A179424 -id:A179424 - cp's OEIS frontend

A179420 E.g.f. A(x) satisfies: A(A(x)) = x*A'(x) with A(0)=0, A'(0)=1.

0, 1, 2, 12, 132, 2200, 50280, 1482768, 54171376, 2381590944, 123292821600, 7390709937600, 506182300962624, 39180896544097152, 3396777800819754624, 327323946734658720000, 34831825328790915321600

Offset: 0

Paul D. Hanna, Jul 13 2010

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2200*x^5/5! +...
E.g.f. satisfies: A(A(x)) = x*A'(x) where:
A'(x) = 1 + 2*x + 12*x^2/2! + 132*x^3/3! + 2200*x^4/4! +...
A(A(x)) = x + 4*x^2/2! + 36*x^3/3! + 528*x^4/4! + 11000*x^5/5! +...
Related expansions begin:
A*Dx(A)/2! = 2*x^2/2! + 15*x^3/3! + 180*x^4/4! + 3150*x^5/5! +...
A*Dx(A*Dx(A))/3! = 6*x^3/3! + 104*x^4/4! + 2140*x^5/5! +...
A*Dx(A*Dx(A*Dx(A)))/4! = 24*x^4/4! + 770*x^5/5! + 24600*x^6/6! +...
A*Dx(A*Dx(A*Dx(A*Dx(A))))/5! = 120*x^5/5! + 6264*x^6/6! +...
which generate iterations of A=A(x) as illustrated by:
A(A(x))/x = 1 + 2*A + 2^2*A*Dx(A)/2! + 2^3*A*Dx(A*Dx(A))/3! +...
A(A(A(x)))/x = 1 + 3*A + 3^2*A*Dx(A)/2! + 3^3*A*Dx(A*Dx(A))/3! +...
A_{-1}(x)/x = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! +-...(inverse).
Illustrate a main property of the iterations A_n(x) of A(x) by:
A(x) = A(A(x)) * A(x)/[x*d/dx A(x)];
A(x) = A_3(x) * A_2(x)/[x*d/dx A_2(x)];
A(x) = A_4(x) * A_3(x)/[x*d/dx A_3(x)]; ...
which can be shown consistent by the chain rule of differentiation.
...
The RIORDAN ARRAY (A(x)/x, A(x)) begins:
. 1;
. 1, 1;
. 4/2!, 2, 1;
. 33/3!, 10/2!, 3, 1;
. 440/4!, 90/3!, 18/2!, 4, 1;
. 8380/5!, 1240/4!, 177/3!, 28/2!, 5, 1;
. 211824/6!, 23800/5!, 2544/4!, 300/3!, 40/2!, 6, 1;
. 6771422/7!, 598788/6!, 49680/5!, 4520/4!, 465/3!, 54/2!, 7, 1; ...
where the e.g.f. of column k = A(x)^(k+1)/x for k>=0.
...
The MATRIX LOG of the above Riordan array (A(x)/x, A(x)) begins:
. 0;
. 1, 0;
. 2/2!, 2, 0;
. 12/3!, 4/2!, 3, 0;
. 132/4!, 24/3!, 6/2!, 4, 0;
. 2200/5!, 264/4!, 36/3!, 8/2!, 5, 0;
. 50280/6!, 4400/5!, 396/4!, 48/3!, 10/2!, 6, 0;
. 1482768/7!, 100560/6!, 6600/5!, 528/4!, 60/3!, 12/2!, 7, 0; ...
where the e.g.f. of column k = (k+1)*A(x) for k>=0.

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A193202, A179421, A179422, A179423, A179424, variant: A179320.

a(n)/n! = A221019(n)/A221020(n).

a[n_] := a[n] = Module[{A}, A[x_] = x+x^2+Sum[a[m]*x^m/m!, {m, 3, n-1}]; If[n<3, n!*Coefficient[A[x], x, n], n!*Coefficient[A[A[x]], x, n]/(n-2)] ]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jan 15 2018, translated from PARI *)

Co(n, k, F):=if k=1  then F(n) else sum(F(i+1)*Co(n-i-1, k-1, F), i, 0, n-k);
a(n):=if n=0 then 0 else if n<3 then 1 else sum(Co(n,k,a)*a(k),k,2,n-1)/(n-2); /* Vladimir Kruchinin, Jun 29 2011 */

