A143768 -id:A143768 - cp's OEIS frontend

A088695 E.g.f. satisfies A(x) = f(x*A(x)), where f(x) = exp(x+x^2).

1, 1, 5, 40, 485, 7776, 156457, 3788800, 107414505, 3491200000, 128019454541, 5229222395904, 235490648957005, 11592449531084800, 619331166211640625, 35691050995648823296, 2206955604752999720273, 145757527499874820423680, 10240455593560436925898645

Offset: 0

Paul D. Hanna, Oct 07 2003

nonn

Radius of convergence of A(x): r = (1/2)*exp(-3/4) = 0.23618..., where A(r) = exp(3/4) and r = limit a(n)/a(n+1)*(n+1) as n->infinity. Radius of convergence is from a general formula yet unproved.

Seiichi Manyama, Table of n, a(n) for n = 0..364
Index entries for reversions of series

Cf. A143768, A362771.

Cf. A365031, A365032.

Table[n!*SeriesCoefficient[(E^(x+x^2))^(n+1)/(n+1),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 24 2014 *)

a(n)=n!*polcoeff(exp(x+x^2)^(n+1)+x*O(x^n),n,x)/(n+1)

a(n) = n! * [x^n] exp(x+x^2)^(n+1)/(n+1).
a(n) = n! * Sum_{k=floor(n/2)..n} binomial(k,n-k)*(n+1)^(k-1)/k!. - Vladimir Kruchinin, Aug 04 2011
a(n) ~ 2^(n+1/2) * n^(n-1) / (sqrt(3) * exp(n/4 - 3/4)). - Vaclav Kotesovec, Jan 24 2014
E.g.f.: (1/x) * Series_Reversion( x*exp(-x*(1 + x)) ). - Seiichi Manyama, Sep 23 2024

A362694 E.g.f. satisfies A(x) = exp(x + x * A(x)^2).

Original entry on oeis.org

1, 2, 12, 152, 2960, 78112, 2607808, 105432448, 5008584960, 273482293760, 16878251101184, 1161918967060480, 88277165100666880, 7337286679766179840, 662287143981044121600, 64516370031367063175168, 6746443728505612426870784, 753763691778003738319519744

Offset: 0

Seiichi Manyama, May 01 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..339
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A349562, A362693, A362734, A362735.

Cf. A143768, A202617.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-2*x*exp(2*x))/2)))

E.g.f.: sqrt( -LambertW(-2*x*exp(2*x)) / (2*x) ) = exp( x - LambertW(-2*x*exp(2*x))/2 ).
a(n) = Sum_{k=0..n} (2*k+1)^(n-1) * binomial(n,k) = 2^n * A202617(n).
a(n) ~ sqrt(1 + 1/LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (exp(n) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Nov 10 2023

A362474 E.g.f. satisfies A(x) = exp(x + x^2/2 * A(x)^2).

Original entry on oeis.org

1, 1, 2, 10, 70, 646, 7576, 106744, 1761628, 33361948, 712950616, 16976294776, 445751093800, 12795850109992, 398697898011232, 13401365473319776, 483376669737381136, 18623161719254837008, 763300232417720682784, 33163224556779213475744

Offset: 0

Seiichi Manyama, Apr 21 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..392
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=1 of A362483.
Cf. A362478, A362491.
Cf. A143768, A362475.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x + x^2/2*A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^2*exp(2*x))/2)))

E.g.f.: exp(x - LambertW(-x^2 * exp(2*x))/2) = sqrt(-LambertW(-x^2*exp(2*x))/x^2).
a(n) = n! * Sum_{k=0..floor(n/2)} (1/2)^k * (2*k+1)^(n-k-1) / (k! * (n-2*k)!).
a(n) ~ sqrt(1 + LambertW(exp(-1/2))) * n^(n-1) / (sqrt(2) * exp(n) * LambertW(exp(-1/2))^(n+1)). - Vaclav Kotesovec, Nov 10 2023

A362472 E.g.f. satisfies A(x) = exp(x + x^3 * A(x)^3).

