hanna_, Jul 16 2010: (Start)
Given e.g.f. A(x), and A179424 = Riordan array (A(x),x*A(x)) where the g.f. of column k in A179424 equals A(x)^(k+1):
1;
1, 1;
4/2!, 2, 1;
27/3!, 10/2!, 3, 1;
260/4!, 78/3!, 18/2!, 4, 1;
3270/5!, 832/4!, 159/3!, 28/2!, 5, 1;
...
then the matrix log of A179424 equals the triangular matrix:
0;
1, 0;
1, 2, 0;
1, 2, 3, 0;
1, 2, 3, 4, 0;
1, 2, 3, 4, 5, 0; ...
in which column k equals k+1 in row n for n>k>=0.
(End)
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CoefficientList[1/(1-InverseSeries[Series[x/(1-Log[1-x]),{x,0,20}],x]),x]*Range[0,20]! (* Vaclav Kotesovec, Jan 11 2014 *)
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{a(n)=n!*polcoeff(1/(1-serreverse(x/(1-log(1-x+x*O(x^n))))),n)}
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/* Using matrix log of Riordan array (A(x),x*A(x)): */
{a(n)=local(L=matrix(n+1,n+1,r,c,if(r>c,c)),M=sum(m=0,#L,L^m/m!));n!*M[n+1,1]} \\ Paul D. Hanna, Jul 16 2010
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/* From A(x) = (1 + x*A'(x)/A(x))*(1 - x*A(x))/(1-x): */
{a(n)=local(A=1+x);for(k=2,n,A=A-polcoeff((1+x*deriv(A)/A)*(1-x*A)/(1-x+x*O(x^n)),k)*x^k/(k-1));n!*polcoeff(A,n)} \\ Paul D. Hanna, Jul 16 2010
A375904
E.g.f. satisfies A(x) = (1 - log(1 - x * A(x)^(1/2)))^2.
Original entry on oeis.org
1, 2, 8, 52, 482, 5868, 88916, 1616032, 34300584, 833331600, 22815541872, 695254540704, 23343144376944, 856304206749600, 34078074674281920, 1462430515496217600, 67322828681189917440, 3309486192768294286848, 173034403244058605806080
Offset: 0
-
my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x/(1-log(1-x)))/x)^2))
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a(n) = 2*(n+1)!*sum(k=0, n, abs(stirling(n, k, 1))/(n-k+2)!);
A375905
E.g.f. satisfies A(x) = (1 - log(1 - x * A(x)^(1/3)))^3.
Original entry on oeis.org
1, 3, 15, 111, 1116, 14352, 226176, 4233492, 91936080, 2274815712, 63220205736, 1950659365608, 66187523184048, 2450020566119760, 98269427218682880, 4246150991775421824, 196657057172519603712, 9719485198364207149056, 510628699670802850684800
Offset: 0
-
my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x/(1-log(1-x)))/x)^3))
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a(n) = 3*(n+2)!*sum(k=0, n, abs(stirling(n, k, 1))/(n-k+3)!);
A377737
Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 - log(1-2*x) / 2) ).
Original entry on oeis.org
1, 1, 4, 32, 392, 6504, 136464, 3466224, 103425664, 3546396288, 137423600640, 5939224680960, 283254408582144, 14777481937449984, 837175325044101120, 51182161648716349440, 3358765321328869539840, 235492308312669671424000, 17568539556367396687183872
Offset: 0
-
my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1-log(1-2*x)/2))/x))
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a(n) = n!*sum(k=0, n, 2^(n-k)*abs(stirling(n, k, 1))/(n-k+1)!);
A377803
Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 - log(1-3*x) / 3) ).
Original entry on oeis.org
1, 1, 5, 51, 798, 16914, 453294, 14704542, 560274336, 24529661568, 1213557885720, 66956662956600, 4076440417367856, 271472012197547472, 19631093304600307152, 1531919987372848152240, 128314172533501646058240, 11482569303348317402868480, 1093343670892117401737893632
Offset: 0
-
my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1-log(1-3*x)/3))/x))
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a(n) = n!*sum(k=0, n, 3^(n-k)*abs(stirling(n, k, 1))/(n-k+1)!);