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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. 35
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

T. D. Noe, Rows n=0..100 of triangle, flattened

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].

T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms

Bill Gosper, Colored illustrations of triangle of Stirling numbers of first kind read mod 2, 3, 4, 5, 6, 7

Eric Weisstein's World of Mathematics, Stirling Number of the First Kind

Wikipedia, Stirling numbers and exponential generating functions

OEIS Wiki, Factorial polynomials

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

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).

Cf. A054654, A054655, A084938, A145324, A094216, A003422, A000166, A000110, A000204, A000045, A000108.

Sequence in context: A156367 A193593 A181853 * A094638 A196844 A196843

Adjacent sequences:  A008273 A008274 A008275 * A008277 A008278 A008279

KEYWORD

sign,tabl,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 11 21:15 EST 2017. Contains 295919 sequences.