A299044 G.f. Sum_{n>=0} Series_Reversion( x/(1+x)^n )^n.

1, 1, 2, 6, 25, 129, 784, 5472, 42993, 374190, 3564176, 36808647, 409067204, 4861490200, 61457674398, 822732344816, 11618029697489, 172476856415121, 2683881876383377, 43660291710726058, 740764460615030663, 13080604188895285878, 239939914279952537597, 4564083798329838120034, 89886989241387131773525, 1830230258908641519168564

Offset

0,3

Comments

Antidiagonal sums of square table A299427.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n equals the following expressions.

(1) A(x) = Sum_{n>=0} Series_Reversion( x/(1+x)^n )^n.

(2) A(x) = Sum_{n>=1} x^n * R(x,n)^(n^2), where

(2.a) R(x,n) = 1 + x*R(x,n)^n,

(2.b) R(x,n)^n = Series_Reversion( x/(1+x)^n ) / x,

(2.c) R(x,n)^n = Sum_{k>=0} C(n*(k+1), k)/(k+1) * x^k;

(2.d) R(x,n)^(n^2) = Sum_{k>=0} C(n*(n+k), k) * n/(n+k) * x^k.

FORMULAS FOR TERMS.

a(n) = Sum_{k=0..n} binomial(n*(n-k), k) * (n-k)/n.

a(n)^(1/n) ~ n^(n/w) * (n+1-w)^(1 - (n+1)/w) * (w-1)^(1/w - 1), where w = LambertW(exp(1)*n). - Vaclav Kotesovec, Feb 19 2018

Example

G.f. A(x) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 129*x^5 + 784*x^6 + 5472*x^7 + 42993*x^8 + 374190*x^9 + 3564176*x^10 + ...
such that
A(x) = 1 + x*R(x,1) + x^2*R(x,2)^4 + x^3*R(x,3)^9 + x^4*R(x,4)^16 + x^5*R(x,5)^25 + x^6*R(x,6)^36 + ...
where series R(x,n) = 1 + x*R(x,n)^n begin:
R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + ...
R(x,2) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ...
R(x,3) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ...
R(x,4) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + ...
R(x,5) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + ...
R(x,6) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + ...
...
and series R(x,n)^(n^2) begin:
R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + ...
R(x,2)^4 = 1 + 4*x + 14*x^2 + 48*x^3 + 165*x^4 + 572*x^5  + ...
R(x,3)^9 = 1 + 9*x + 63*x^2 + 408*x^3 + 2565*x^4 + 15939*x^5 + ...
R(x,4)^16 = 1 + 16*x + 184*x^2 + 1872*x^3 + 17980*x^4 + 167552*x^5 + ...
R(x,5)^25 = 1 + 25*x + 425*x^2 + 6175*x^3 + 82775*x^4 + 1059380*x^5 + ...
R(x,6)^36 = 1 + 36*x + 846*x^2 + 16536*x^3 + 292581*x^4 + 4874688*x^5 + ...
...
demonstrating that A(x) = Sum_{n>=1} x^n * R(x,n)^(n^2).

Mathematica

a[0]=1; a[n_] := Sum[Binomial[n*(n-k), k]*(n-k)/n, {k, 0, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 23 2018 *)

Prog

(PARI) {a(n) = my(A, Ox=x^2*O(x^n)); A = sum(m=0, n+1, serreverse( x/(1+x +Ox)^m +Ox)^m ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = if(n==0, 1, sum(k=0, n, binomial(n*(n-k), k) * (n-k)/n ) )}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A299427, A099237, A299043.

Sequence in context: A030921 A132290 A030933 * A100325 A353180 A352428

Adjacent sequences: A299041 A299042 A299043 * A299045 A299046 A299047

Keyword

nonn

Author

Paul D. Hanna, Feb 18 2018

Status

approved