A201201 -id:A201201 - cp's OEIS frontend

A201203 Alternating row sums of triangle A201201: first associated monic Laguerre-Sonin(e) polynomials with parameter alpha=1 evaluated at x=-1.

1, -5, 29, -201, 1631, -15173, 159093, -1854893, 23788271, -332613321, 5033396573, -81929955953, 1426898945343, -26468817431501, 520884561854501, -10836674357638293, 237603001692915983, -5475288709200573713, 132276033079845108621

Offset: 0

Wolfdieter Lang, Dec 06 2011

G. C. Greubel, Table of n, a(n) for n = 0..440
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, Table 2 on p. 13.

Cf. A201201, A201202 (row sums), A073003, A002793.

A201203 := proc(n)
    add((-1)^k*A201201(n,k),k=0..n) ;
end proc:
seq(A201203(n),n=0..20) ; # R. J. Mathar, Dec 07 2011

Flatten[{1,RecurrenceTable[{n*(1+n)*a[-2+n]+(3+2*n)*a[-1+n] +a[n]==0, a[1]==-5,a[2]==29}, a, {n, 20}]}] (* Vaclav Kotesovec, Oct 19 2013 *)

a(n) = Sum_{k=0..n} ((-1)^k)*A201201(n,k), n>=0.
a(n)+(2*n+3)*a(n-1)+n*(n+1)*a(n-2)=0, a(-1)=0, a(0)=1. - R. J. Mathar, Dec 07 2011
From Wolfdieter Lang, Dec 11 2011: (Start)
E.g.f. from A201201 with x=-1, z->x: g(x) = exp(1/(1+x))*(3+2*x)*(exp(-1) + (Ei(1,1/(1+x))-Ei(1,1)))/(1+x)^4-(2+x)/(1+x)^3, with the exponential integral Ei.
This e.g.f. satisfies the homogeneous ordinary second-order differential equation (1+x)^2*(d^2(g(x))/dx^2) + (7+6*x)*(d(g(x))/dx)+6*g(x), with g(0)=1 and (d(g(x))/dx){x=0} = -5. This is equivalent to the recurrence conjectured above by _R. J. Mathar, which is thus proved.
(End)
Let G denote Gompertz's constant A073003. The unsigned sequence is the sequence of numerators in the convergents coming from the infinite continued fraction expansion 1 - G = 1/(3 - 2/(5 - 6/(7 - ... - n*(n+1)/((2*n+3) - ...)))). The sequence of convergents begins [1/3, 5/13, 29/73, 201/501, ...]. The denominators are in A000262. - Peter Bala, Aug 19 2013
a(n) ~ (-1)^n * 2^(-1/2)*(exp(-1)-Ei(1,1)) * exp(2*sqrt(n)-n+1/2) * n^(n+7/4) * (1+91/(48*sqrt(n))), where Ei(1,1) = 0.21938393439552... = G / exp(1), where G = 0.596347362323194... is the Gompertz constant (see A073003). - Vaclav Kotesovec, Oct 19 2013

R. J. Mathar conjecture corrected and proved by Wolfdieter Lang, Dec 11 2011

A201202 Row sums of triangle A201201: first associated monic Laguerre polynomials with parameter alpha=1 evaluated at x=1.

Original entry on oeis.org

1, -3, 9, -27, 63, 117, -4167, 55953, -651177, 7336593, -82438983, 927666333, -10331176977, 110106505773, -1023541502247, 5304225184137, 103363857534663, -5240827920059127, 158560193765332953, -4192332947225516907, 105290369454806352927

Offset: 0

Wolfdieter Lang, Dec 06 2011

Cf. A201201, A201203 (alternating row sums).

A201202 := proc(n)
    add(A201201(n,k),k=0..n) ;
end proc:
seq(A201202(n),n=0..20) ; # R. J. Mathar, Dec 07 2011

a[n_, k_] := (-1)^(n-k)*((n+1)*(n+1)!/((k+1)*(k+1)!))*Binomial[n, k]*HypergeometricPFQ[{-(n-k), k, 1}, {-(n+1), k+2}, 1]; Table[Sum[a[n, k], {k, 0, n}], {n, 0, 20}]  (* Jean-François Alcover, Jun 21 2013 *)

a(n)=sum(A201201(n,k),k=0..n), n>=0.
Apparently a(n)+(2*n+1)*a(n-1)+n*(n+1)*a(n-2)=0, a(-1)=0, a(1)=1. - R. J. Mathar, Dec 07 2011
From Wolfdieter Lang, Dec 12 2011: (Start)
E.g.f. from A201201 with x=1, z->x: g(x)=(1+2*x)*exp(-1/(1+x))*(exp(1)-((Ei(1,-1/(1+x)) - Ei(1,-1))))/(1+x)^4 - x/(1+x)^3, with the exponential integral Ei. In order to obtain the series use first Ei(1,-y/(1+x)) - Ei(1,-y) and put y=1 afterwards.
This e.g.f. satisfies the homogeneous second-order differential equation: (1+x)^2*(d^2/dx^2)g(x) + (5+6*x)*(d/dx)g(x) + 6*g(x) = 0, with g(0)=1 and (d/dx)g(x)|{x=0} = -3. This is equivalent to the recurrence conjectured by _R. J. Mathar, which is thus proved.
(End)

cp's OEIS Frontend

A201203 Alternating row sums of triangle A201201: first associated monic Laguerre-Sonin(e) polynomials with parameter alpha=1 evaluated at x=-1.

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