Original entry on oeis.org

1, 1, 1, 7, 97, 961, 10201, 177241, 3801505, 80718625, 1887205681, 52896262321, 1648697978401, 54216677033377, 1928791931034697, 75326014326206281, 3159713152034201281, 140373558362282197441, 6632746205445950124385, 333591744669464008432225

Offset: 0

Seiichi Manyama, Apr 21 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..384
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=6 of A362490.
Cf. A143768, A349562, A362473.
Cf. A362392, A362481.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-3*x^3*exp(3*x))/3)))

E.g.f.: exp(x - LambertW(-3*x^3 * exp(3*x))/3) = ( -LambertW(-3*x^3 * exp(3*x))/(3*x^3) )^(1/3).

a(n) = n! * Sum_{k=0..floor(n/3)} (3*k+1)^(n-2*k-1) / (k! * (n-3*k)!).

A363354 E.g.f. satisfies A(x) = exp(x * (1 + x * A(x)^3)).

Original entry on oeis.org

1, 1, 3, 25, 277, 4221, 81421, 1891429, 51638217, 1618907257, 57332786041, 2264047223241, 98641443498973, 4700569138096885, 243213757144477029, 13579261873673960941, 813757288951509415441, 52098716516012891238129, 3548972379593741013388657

Offset: 0

Seiichi Manyama, Aug 17 2023

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A125500, A143768, A363529.

Cf. A212917.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-3*x^2*exp(3*x))/3)))

E.g.f.: exp( x - LambertW(-3*x^2*exp(3*x))/3 ).

a(n) = n! * Sum_{k=0..n} (3*n-3*k+1)^(k-1) * binomial(k,n-k)/k!.

A363529 E.g.f. satisfies A(x) = exp(x * (1 + x * A(x)^4)).

Original entry on oeis.org

1, 1, 3, 31, 409, 7361, 170251, 4732351, 154694961, 5814634753, 246946119571, 11698927124831, 611660759515081, 34984757221103041, 2173041881789331099, 145669007565799127551, 10482025117382045382241, 805892200757926620144641

Offset: 0

Seiichi Manyama, Aug 17 2023

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A125500, A143768, A363354.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-4*x^2*exp(4*x))/4)))

E.g.f.: exp( x - LambertW(-4*x^2*exp(4*x))/4 ).

a(n) = n! * Sum_{k=0..n} (4*n-4*k+1)^(k-1) * binomial(k,n-k)/k!.

A362473 E.g.f. satisfies A(x) = exp(x + x^4 * A(x)^4).

Original entry on oeis.org

1, 1, 1, 1, 25, 601, 9001, 105001, 1231441, 24146641, 740098801, 22443260401, 607394284201, 16102368745321, 497289446373721, 19072987370400601, 806135144596672801, 33945128330918599201, 1426006261391514829921, 63478993000497055809121

Offset: 0

Seiichi Manyama, Apr 21 2023

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A143768, A349562, A362472.

Cf. A362393, A362482, A362491.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-4*x^4*exp(4*x))/4)))

E.g.f.: exp(x - LambertW(-4*x^4 * exp(4*x))/4) = ( -LambertW(-4*x^4 * exp(4*x))/(4*x^4) )^(1/4).

a(n) = n! * Sum_{k=0..floor(n/4)} (4*k+1)^(n-3*k-1) / (k! * (n-4*k)!).

A362475 E.g.f. satisfies A(x) = exp(x + 3x^2/2 A(x)^2).