{a(n)=local(A=x+x^2+sum(m=3,n-1,a(m)*x^m/m!)+x*O(x^n));if(n<3,n!*polcoeff(A,n),n!*polcoeff(subst(A,x,A),n)/(n-2))}

E.g.f. A(x) equals the e.g.f. of column 0 in the matrix log of the Riordan array (A(x)/x, A(x)).
Let A_n(x) denote the n-th iteration of e.g.f. A(x) with A_0(x)=x,
then A=A(x) satisfies:
A(x)/x = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +...
A_{-1}(x)/x = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+...
A_n(x)/x = 1 + n*A + n^2*A*Dx(A)/2! + n^3*A*Dx(A*Dx(A))/3! + n^4*A*Dx(A*Dx(A*Dx(A)))/4! +...
where Dx(F) = d/dx(x*F).
Further, we have: A(x) = A_{n+1}(x) * A_n(x)/[x*d/dx A_n(x)] which holds for all n.
a(n)=sum(k=2..n-1, R(n-1,k-1)*a(k))/(n-2), n>2, a(1)=1, a(2)=1, where R is the Riordan array (A(x)/x, A(x)). [Vladimir Kruchinin, Jun 29 2011]
E.g.f. satisfies: A(x) = Series_Reversion(-G(-x)) where G(x) is the e.g.f. of A193202 and satisfies: G(G(x)) = x*G'(G(x)). [Paul D. Hanna, Jul 22 2011]

A179421 E.g.f. A(x) satisfies: xA(x) equals column 0 in the matrix log of the Riordan array (A(x), xA(x)).

Original entry on oeis.org

1, 1, 4, 33, 440, 8380, 211824, 6771422, 264621216, 12329282160, 671882721600, 42181858413552, 3013915118776704, 242626985772839616, 21821596448977248000, 2176989083049432207600, 239420370429753669425664

Offset: 0

Paul D. Hanna, Jul 15 2010

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 440*x^4/4! +...
x*A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2200*x^5/5! +...
...
The RIORDAN ARRAY (A(x), x*A(x)) begins:
. 1;
. 1, 1;
. 4/2!, 2, 1;
. 33/3!, 10/2!, 3, 1;
. 440/4!, 90/3!, 18/2!, 4, 1;
. 8380/5!, 1240/4!, 177/3!, 28/2!, 5, 1;
. 211824/6!, 23800/5!, 2544/4!, 300/3!, 40/2!, 6, 1;
. 6771422/7!, 598788/6!, 49680/5!, 4520/4!, 465/3!, 54/2!, 7, 1; ...
where the e.g.f. of column k = A(x)^(k+1) for k>=0.
...
The MATRIX LOG of the above Riordan array (A(x), x*A(x)) begins:
. 0;
. 1, 0;
. 2/2!, 2, 0;
. 12/3!, 4/2!, 3, 0;
. 132/4!, 24/3!, 6/2!, 4, 0;
. 2200/5!, 264/4!, 36/3!, 8/2!, 5, 0;
. 50280/6!, 4400/5!, 396/4!, 48/3!, 10/2!, 6, 0;
. 1482768/7!, 100560/6!, 6600/5!, 528/4!, 60/3!, 12/2!, 7, 0; ...
where the e.g.f. of column k = (k+1)*x*A(x) for k>=0.

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A179420, A179422, A179423, A179424.

{a(n)=local(A=1+x+sum(m=2,n-1,a(m)*x^m/m!)+x*O(x^n));if(n<2,n!*polcoeff(A,n),n!*polcoeff(subst(x*A,x,x*A)/x,n)/(n-1))}

a(n) = A179420(n+1)/(n+1).
E.g.f. A(x) satisfies: A(x) = x*A'(x)/(A(x*A(x)) - 1).
Let L_{n}(x) denote the n-th iteration of L=L(x)=x*A(x) then:
. L(L(x)) = x*L'(x),
. L_{n+1}(x) = L(x)*x*[d/dx L_n(x)]/L_n(x),
. L(x)/x = 1 + L + L*Dx(L)/2! + L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! +...
. L_{n}(x)/x = 1 + n*L + n^2*L*Dx(L)/2! + n^3*L*Dx(L*Dx(L))/3! + n^4*L*Dx(L*Dx(L*Dx(L)))/4! +...
where Dx(F) = d/dx(x*F).

