A138737 -id:A138737 - cp's OEIS frontend

A138734 a(n) = n-th term of the (n+1)-th inverse binomial transform of A138737 for n>=0.

1, 0, 4, 0, 176, 0, 24192, 0, 6966528, 0, 3459768320, 0, 2642804895744, 0, 2880385956446208, 0, 4248075487709560832, 0, 8150896954121845211136, 0, 19738484719911211593891840, 0, 58891248341123034677673721856, 0

Offset: 0

Paul D. Hanna, Apr 05 2008

nonn

The n-th term of the n-th inverse binomial transform of A138737 = (n+1)^(n-1) for n>=0.

			E.g.f.: A(x) = 1 + 4*x^2/2! + 176*x^4/4! + 24192*x^6/6! + 6966528*x^8/8! +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A138737, A215880.

CoefficientList[Series[-LambertW[x]*LambertW[-x]/x^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)

{a(n)=local(A=[1]);for(k=1,n,A=concat(A,0); A[k+1]=(k+1)^(k-1)-polcoeff(subst(Ser(A),x,x/(1+k*x+x*O(x^k)))/(1+k*x),k)); polcoeff(subst(Ser(A),x,x/(1+(n+1)*x+x*O(x^n)))/(1+(n+1)*x),n)}

{a(n)=sum(k=0, n, (-1)^k*binomial(n,k)*(k+1)^(k-1)*(n-k+1)^(n-k-1))}

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

{a(n)=n!*polcoeff(exp(sum(m=1,n,2*(2*m)^(2*m-1)*x^(2*m)/(2*m)!)+x*O(x^n)),n)}

x='x+O('x^50); Vec(serlaplace(-lambertw(x)*lambertw(-x)/x^2)) \\ G. C. Greubel, Nov 15 2017

From Paul D. Hanna, Apr 20 2011: (Start)
a(n) = Sum_{k=0..n} (-1)^k*C(n,k) * (k+1)^(k-1) * (n-k+1)^(n-k-1).
E.g.f.: -LambertW(x)*LambertW(-x)/x^2.
E.g.f.: exp( Sum_{n>=1} 2*(2n)^(2n-1) * x^(2n)/(2n)! ). (End)
Let Chw(x) = Sum_{n>=0} (2*n+1)^(2*n-1) * x^(2*n)/(2*n)!
and Shw(x) = Sum_{n>=0} (2*n+2)^(2*n) * x^(2*n+1)/(2*n+1)!
then e.g.f. A(x) = Chw(x)^2 - Shw(x)^2. - Paul D. Hanna, Aug 22 2012
If n is even, a(n) ~ 2*exp(2) * LambertW(exp(-1)) * n^(n-1). - Vaclav Kotesovec, Oct 08 2013

A138736 Inverse binomial transform of A138737.

Original entry on oeis.org

1, 1, 4, 36, 368, 5200, 90432, 1884736, 45817088, 1273874688, 39891461120, 1389816423424, 53334303584256, 2235679577657344, 101651458028158976, 4983219643056537600, 262026143585449607168, 14711289584591513387008

Offset: 0

Paul D. Hanna, Apr 05 2008

nonn

The n-th term of the n-th inverse binomial transform of A138737 equals (n+1)^(n-1) for n>=0.

Related to LambertW(-x)/(-x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A138737.

{a(n)=local(A=[1]);for(k=1,n,A=concat(A,0); A[k+1]=(k+1)^(k-1)-Vec(subst(Ser(A),x,x/(1+(k-1)*x+x*O(x^k)))/(1+(k-1)*x))[k+1]);A[n+1]}

O.g.f. satisfies: [x^n] A( x/(1+(n-1)*x) )/(1+(n-1)*x) = (n+1)^(n-1) for n>=0.
E.g.f. satisfies: [x^n] A(x)*exp(-(n-1)*x) = (n+1)^(n-1)/n! for n>=0.
a(n) ~ (1 + LambertW(exp(-1)))^(3/2)*n^(n-1) / (exp(n-2)*LambertW(exp(-1))^(n-1)). - Vaclav Kotesovec, Oct 30 2017

A215882 Expansion of e.g.f.: -LambertW(-x) / LambertW(x).

Original entry on oeis.org

1, 2, 4, 26, 160, 2002, 21184, 395866, 5980160, 149083874, 2933576704, 91549564570, 2222207205376, 83345185392562, 2407376957456384, 105482963294851418, 3534260251308064768, 177194291803516980418, 6757029862401745616896, 381514700506253250858778

Offset: 0

Paul D. Hanna, Aug 25 2012

nonn

			E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 26*x^3/3! + 160*x^4/4! + 2002*x^5/5! +... such that A(x) = -LambertW(-x)/LambertW(x) where LambertW(x) = x - 2*x^2/2! + 9*x^3/3! - 64*x^4/4! + 625*x^5/5! - 7776*x^6/6! + 117649*x^7/7! - 2097152*x^8/8! +...+ (-n)^(n-1)*x^n/n! +... .
Related expansions: log(A(x)) = 2*x + 18*x^3/3! + 1250*x^5/5! + 235298*x^7/7! + 86093442*x^9/9! +...+ 2*(2*n+1)^(2*n)*x^(2*n+1)/(2*n+1)! +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A215880, A215881, A138737, A215890.