Original entry on oeis.org

1, 1, 4, 28, 298, 4186, 74116, 1578340, 39394972, 1127378332, 36411516496, 1310173698736, 51982859674648, 2254757407407064, 106150698182657584, 5390926011965379376, 293782337188718257936, 17100576708082841577232, 1058920120014192744673600

Offset: 0

Seiichi Manyama, Apr 21 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..370
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=3 of A362483.
Cf. A143768, A362474.
Cf. A362380.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-3*x^2*exp(2*x))/2)))

E.g.f.: exp(x - LambertW(-3*x^2 * exp(2*x))/2) = sqrt( -LambertW(-3*x^2 * exp(2*x))/(3*x^2) ).

a(n) = n! * Sum_{k=0..floor(n/2)} (3/2)^k * (2*k+1)^(n-k-1) / (k! * (n-2*k)!).

A362480 E.g.f. satisfies A(x) = exp(x - x^2 * A(x)^2).

Original entry on oeis.org

1, 1, -1, -17, -47, 961, 14191, -35825, -4258463, -46744703, 1252890271, 49630926511, 61171154353, -41944148256191, -1033550755723121, 24977027757497551, 2117415434541888961, 20487158235798909697, -3240242006475108681665, -146763820123398901335185

Offset: 0

Seiichi Manyama, Apr 21 2023

sign

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362481, A362482.

Cf. A143768, A362492.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(2*x^2*exp(2*x))/2)))

E.g.f.: exp(x - LambertW(2*x^2 * exp(2*x))/2) = sqrt( LambertW(2*x^2 * exp(2*x))/(2*x^2) ).

a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * (2*k+1)^(n-k-1) / (k! * (n-2*k)!).

A362483 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (k/2)^j * (2j+1)^(n-j-1) / (j! (n-2*j)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 10, 1, 1, 1, 4, 19, 70, 1, 1, 1, 5, 28, 169, 646, 1, 1, 1, 6, 37, 298, 2041, 7576, 1, 1, 1, 7, 46, 457, 4186, 30811, 106744, 1, 1, 1, 8, 55, 646, 7081, 74116, 560827, 1761628, 1, 1, 1, 9, 64, 865, 10726, 141901, 1578340, 11957905, 33361948, 1

Offset: 0

Seiichi Manyama, Apr 21 2023

			Square array begins:
  1,   1,    1,    1,    1,     1, ...
  1,   1,    1,    1,    1,     1, ...
  1,   2,    3,    4,    5,     6, ...
  1,  10,   19,   28,   37,    46, ...
  1,  70,  169,  298,  457,   646, ...
  1, 646, 2041, 4186, 7081, 10726, ...

Eric Weisstein's World of Mathematics, Lambert W-Function.

Columns k=0..3 give A000012, A362474, A143768, A362475.

Cf. A362377, A362490.

T(n, k) = n! * sum(j=0, n\2, (k/2)^j*(2*j+1)^(n-j-1)/(j!*(n-2*j)!));

E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x + k*x^2/2 * A_k(x)^2).
A_k(x) = exp(x - LambertW(-k*x^2 * exp(2*x))/2).
A_k(x) = sqrt( -LambertW(-k*x^2 * exp(2*x))/(k*x^2) ) for k > 0.

cp's OEIS Frontend

A088695 E.g.f. satisfies A(x) = f(x*A(x)), where f(x) = exp(x+x^2).

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A362694 E.g.f. satisfies A(x) = exp(x + x * A(x)^2).

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A362474 E.g.f. satisfies A(x) = exp(x + x^2/2 * A(x)^2).

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A362472 E.g.f. satisfies A(x) = exp(x + x^3 * A(x)^3).

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A363354 E.g.f. satisfies A(x) = exp(x * (1 + x * A(x)^3)).

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A363529 E.g.f. satisfies A(x) = exp(x * (1 + x * A(x)^4)).

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A362473 E.g.f. satisfies A(x) = exp(x + x^4 * A(x)^4).

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A362475 E.g.f. satisfies A(x) = exp(x + 3*x^2/2 * A(x)^2).

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A362480 E.g.f. satisfies A(x) = exp(x - x^2 * A(x)^2).

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A362475 E.g.f. satisfies A(x) = exp(x + 3x^2/2 A(x)^2).

A362483 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (k/2)^j * (2j+1)^(n-j-1) / (j! (n-2*j)!).