A143168 -id:A143168 - cp's OEIS frontend

A143169 Fourth column of triangle A000369: |S2(-3;n+4,4)|.

1, 30, 825, 24150, 775845, 27478710, 1069801425, 45547251750, 2108878296525, 105616706545350, 5693005525232025, 328784072492625750, 20261087389388971125, 1327378299252353097750, 92142485069345244158625, 6756933615539839013031750, 522007423480304780922028125

Offset: 0

Wolfdieter Lang, Sep 15 2008

Also third column (m=3) of triangle A049029 (S2(5)).

Third column of A000369 is A143168, fifth one is A143170.

Also third column (m=3) of triangle A049029 (S2(5)). - Wolfdieter Lang, Nov 17 2008

a(n) = A000369(n+4,4) = |S2(-3;n+4,4)|, n >= 0.

E.g.f.: d^4/dx^4 ((1-(1-4*x)^(1/4))^4)/4! = (-1/2)*(-45*(1-4*x)^(1/2)+120*(1-4*x)^(1/4)-77)/(1-4*x)^(15/4).

A143167 Second column of triangle A000369: |S2(-3;n+2,2)|.

Original entry on oeis.org

1, 9, 111, 1785, 35595, 848925, 23586255, 748471185, 26715409875, 1059544210725, 46230843633975, 2201008238854425, 113546715232225275, 6309834090304870125, 375777507964741257375, 23876826206710426574625, 1612323634555365676819875

Offset: 0

Wolfdieter Lang, Sep 15 2008

Robert Israel, Table of n, a(n) for n = 0..364

First column of A000369 is A008545, third one is A143168.

f:= gfun:-rectoproc({a(n) = (8*n+1)*a(n-1) - 2*(4*n-1)*(2*n-1)*a(n-2),a(0)=1, a(1)=9}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Jan 09 2019

x = 'x + O('x^40); serlaplace((3 - 2*(1-4*x)^(1/4))/(1-4*x)^(7/4)) \\ Michel Marcus, Jun 18 2017

a(n) = A000369(n+2,2) = |S2(-3;n+2,2)|, n >= 0.
E.g.f.: d^2/dx^2 ((1-(1-4*x)^(1/4))^2 )/2! = (3 - 2*(1-4*x)^(1/4))/(1-4*x)^(7/4).
From Robert Israel, Jan 09 2019: (Start)
a(n) = (8*n+1)*a(n-1) - 2*(4*n-1)*(2*n-1)*a(n-2).
a(n) = 4^(n+1)*(Gamma(n+7/4)/Gamma(3/4) - Gamma(n+3/2)/Gamma(1/2)). (End)

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A143169 Fourth column of triangle A000369: |S2(-3;n+4,4)|.

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A143167 Second column of triangle A000369: |S2(-3;n+2,2)|.

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