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

1, 1, 2, 1, 9, 10, 1, 24, 107, 90, 1, 50, 575, 1750, 1248, 1, 90, 2135, 16050, 38244, 24360, 1, 147, 6265, 95445, 537334, 1078728, 631440, 1, 224, 15610, 424340, 4734289, 21569996, 38105220, 20865600, 1, 324, 34482, 1529640, 30128049

Offset: 1

Author

Vladeta Jovovic, Jan 29 2001

Keywords

• nonn
• tabl

Examples

			[1],
[1, 2],
[1, 9, 10],
[1, 24, 107, 90],
[1, 50, 575, 1750, 1248],
[1, 90, 2135, 16050, 38244, 24360],
[1, 147, 6265, 95445, 537334, 1078728, 631440],
...
P(2,k) = k + 2,
P(3,k) = (1/2!)*(k^2 + 9*k + 10),
P(4,k) = (1/3!)*(k^3 + 24*k^2 + 107*k + 90).
		

Links

• Vladeta Jovovic, More information

Crossrefs

Cf. A059605, A059606, A320962.

Programs

Maple

P := (n, k) -> (n-1)!*add(Stirling2(n,i)*binomial(k+i-1,k), i=0..n):
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

Mathematica

row[n_] := (n-1)! CoefficientList[Sum[StirlingS2[n,i] Binomial[k+i-1,k] // FunctionExpand, {i,0,n}], k] // Reverse;
Array[row,10] // Flatten (* Jean-François Alcover, Jun 03 2019 *)

PARI

row(n)={Vec((n-1)!*sum(i=0, n, stirling(n,i,2)*binomial(x+i-1,i-1)))}
for(n=1, 10, print(row(n))) \\ Andrew Howroyd, Nov 07 2018