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
Examples
First 3 rows of A249074: 1 4 1 6 4 1 so that a(0) = 1, a(1) = 5, a(2) = 11.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..720 (terms 0..100 from Clark Kimberling)
Crossrefs
Cf. A249074.
Programs
-
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 *)
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)