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A059604 Coefficients of polynomials (n-1)!*P(n,k), P(n,k) = Sum_{i=0..n} Stirling2(n,i)*binomial(k+i-1,k).

%I A059604 #21 Jun 03 2019 06:45:55
%S A059604 1,1,2,1,9,10,1,24,107,90,1,50,575,1750,1248,1,90,2135,16050,38244,
%T A059604 24360,1,147,6265,95445,537334,1078728,631440,1,224,15610,424340,
%U A059604 4734289,21569996,38105220,20865600,1,324,34482,1529640,30128049
%N A059604 Coefficients of polynomials (n-1)!*P(n,k), P(n,k) = Sum_{i=0..n} Stirling2(n,i)*binomial(k+i-1,k).
%H A059604 Vladeta Jovovic, <a href="/A059604/a059604.pdf">More information</a>
%e A059604 [1],
%e A059604 [1, 2],
%e A059604 [1, 9, 10],
%e A059604 [1, 24, 107, 90],
%e A059604 [1, 50, 575, 1750, 1248],
%e A059604 [1, 90, 2135, 16050, 38244, 24360],
%e A059604 [1, 147, 6265, 95445, 537334, 1078728, 631440],
%e A059604 ...
%e A059604 P(2,k) = k + 2,
%e A059604 P(3,k) = (1/2!)*(k^2 + 9*k + 10),
%e A059604 P(4,k) = (1/3!)*(k^3 + 24*k^2 + 107*k + 90).
%p A059604 P := (n, k) -> (n-1)!*add(Stirling2(n,i)*binomial(k+i-1,k), i=0..n):
%p A059604 for n from 1 to 8 do seq(coeff(expand(P(n,x)),x,n-k), k=1..n) od; # _Peter Luschny_, Nov 07 2018
%t A059604 row[n_] := (n-1)! CoefficientList[Sum[StirlingS2[n,i] Binomial[k+i-1,k] // FunctionExpand, {i,0,n}], k] // Reverse;
%t A059604 Array[row,10] // Flatten (* _Jean-François Alcover_, Jun 03 2019 *)
%o A059604 (PARI) row(n)={Vec((n-1)!*sum(i=0, n, stirling(n,i,2)*binomial(x+i-1,i-1)))}
%o A059604 for(n=1, 10, print(row(n))) \\ _Andrew Howroyd_, Nov 07 2018
%Y A059604 Cf. A059605, A059606, A320962.
%K A059604 nonn,tabl
%O A059604 1,3
%A A059604 _Vladeta Jovovic_, Jan 29 2001