A179425 Number of ways to place 5 nonattacking kings on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 10, 14940, 229908, 1678336, 8155404, 30614620, 96011322, 263506752, 652150382, 1485650012, 3161648520, 6355083264, 12167739256, 22339050588, 39536586430, 67748508480, 112804636266, 183057635420, 290261282204, 450688785408, 686540794500, 1027700020828, 1513897376994, 2197363228480, 3146046781446, 4447496831580

Offset: 1

Vaclav Kotesovec, Jan 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A061998, A179403, A179404, A179424.

CoefficientList[Series[- 2 x^4 (260 x^11 - 1932 x^10 + 6567 x^9 - 16223 x^8 + 38507 x^7 - 77869 x^6 + 102208 x^5 - 61576 x^4 - 15301 x^3 + 33059 x^2 + 7415 x + 5) / (x - 1)^11, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 01 2013 *)

Explicit formula: a(n) = 1/120*n^2*(n^8-90n^6+3155n^4-51450n^2+332544), n>=6.

G.f.: -2x^5*(260x^11 - 1932x^10 + 6567x^9 - 16223x^8 + 38507x^7 - 77869x^6 + 102208x^5 - 61576x^4 - 15301x^3 + 33059x^2 + 7415x + 5)/(x-1)^11.

A179426 Number of ways to place 6 nonattacking kings on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 0, 10596, 486668, 7063520, 55345356, 299491100, 1263811604, 4455716184, 13701863604, 37823872044, 95648273100, 224887404416, 497181121100, 1042609380588, 2088337713332, 4017815773400, 7459198321428, 13414493857116, 23444476061772, 39928736913120, 66425550447500, 108162598959740, 172697249542932, 270794133842456, 417578468928308, 634036069773900

Offset: 1

Vaclav Kotesovec, Jan 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A172158, A179403, A179404, A179424, A179425.

CoefficientList[Series[4 x^5 (426 x^13 - 4263 x^12 + 22311 x^11 - 82449 x^10 + 220918 x^9 - 391803 x^8 + 369356 x^7 + 10716 x^6 - 382230 x^5 + 163719 x^4 + 387689 x^3 - 390831 x^2 - 87230 x - 2649) / (x - 1)^13, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 01 2013 *)

a(n) = 1/720*n^2*(n^10 -135*n^8 +7525*n^6 -217665*n^4 +3289354*n^2 -20949480), n>=7.

G.f.: 4*x^6*(426*x^13 - 4263*x^12 + 22311*x^11 - 82449*x^10 + 220918*x^9 - 391803*x^8 + 369356*x^7 + 10716*x^6 - 382230*x^5 + 163719*x^4 + 387689*x^3 - 390831*x^2 - 87230*x - 2649)/(x-1)^13.

A179427 Number of ways to place 7 nonattacking kings on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 0, 3420, 576856, 19760512, 270487188, 2209065700, 12914201256, 59659859232, 231216019632, 781647658596, 2367858314700, 6553746728448, 16815788711212, 40446802230372, 92003239814224, 199311860224800, 413589922308360, 825997764087012, 1594007700404532, 2982430581363072, 5425904270482500, 9622254525739492, 16669554533555832, 28264133502586912, 46982453295836640, 76676963241363300

Offset: 1

Vaclav Kotesovec, Jan 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (15, -105, 455, -1365, 3003, -5005, 6435, -6435, 5005, -3003, 1365, -455, 105, -15, 1).

Cf. A179403, A179404, A179424, A179425, A179426.

CoefficientList[Series[- 4 x^5 (1379 x^16 - 18219 x^15 + 124755 x^14 - 553765 x^13 + 1657983 x^12 - 3369984 x^11 + 4870575 x^10 - 6400905 x^9 + 10992208 x^8 - 19069951 x^7 + 21246441 x^6 - 8631071 x^5 - 7797385 x^4 + 8273322 x^3 + 2866693 x^2 + 131389 x + 855) / (x - 1)^15, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 01 2013 *)

Explicit formula: a(n) = 1/5040*n^2*(n^12 -189*n^10 +15295*n^8 -681135*n^6 +17692024*n^4 -255655596*n^2 +1617230880), n>=8.

G.f.: -4*x^6*(1379*x^16 - 18219*x^15 + 124755*x^14 - 553765*x^13 + 1657983*x^12 - 3369984*x^11 + 4870575*x^10 - 6400905*x^9 + 10992208*x^8 - 19069951*x^7 + 21246441*x^6 - 8631071*x^5 - 7797385*x^4 + 8273322*x^3 + 2866693*x^2 + 131389*x + 855)/(x-1)^15.

