A249075 Sum of the numbers in row n of the array at A249074.

1, 5, 11, 51, 161, 773, 3027, 15395, 69881, 377781, 1915163, 10981907, 60776145, 368269541, 2191553891, 13976179203, 88489011497, 591631462805, 3954213899691, 27619472411891, 193696456198913, 1408953242322117, 10318990227472115, 77948745858933731

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..720 (terms 0..100 from Clark Kimberling)

Formula

a(n) = a(n-1) + 2*(n+1)*a(n-2). - Vaclav Kotesovec, Aug 10 2021, following a suggestion from John M. Campbell

From Vaclav Kotesovec, Aug 10 2021: (Start)

E.g.f. A(x) satisfies the differential equation 6*A(x) + (2*x + 1)*A'(x) - A''(x) = 0, A(0) = 1, A'(0) = 5.

E.g.f.: 1 + 2*x + sqrt(Pi) * (3 + 4*x*(1 + x)) * exp((x + 1/2)^2) * (erf(x + 1/2) - erf(1/2))/2.

a(n) ~ sqrt(Pi) * erfc(1/2) * 2^((n-1)/2) * n^(n/2 + 1) * exp(1/8 + sqrt(n/2) - n/2). (End)

Example

First 3 rows of A249074:
1
4    1
6    4    1
so that a(0) = 1, a(1) = 5, a(2) = 11.

Mathematica

z = 11; p[x_, n_] := x + 2 n/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249074 array *)
Flatten[CoefficientList[u, x]] (* A249074 sequence  *)
u /. x -> 1  (* A249075 *)
RecurrenceTable[{a[n] == a[n-1] + 2*(n+1)*a[n-2], a[0] == 1, a[1] == 5}, a, {n, 0, 25}] (* Vaclav Kotesovec, Aug 10 2021 *)
nmax = 25; CoefficientList[Series[1 + 2*x + Sqrt[Pi]*(3 + 4*x*(1 + x)) * E^((1/2 + x))^2 * (Erf[1/2 + x] - Erf[1/2])/2, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 10 2021 *)

Crossrefs

Cf. A249074.

Sequence in context: A149524 A149525 A149526 * A089004 A197958 A094107

Adjacent sequences: A249072 A249073 A249074 * A249076 A249077 A249078

Keyword

nonn,easy

Author

Clark Kimberling, Oct 20 2014

Status

approved