cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A187044 Row sums of number triangle A070895.

Original entry on oeis.org

1, 2, 4, 9, 22, 59, 170, 525, 1716, 5917, 21362, 80533, 315516, 1281913, 5383622, 23330405, 104084736, 477371217, 2246811730, 10839493637, 53528916508, 270318789249, 1394426035918, 7341439399397, 39413238225512, 215607783811041
Offset: 0

Views

Author

Paul Barry, Mar 02 2011

Keywords

Links

Crossrefs

Cf. A070895, A193361.

Programs

  • Mathematica
    CoefficientList[Series[E^(x+x^2/2)*(Sqrt[Pi/2]*Erf[x/Sqrt[2]]+1), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Aug 15 2013 *)

Formula

E.g.f.: exp(x+x^2/2)*(sqrt(pi/2)*ERF(x/sqrt(2)) + 1).
Conjecture: a(n) -2*a(n-1) +(2-n)*a(n-2) +(n-2)*a(n-3)=0. - R. J. Mathar, Sep 29 2012
a(n) ~ (sqrt(2)+sqrt(Pi))/2 * exp(sqrt(n)-n/2-1/4)*n^(n/2) * (1+7/(24*sqrt(n))). - Vaclav Kotesovec, Aug 15 2013
a(n) = A193361(n+2). - Vaclav Kotesovec, Feb 14 2014

A187830 a(n)=2*a(n-1)+(n+3)*a(n-2)-(n+3)*a(n-3), a(0)=0, a(1)=0, a(2)=1.

Original entry on oeis.org

0, 0, 1, 2, 11, 30, 141, 472, 2165, 8302, 38613, 163144, 780953, 3554402, 17611557, 85145196, 437376337, 2225425454, 11847704869, 63032490312, 347377407169, 1923189664970, 10955002251365, 62881123205556, 369621186243777, 2193173759204902, 13281809346518213
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 27 2012

Keywords

Comments

This is case k=3. In general case, recurrence a(n)=2*a(n-1)+(n+k)*(a(n-2)-a(n-3)) is asymptotic to a(n) ~ c * n^(n/2+k/2+1)*exp(sqrt(n)-n/2-1/4) * (1+(12*k+31)/(24*sqrt(n))), where c is constant dependent only on k.
EGF is solution of the equation DSolve[{(3+k)*f[x] + (x-3-k)*f'[x] - (x+2)*f''[x] + f'''[x]==0, f[0]==0, f'[0]==0, f''[0]==1}, f, x]

Links

Crossrefs

Cf. A220700 (k=2), A213720 (k=1), A185309 (k=0), A185308 (k=-1), A186738 (k=-2), A186739 (k=-3), A193361 (k=-4), A220699 (k=-5).

Programs

  • Mathematica
    RecurrenceTable[{(3+n)*a[-3+n]+(-3-n)*a[-2+n]-2*a[-1+n]+a[n]==0,a[0]==0,a[1]==0,a[2]==1},a,{n,20}]
FullSimplify[CoefficientList[Series[1/30*E^(-(x^2/2))*((8*Sqrt[2*E*Pi]*Erf[1/Sqrt[2]]-27)*E^(x^2+x)*(x+1)*(x*(x+2)*(x*(x+2)+12)+26)+Sqrt[2*Pi]*E^(x^2+x)*(x+1)*(x*(x+2)*(x*(x+2)+12)+26)*(15*Erf[x/Sqrt[2]]-8*Sqrt[E]*Erf[(x+1)/Sqrt[2]])-16*E^(x^2/2)*(x*(x+2)+2)*(x*(x+2)+9)+30*E^(1/2*x*(x+2))*(x*(x*(x*(x+5)+19)+35)+33)), {x, 0, 20}], x]* Range[0, 20]!]

Formula

E.g.f.: 1/30*exp(-(x^2/2))*((8*sqrt(2*exp(1)*Pi)*erf(1/sqrt(2))-27)*exp(x^2+x)*(x+1)*(x*(x+2)*(x*(x+2)+12)+26)+sqrt(2*Pi)*exp(x^2+x)*(x+1)*(x*(x+2)*(x*(x+2)+12)+26)*(15*erf(x/sqrt(2))-8*sqrt(exp(1))*erf((x+1)/sqrt(2)))-16*exp(x^2/2)*(x*(x+2)+2)*(x*(x+2)+9)+30*exp(1/2*x*(x+2))*(x*(x*(x*(x+5)+19)+35)+33))
a(n) ~ (1/2*sqrt(Pi)-9/(10*sqrt(2))+4/15*sqrt(Pi)*exp(1/2)*(erf(1/sqrt(2))-1)) * n^(n/2+5/2)*exp(sqrt(n)-n/2-1/4) * (1+(67/(24*sqrt(n))))
Showing 1-2 of 2 results.

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