A138909 -id:A138909 - cp's OEIS frontend

A138910 Inverse binomial transform of A138909.

1, 1, 3, 20, 129, 1164, 12265, 151458, 2136337, 33901640, 597761361, 11593851510, 245310524041, 5622982528188, 138803996674057, 3671135646515834, 103568483199034785, 3104443346427521808, 98528857134710517793

Offset: 0

Paul D. Hanna, Apr 05 2008

nonn

Cf. A138909.

Range[0,19]! CoefficientList[Series[(x + 1) / (Exp[x] - x Exp[2 x]), {x, 0, 19}], x] (* Vincenzo Librandi, Nov 07 2016 *)

{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-1)*x+x*O(x^k)))/(1+(k-1)*x),k));A[n+1]}

O.g.f. satisfies: [x^n] A( x/(1+(n-1)*x) )/(1+(n-1)*x) = n! for n>=0.
E.g.f. satisfies: [x^n] A(x)*exp(-(n-1)*x) = 1 for n>=0.
E.g.f.: (x+1)/(exp(x)-x*exp(2*x)). - Vladimir Kruchinin, Nov 07 2016
a(n) ~ n! / LambertW(1)^(n-1). - Vaclav Kotesovec, Oct 30 2017

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

Original entry on oeis.org

1, 2, 7, 52, 541, 7446, 127939, 2641192, 63746169, 1762380010, 54938528191, 1906911695580, 72949449568021, 3049813346508670, 138352912908850683, 6769028553912294736, 355311287187804226033, 19918243846821103623378

Offset: 0

Paul D. Hanna, Apr 05 2008

nonn

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

			If the successive inverse binomial transforms are placed in a table,
then we see that the diagonal consists of terms (n+1)^(n-1):
n=0:[(1),2,7,52,541,7446,127939,2641192,63746169,1762380010,...];
n=1:[1,(1),4,36,368,5200,90432,1884736,45817088,1273874688,...];
n=2:[1, 0,(3),26,245,3684,64087,1349214,33003945,922386824,...];
n=3:[1,-1, 4,(16),160,2688,45184,970240,23814144,668975104,...];
n=4:[1,-2,7, 0,(125),2002,31203,705268,17177273,486100710,...];
n=5:[1,-3,12,-28, 176,(1296),21184,524352,12305664,354510080,...];
n=6:[1,-4,19,-74,373, 0,(16807),395866,8645673,260994628,...];
n=7:[1,-5,28,-144,800,-2816, 24192,(262144),5980160,195969024,...];
n=8:[1,-6,39,-244,1565,-8562,56419, 0,(4782969),149083874,...];
n=9:[1,-7,52,-380,2800,-19248,136768,-638912, 6966528,(100000000),..];
n=10:[1,-8,67,-558,4661,-37604,302679,-2112938,17204009, 0,...].
Notice the occurrence of zeros in the secondary diagonal = A138734.

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

Cf. A138736 (inverse binomial transform), A138734; variants: A138909, A138911.

{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*x+x*O(x^k)))/(1+k*x))[k+1]);A[n+1]}

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

cp's OEIS Frontend

A138910 Inverse binomial transform of A138909.

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