A179428 Number of ways to place 8 nonattacking kings on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 0, 486, 346381, 36285336, 956078397, 12428297150, 104000525596, 643409498286, 3191250652226, 13361641961066, 48905750870775, 160414160371552, 480243686391743, 1330654487994234, 3449609146025210, 8439769551278350, 19624142987739108, 43616849672119790, 93112709811981557, 191696927842663704, 381920049400830625, 738532765420347014, 1389708580432837752, 2550402748009811870, 4573836436177381798, 8029626473495462850

Offset: 1

Vaclav Kotesovec, Jan 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Cf. A179403, A179404, A179424, A179425, A179426, A179427.

CoefficientList[Series[x^5 (17728 x^19 - 301964 x^18 + 2573500 x^17 - 13833040 x^16 + 51521058 x^15 - 143708688 x^14 + 325486412 x^13 - 629393865 x^12 + 996601251 x^11 - 1090603627 x^10 + 426710617 x^9 + 807953488 x^8 - 1328885640 x^7 + 262625618 x^6 + 1106513030 x^5 - 875387697 x^4 - 386005021 x^3 - 30462955 x^2 - 338119 x - 486) / (x - 1)^17, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 01 2013 *)

a(n) = 1/40320*n^2 * (n^14 -252*n^12 +27874*n^10 -1759800*n^8 +68745649*n^6 -1669136028*n^4 +23447322156*n^2 -147931524720), n>=9.

G.f.: x^6*(17728x^19 - 301964x^18 + 2573500x^17 - 13833040x^16 + 51521058x^15 - 143708688x^14 + 325486412x^13 - 629393865x^12 + 996601251x^11 - 1090603627x^10 + 426710617x^9 + 807953488x^8 - 1328885640x^7 + 262625618x^6 + 1106513030x^5 - 875387697x^4 - 386005021x^3 - 30462955x^2 - 338119x - 486)/(x-1)^17.

A180067 Number of ways to place 9 nonattacking kings on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 0, 28, 81095, 42752576, 2436444603, 53633024900, 666519047964, 5655962632720, 36502953719310, 191587564345044, 854990702601025, 3346890268570368, 11756179090049177, 37692541754516628, 111774885566128630, 309788198526691600

Offset: 1

Vaclav Kotesovec, Jan 15 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).

Cf. A179403, A179404, A179424, A179425, A179426, A179427, A179428.

CoefficientList[Series[- x^5 (56520 x^22 - 1215064 x^21 + 12642984 x^20 - 82438064 x^19 + 378510176 x^18 - 1315100032 x^17 + 3593010018 x^16 - 7742517098 x^15 + 12798616135 x^14 - 15614945085 x^13 + 14742135008 x^12 - 17197088896 x^11 + 33440162097 x^10 - 55183782403 x^9 + 50601858342 x^8 - 7249042450 x^7 - 32800069391 x^6 + 23010354469 x^5 + 14572795412 x^4 + 1637985772 x^3 + 41216559 x^2 + 80563 x + 28) / (x - 1)^19, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 01 2013 *)

a(n) = 1/362880*n^2 * (n^16 -324*n^14 +46914*n^12 -3975048*n^10 +216203169*n^8 -7756575876*n^6 +179987135516*n^4 -2481599151792*n^2 +15651056776320), n>=10.
G.f.: -x^6*(56520x^22 - 1215064x^21 + 12642984x^20 - 82438064x^19 + 378510176x^18 - 1315100032x^17 + 3593010018x^16 - 7742517098x^15 + 12798616135x^14 - 15614945085x^13 + 14742135008x^12 - 17197088896x^11 + 33440162097x^10 - 55183782403x^9 + 50601858342x^8 - 7249042450x^7 - 32800069391x^6 + 23010354469x^5 + 14572795412x^4 + 1637985772x^3 + 41216559x^2 + 80563x + 28)/(x-1)^19.
General asymptotic formula for number of ways to place k nonattacking kings on an n X n toroidal board: n^2k/k! - 9/2*n^(2k-2)/(k-2)! + (243k+47)*n^(2k-4)/(24*(k-3)!) - (243k^2+141k+80)*n^(2k-6)/(16*(k-4)!) + (98415k^3+114210k^2+140645k+101762)*n^(2k-8)/(5760*(k-5)!)-...

