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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.

%I A008276 #104 Feb 16 2025 08:32:31
%S A008276 1,1,-1,1,-3,2,1,-6,11,-6,1,-10,35,-50,24,1,-15,85,-225,274,-120,1,
%T A008276 -21,175,-735,1624,-1764,720,1,-28,322,-1960,6769,-13132,13068,-5040,
%U A008276 1,-36,546,-4536,22449,-67284,118124,-109584,40320,1,-45
%N 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.
%C A008276 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
%C A008276 From _Daniel Forgues_, Jan 16 2016: (Start)
%C A008276 For n >= 1, the row sums [of either signed or absolute values] are
%C A008276   Sum_{k=1..n} T(n,k) = 0^(n-1),
%C A008276   Sum_{k=1..n} |T(n,k)| = T(n+1,1) = n!. (End)
%C A008276 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
%C A008276 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
%D A008276 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.
%D A008276 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
%D A008276 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), p. 257.
%H A008276 T. D. Noe, <a href="/A008276/b008276.txt">Rows n=0..100 of triangle, flattened</a>
%H A008276 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A008276 T. Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>
%H A008276 Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, <a href="https://arxiv.org/abs/2407.04409">The Fibonacci-Fubini and Lucas-Fubini numbers</a>, arXiv:2407.04409 [math.CO], 2024. See pp. 10, 12.
%H A008276 Bill Gosper, <a href="/A008275/a008275.png">Colored illustrations of triangle of Stirling numbers of first kind read mod 2, 3, 4, 5, 6, 7</a>
%H A008276 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html">Stirling Number of the First Kind</a>
%H A008276 Wikipedia, <a href="http://en.wikipedia.org/wiki/Stirling_numbers_and_exponential_generating_functions">Stirling numbers and exponential generating functions </a>
%F A008276 n!*binomial(x, n) = Sum_{k=1..n-1} T(n, k)*x^(n-k).
%F A008276 |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
%F A008276 |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).
%F A008276 A094638 formula for unsigned T(n, k).
%F A008276 |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.
%F A008276 |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.
%F A008276 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
%F A008276 Product_{i=1..n} (x-i) = Sum_{k=0..n} T(n,k)*x^k. - _Reinhard Zumkeller_, Dec 29 2007
%F A008276 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
%F A008276 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
%F A008276 For n>=2, Sum_{k=1..n} k*T(n,k) = (-1)^(n-1)*(n-2)!. - _Zizheng Fang_, Dec 27 2020
%e A008276 3!*binomial(x,3) = x*(x-1)*(x-2) = x^3 - 3*x^2 + 2*x.
%e A008276 Triangle begins
%e A008276   1;
%e A008276   1,  -1;
%e A008276   1,  -3,   2;
%e A008276   1,  -6,  11,   -6;
%e A008276   1, -10,  35,  -50,  24;
%e A008276   1, -15,  85, -225, 274, -120;
%e A008276 ...
%p A008276 seq(seq(coeff(expand(n!*binomial(x,n)),x,j),j=n..1,-1),n=1..15); # _Robert Israel_, Jan 24 2016
%p A008276 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
%t A008276 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 *)
%t A008276 Flatten@Table[CoefficientList[Product[1-k x, {k, 1, n}], x], {n, 0, 8}] (* _Oliver Seipel_, Jun 14 2024 *)
%t A008276 Flatten@Table[Coefficient[Product[x-k, {k, 0, n-1}], x, Reverse@Range[n]], {n, Range[9]}] (* _Oliver Seipel_, Jun 14 2024, after  _Ralf Stephan_ *)
%o A008276 (PARI) T(n,k)=if(n<1,0,n!*polcoeff(binomial(x,n),n-k+1))
%o A008276 (PARI) T(n,k)=if(n<1,0,n!*polcoeff(polcoeff(y*(1+y*x+x*O(x^n))^(1/y),n),k))
%o A008276 (Haskell)
%o A008276 a008276 n k = a008276_tabl !! (n-1) !! (k-1)
%o A008276 a008276_row n = a008276_tabl !! (n-1)
%o A008276 a008276_tabl = map init $ tail a054654_tabl
%o A008276 -- _Reinhard Zumkeller_, Mar 18 2014
%o A008276 (Sage) def T(n,k): return falling_factorial(x,n).expand().coefficient(x,n-k+1) # _Ralf Stephan_, Dec 11 2016
%Y A008276 See A008275 and A048994, which are the main entries for this triangle of numbers.
%Y A008276 See A008277 triangle of Stirling numbers of the second kind, S2(n,k).
%Y A008276 Cf. A054654, A054655, A084938, A145324, A094216, A003422, A000166, A000110, A000204, A000045, A000108.
%K A008276 sign,tabl,nice
%O A008276 1,5
%A A008276 _N. J. A. Sloane_