List([0..20],n->Sum([0..n],k->-(-1)^k*Binomial(n,k)*(k-1)^(k-1)*(n-k+1)^(n-k-1))); # Muniru A Asiru, Feb 20 2018

CoefficientList[Series[-LambertW[-x]/LambertW[x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)

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

{a(n)=n!*polcoeff(exp(sum(m=0, n, 2*(2*m+1)^(2*m)*x^(2*m+1)/ (2*m+1)!)+x*O(x^n)), n)}

{a(n)=sum(k=0, n, -(-1)^k*binomial(n, k)*(k-1)^(k-1)*(n-k+1)^(n-k-1))}
for(n=0,21,print1(a(n),", "))

x='x+O('x^30); Vec(serlaplace(-lambertw(-x)/lambertw(x))) \\ G. C. Greubel, Feb 19 2018

E.g.f.: exp( Sum_{n>=0} 2*(2*n+1)^(2*n) * x^(2*n+1)/(2*n+1)! ).
a(n) = Sum_{k=0..n} -(-1)^k*C(n,k) * (k-1)^(k-1) * (n-k+1)^(n-k-1).
a(n) ~ c * n^(n-1), where c = (1-LambertW(exp(-1))^2)/LambertW(exp(-1)) = 3.31265693390754834... if n is even and c = (1+LambertW(exp(-1))^2)/ LambertW(exp(-1)) = 3.86958601942969593... if n is odd. - Vaclav Kotesovec, Nov 27 2012

A138909 Expansion of e.g.f.: (1+x)/(1-x*exp(x)).

Original entry on oeis.org

1, 2, 6, 33, 232, 2045, 21636, 267043, 3766848, 59776137, 1053986500, 20442543671, 432537117552, 9914571191005, 244742376434388, 6473030199939675, 182614314495736576, 5473825396372806545, 173728330742517310980

Offset: 0

Paul D. Hanna, Apr 05 2008, Apr 06 2008

nonn

The n-th term of the n-th inverse binomial transform of this sequence equals n! for n>=0.

			If the successive inverse binomial transforms are placed in a table,
then we see that the diagonal consists of the factorials:
n=0:[(1),2,6,33,232,2045,21636,267043,3766848,59776137,1053986500,...];
n=1:[1,(1),3,20,129,1164,12265,151458,2136337,33901640,597761361,...];
n=2:[1,0,(2),13,64,693,6856,86175,1210896,19228825,339012304,...];
n=3:[1,-1,3,(6),25,482,3429,50908,678465,10937430,192150469,...];
n=4:[1,-2,6,-7,(24),381,844,36291,341728,6433865,107801436,...];
n=5:[1,-3,11,-32,97,(120),-839,37158,55953,4638052,54573025,...];
n=6:[1,-4,18,-75,304,-811,(720),40783,-262608,5542425,6069736,...];
n=7:[1,-5,27,-142,729,-3282,11941,(5040),-497279,9166130,...];
n=8:[1,-6,38,-239,1480,-8643,45844,-178557,(40320),12301705,...];
n=9:[1,-7,51,-372,2689,-18844,125289,-741974,3354513,(362880),...].

Cf. A006153.

Cf. A138910 (inverse binomial transform); variants: A138911, A138737.

With[{nn=20},CoefficientList[Series[(1+x)/(1-x*Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 09 2018 *)

{a(n)=local(A=[1]);for(k=1,n,A=concat(A,0); A[k+1]=k!-polcoeff(subst(Ser(A),x,x/(1+k*x+x*O(x^k)))/(1+k*x),k));A[n+1]}

{a(n)=n!+sum(k=0,n-1,k!*binomial(n,k)*n*k^(n-k-1))}

E.g.f. (1+x)/(1-x*exp(x)) - Olivier Gérard, Sep 15 2016
O.g.f. satisfies: [x^n] A( x/(1+n*x) )/(1+n*x) = n! for n>=0.
E.g.f. satisfies: [x^n] A(x)*exp(-n*x) = 1 for n>=0.
a(n) = n! + Sum_{k=0..n-1} k!*C(n,k)*n*k^(n-k-1) for n>1 with a(0)=1.
Equivalent to the sum above by properties of the binomial triangle:
a(n) = A006153(n)+n*A006153(n-1).
a(n) = n! ( Sum_{k=0..n-1} ((n-1-k)^k + (n-k)^k)/k!) for n>1 with a(0)=1.
a(n) ~ n! / LambertW(1)^n. - Vaclav Kotesovec, Oct 30 2017

Name change and e.g.f. by Olivier Gérard, Sep 15 2016

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A138734 a(n) = n-th term of the (n+1)-th inverse binomial transform of A138737 for n>=0.

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A138736 Inverse binomial transform of A138737.

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A215882 Expansion of e.g.f.: -LambertW(-x) / LambertW(x).

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A138909 Expansion of e.g.f.: (1+x)/(1-x*exp(x)).

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