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

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 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. 37
 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]. 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). Cf. A054654, A054655, A084938, A145324, A094216, A003422, A000166, A000110, A000204, A000045, A000108. Sequence in context: A193593 A308616 A181853 * A094638 A196844 A196843 Adjacent sequences: A008273 A008274 A008275 * A008277 A008278 A008279 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.

Last modified December 5 19:57 EST 2022. Contains 358588 sequences. (Running on oeis4.)