Rationals z(n):=A130189(n)/A130190(n)

z(n), n=0..30:

 [1,-1/2, 5/6,-7/4,68/15,-167/12,2057/42,-4637/24,75703/90,-39941/10,676360/33, -902547/8,602501827/910,-432761746/105,2438757091/90,-8997865117/48,346824403906/255,-1857709421899/180,325976550837563/3990,-282728710837871/420]


Numerator sequence A130189(n), n=0..30:

 [1, -1, 5, -7, 68, -167, 2057, -4637, 75703, -39941, 676360, -902547, 602501827, -432761746, 2438757091, -8997865117, 346824403906, -1857709421899, 325976550837563, -282728710837871, 39928855264303811, -16874802689368067, 162083496666375118, -3212329557624761759, 596987966249847103201, -120662218572021796466, 1738610608872726996923, -1378574280287140193083, 78528781614512428940448, -122321894309073168113021]


Denominator sequence A130190(n), n=0..30:

[1, 2, 6, 4, 15, 12, 42, 24, 90, 10, 33, 8, 910, 105, 90, 48, 255, 180, 3990, 420, 6930, 330, 345, 720, 13650, 273, 378, 28, 145, 20, 14322, 2464, 117810, 3570, 7, 24, 1919190, 1729, 2730, 840, 9471, 13860, 99330, 1540, 217350, 4830, 4935, 10080, 324870, 16575] 

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Recurrence relation for the first column (m=0) entries of the Sheffer triangle (matrix) 

S(n,m):=A094816(n,m) (coefficient triangle of certain Poisson-Charlier polynomials 

(a=1, x->-x)): 

S(n,0) = n*sum(z(j)*S(n-1,j),j=0..n-1). n>=1. S(0,0):=1.  


See the W. Lang link under A006232 with a summary on a- and z-sequences for Sheffer matrices.    
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The a-sequence for this Sheffer triangle A094816(n,m) is given by the Bernoulli numbers 

a(n)=B(n)=A027641(n)/A027642(n), leading to the recurrence

S(n,m):=A094816(n,m) =(n/m)*sum(binomial(m-1+j,m-1)*B(j)*S(n-1,m-1+j) ,j=0..n-m), n>=m>=1. 

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The recurrence for the row polynomials is:





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