A194652 Number of ways to place 4 nonattacking kings on an n X n cylindrical chessboard.

Original entry on oeis.org

0, 0, 0, 32, 1205, 13260, 74494, 291708, 908973, 2416410, 5711530, 12327414, 24743693, 46797968, 84216990, 145288600, 241697109, 389546478, 610597338, 933745570, 1396771845, 2048393204, 2950649438, 4181658708, 5838778525, 8042209890, 10939084074, 14708073198

Offset: 1

Vaclav Kotesovec, Aug 31 2011

nonn

V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A061997, A179424, A194650, A194651.

CoefficientList[Series[x^3*(54*x^9 - 384*x^8 + 1052*x^7 - 1263*x^6 + 657*x^5 - 1434*x^4 + 4154*x^3 - 3567*x^2 - 917*x - 32)/(x - 1)^9, {x, 0, 30}], x] (* Wesley Ivan Hurt, Dec 27 2023 *)

a(n) = 1/24*n*(n^7 - 54*n^5 + 36*n^4 + 1019*n^3 - 1236*n^2 - 6690*n + 10884), n>=5.

G.f.: x^4*(54*x^9 - 384*x^8 + 1052*x^7 - 1263*x^6 + 657*x^5 - 1434*x^4 + 4154*x^3 - 3567*x^2 - 917*x - 32)/(x-1)^9.

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

Paul D. Hanna, May 15 2010

nonn

			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)

Cf. A177380, A138013, A226572.

Cf. A179424. [From Paul D. Hanna, Jul 16 2010]

CoefficientList[1/(1-InverseSeries[Series[x/(1-Log[1-x]),{x,0,20}],x]),x]*Range[0,20]! (* Vaclav Kotesovec, Jan 11 2014 *)

{a(n)=n!*polcoeff(1/(1-serreverse(x/(1-log(1-x+x*O(x^n))))),n)}

/* 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

/* 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

E.g.f.: A(x) = 1/(1 - Series_Reversion(x/(1 - log(1-x)))).
...
Let G(x) = e.g.f. of A138013, then:
. A(x) = exp(G(x) - 1),
. A(x) = 1/(1 - x*G(x))
where G(x) = 1 - log(1 - x*G(x)).
...
Let F(x) = e.g.f. of A177380, then:
. [x^n] A(x)^(-n+1)/(-n+1) = A177380(n)/n! for n>1,
. [x^n] F(x)^(n+1)/(n+1) = a(n)/n! for n>=0,
. A(x) = F(x*A(x)) and A(x/F(x)) = F(x),
. A(x) = (1/x)*Series_Reversion(x/F(x))
where F(x) = 1+x + x*log(F(x)).
Contribution from Paul D. Hanna, Jul 16 2010: (Start)
E.g.f. satisfies: A(x) = (1 + x*A'(x)/A(x))*(1 - x*A(x))/(1-x).
...
Let A_n(x) denote the n-th iteration of x*A(x) with G = x/(1-x), then:
. A(x) = 1 + G + G*Dx(G)/2! + G*Dx(G*Dx(G))/3! + G*Dx(G*Dx(G*Dx(G)))/4! +...
. A_n(x)/x = 1 + n*G + n^2*G*Dx(G)/2! + n^3*G*Dx(G*Dx(G))/3! + n^4*G*Dx(G*Dx(G*Dx(G)))/4! +...
where Dx(F) = d/dx(x*F).
...
Given e.g.f. A(x), the matrix log of the Riordan array (A(x),x*A(x)) equals the matrix L defined by L(n,k)=k+1 and L(n,n)=0, for n>=0, n>k.
(End)
a(n) ~ sqrt(s-1) * n^(n-1) * s^(n+1) / exp(n), where s = -LambertW(-1,-exp(-2)) = 3.14619322062... (see A226572). - Vaclav Kotesovec, Jan 11 2014

cp's OEIS Frontend

A179420 E.g.f. A(x) satisfies: A(A(x)) = x*A'(x) with A(0)=0, A'(0)=1.

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A179421 E.g.f. A(x) satisfies: x*A(x) equals column 0 in the matrix log of the Riordan array (A(x), x*A(x)).

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A179421 E.g.f. A(x) satisfies: xA(x) equals column 0 in the matrix log of the Riordan array (A(x), xA(x)).