A209305 -id:A209305 - cp's OEIS frontend

A209306 Binomial self-convolution of sequence k -> A209305(k+1).

1, 6, 52, 608, 9000, 161320, 3395384, 82067848, 2239857464, 68123898696, 2284557569272, 83741888125064, 3330861429420984, 142875672420718024, 6574169480181294200, 322998830024467434760, 16876498518902786900792, 934400728689236533139016

Offset: 0

Emanuele Munarini, Jan 18 2013

nonn

G. C. Greubel, Table of n, a(n) for n = 0..375

Cf. A209305.

(* Generating series *)
  A[x_] := InverseErf[(2 Exp[x] - 2 + Exp[1] Sqrt[Pi] Erf[1])/(Exp[1] Sqrt[Pi])];
CoefficientList[Series[A'[x]^2, {x, 0, 20}], x] Table[n!, {n, 0, 20}]
(* Recurrences *)
a[n_] := a[n] = a[n-1]+2Sum[Binomial[n-2,k]a[k]b[n-2-k],{k,0,n-2}];
a[1] = 1;
a[0] = 1;
b[n_] := Sum[Binomial[n,k]a[k+1]a[n-k+1],{k,0,n}];
Table[b[n], {n, 0, 100}]

a(n) = Sum_{k=0..n} C(n,k)*b(k+1)*b(n-k+1), where b(n) = A209305(n).

E.g.f.: A(x) = B'(x)^2, where B(x) is the e.g.f. of sequence A209305.

A209307 Binomial self-convolution of sequence A209305.

Original entry on oeis.org

1, 2, 8, 52, 492, 6172, 96572, 1810940, 39585980, 988367804, 27750071036, 865420762876, 29680685363772, 1110252095824444, 44984193111861116, 1962563143587356540, 91727727493033914044, 4572606297018521071292, 242169416254095528953852

Offset: 0

Emanuele Munarini, Jan 18 2013

nonn

G. C. Greubel, Table of n, a(n) for n = 0..375

Cf. A209305, A209306.

(* Expansion of the generating series *) CoefficientList[Series[(InverseErf[(2Exp[x]-2+Exp[1]Sqrt[Pi]Erf[1])/(Exp[1]Sqrt[Pi])])^2,{x,0,40}],x]Table[n!,{n,0,40}]
(* Recurrence *)
a[n_] := a[n] = a[n-1]+2Sum[Binomial[n-2,k]a[k]b[n-2-k],{k,0,n-2}];
a[1] = 1;
a[0] = 1;
b[n_] := Sum[Binomial[n,k]a[k+1]a[n-k+1],{k,0,n}];
Table[Sum[Binomial[n, k]a[k]a[n - k], {k, 0, n}], {n, 0, 12}]

E.g.f.: A(x)^2, where A(x) is the e.g.f. of the sequence A209305.

cp's OEIS Frontend

A209306 Binomial self-convolution of sequence k -> A209305(k+1).

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