login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A008276 Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x. 36
1, 1, -1, 1, -3, 2, 1, -6, 11, -6, 1, -10, 35, -50, 24, 1, -15, 85, -225, 274, -120, 1, -21, 175, -735, 1624, -1764, 720, 1, -28, 322, -1960, 6769, -13132, 13068, -5040, 1, -36, 546, -4536, 22449, -67284, 118124, -109584, 40320, 1, -45 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
n-th row of the triangle = charpoly of an (n-1) X (n-1) matrix with (1,2,3,...) in the diagonal and the rest zeros. - Gary W. Adamson, Mar 19 2009
From Daniel Forgues, Jan 16 2016: (Start)
For n >= 1, the row sums [of either signed or absolute values] are
Sum_{k=1..n} T(n,k) = 0^(n-1),
Sum_{k=1..n} |T(n,k)| = T(n+1,1) = n!. (End)
The moment generating function of the probability density function p(x, m=q, n=1, mu=q) = q^q*x^(q-1)*E(x, q, 1)/(q-1)!, with q >= 1, is M(a, m=q, n=1, mu=q) = Sum_{k=0..q}(A000312(q) / A000142(q-1)) * A008276(q, k) * polylog(k, a) / a^q , see A163931 and A274181. - Johannes W. Meijer, Jun 17 2016
Triangle of coefficients of the polynomial x(x-1)(x-2)...(x-n+1), also denoted as falling factorial (x)_n, expanded into decreasing powers of x. - Ralf Stephan, Dec 11 2016
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. 833.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), p. 257.
LINKS
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].
Eric Weisstein's World of Mathematics, Stirling Number of the First Kind
FORMULA
n!*binomial(x, n) = Sum_{k=1..n-1} T(n, k)*x^(n-k).
|A008276(n, k)| = T(n-1, k-1) where T(n, k) is the triangle, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...]; A008276(n, k) = T(n-1, k-1) where T(n, k) is the triangle, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [ -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, ...]. Here DELTA is the operator defined in A084938. - Philippe Deléham, Dec 30 2003
|T(n, k)| = Sum_{m=0..n} A008517(k, m+1)*binomial(n+m, 2*(k-1)), n >= k >= 1. A008517 is the second-order Eulerian triangle. See the Graham et al. reference p. 257, eq. (6.44).
A094638 formula for unsigned T(n, k).
|T(n, k)| = Sum_{m=0..min(k-1, n-k)} A112486(k-1, m)*binomial(n-1, k-1+m) if n >= k >= 1, else 0. - Wolfdieter Lang, Sep 12 2005, see A112486.
|T(n, k)| = (f(n-1, k-1)/(2*(k-1))!)* Sum_{m=0..min(k-1, n-k)} A112486(k-1, m)*f(2*(k-1), k-1-m)*f(n-k, m) if n >= k >= 1, else 0, where f(n, k) stands for the falling factorial n*(n-1)*...*(n-(k-1)) and f(n, 0):=1. - Wolfdieter Lang, Sep 12 2005, see A112486.
With P(n,t) = Sum_{k=0..n-1} T(n,k+1) * t^k = (1-t)*(1-2*t)*...*(1-(n-1)t) and P(0,t) = 1, exp(P(.,t)*x) = (1+t*x)^(1/t) . Compare A094638. T(n,k+1) = (1/k!) (D_t)^k (D_x)^n ( (1+t*x)^(1/t) - 1 ) evaluated at t=x=0 . - Tom Copeland, Dec 09 2007
Product_{i=1..n} (x-i) = Sum_{k=0..n} T(n,k)*x^k. - Reinhard Zumkeller, Dec 29 2007
E.g.f.: Sum_{n>=0} (Sum_{k=0..n} T(n,n-k)*t^k)/n!) = Sum_{n>=0} (x)_n * t^k/n! = exp(x * log(1+t)), with (x)_n the n-th falling factorial polynomial. - Ralf Stephan, Dec 11 2016
Sum_{j=0..m} T(m, m-j)*s2(j+k+1, m) = m^k, where s2(j, m) are Stirling numbers of the second kind. - Tony Foster III, Jul 25 2019
For n>=2, Sum_{k=1..n} k*T(n,k) = (-1)^(n-1)*(n-2)!. - Zizheng Fang, Dec 27 2020
EXAMPLE
3!*binomial(x,3) = x*(x-1)*(x-2) = x^3 - 3*x^2 + 2*x.
Triangle begins
1;
1, -1;
1, -3, 2;
1, -6, 11, -6;
1, -10, 35, -50, 24;
1, -15, 85, -225, 274, -120;
...
MAPLE
seq(seq(coeff(expand(n!*binomial(x, n)), x, j), j=n..1, -1), n=1..15); # Robert Israel, Jan 24 2016
A008276 := proc(n, k): combinat[stirling1](n, n-k+1) end: seq(seq(A008276(n, k), k=1..n), n=1..9); # Johannes W. Meijer, Jun 17 2016
MATHEMATICA
len = 47; m = Ceiling[Sqrt[2*len]]; t[n_, k_] = StirlingS1[n, n-k+1]; Flatten[Table[t[n, k], {n, 1, m}, {k, 1, n}]][[1 ;; len]] (* Jean-François Alcover, May 31 2011 *)
PROG
(PARI) T(n, k)=if(n<1, 0, n!*polcoeff(binomial(x, n), n-k+1))
(PARI) T(n, k)=if(n<1, 0, n!*polcoeff(polcoeff(y*(1+y*x+x*O(x^n))^(1/y), n), k))
(Haskell)
a008276 n k = a008276_tabl !! (n-1) !! (k-1)
a008276_row n = a008276_tabl !! (n-1)
a008276_tabl = map init $ tail a054654_tabl
-- Reinhard Zumkeller, Mar 18 2014
(Sage) def T(n, k): return falling_factorial(x, n).expand().coefficient(x, n-k+1) # Ralf Stephan, Dec 11 2016
CROSSREFS
See A008275 and A048994, which are the main entries for this triangle of numbers.
See A008277 triangle of Stirling numbers of the second kind, S2(n,k).
Sequence in context: A193593 A308616 A181853 * A094638 A196844 A196843
KEYWORD
sign,tabl,nice
AUTHOR
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 20 20:53 EST 2024. Contains 370217 sequences. (Running on oeis4.)