A000771 Stirling numbers of second kind, S(n,7).

1, 28, 462, 5880, 63987, 627396, 5715424, 49329280, 408741333, 3281882604, 25708104786, 197462483400, 1492924634839, 11143554045652, 82310957214948, 602762379967440, 4382641999117305, 31677463851804540, 227832482998716310, 1631853797991016600

Offset: 7

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

G.f.: x^7/product(1-k*x,k=1..7). E.g.f.: ((exp(x)-1)^7)/7!.

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
• F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=7..200
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 350
• Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Crossrefs

Cf. A008277.

Programs

Mathematica

lst={};Do[f=StirlingS2[n, 7];AppendTo[lst, f], {n, 7, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x) (1 - 6 x) (1 - 7 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
Table[1/720*(7^(n-1)-6^n+3*5^n-5*4^n+5*3^n-3*2^n+1),{n,7,20}] (* Vaclav Kotesovec, Nov 19 2012 *)
LinearRecurrence[{28,-322,1960,-6769,13132,-13068,5040},{1,28,462, 5880, 63987, 627396,5715424},20] (* or *) Drop[StirlingS2[Range[30],7],6] (* Harvey P. Dale, Jul 25 2021 *)

Formula

a(n) = A008277(n, 7) (Stirling2 triangle).

a(n) = 1/720*(7^(n-1)-6^n+3*5^n-5*4^n+5*3^n-3*2^n+1). - Vaclav Kotesovec, Nov 19 2012

a(n) = det(|s(i+7,j+6)|, 1 <= i,j <= n-7), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

Extensions

Two more terms from Neven Juric, Oct 22 2009

Definition corrected by Vaclav Kotesovec, Nov 19 2012