1,5

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. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 19 2009]

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.

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].

A. F. Labossiere, Sobalian Coefficients.

A. F. Labossiere, Miscellaneous.

n!*binomial(x, n)= Sum T(n, k)*x^(n-k), k=1..n-1.

|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 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 30 2003

|T(n, k)| = sum(A008517(k, m+1)*binomial(n+m, 2*(k-1)), m=0..n), n>=k>=1. A008517 is the second-order Eulerian triangle. See the Graham et al. reference p. 257, eq. (6.44).

A111999 formula for signed T(n, k).

|T(n, k)| = sum(A112486(k-1, m)*binomial(n-1, k-1+m), m=0..min(k-1, n-k)) if n>=k>=1, else 0. - W. Lang Sep 12 2005, see A112486.

|T(n, k)| = (f(n-1, k-1)/(2*(k-1))!)* sum(A112486(k-1, m)*f(2*(k-1), k-1-m)*f(n-k, m), m=0..min(k-1, n-k)) 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. - W. Lang Sep 12 2005, see A112486.

With P(n,t) = sum(k=0,...,n-1) T(n,k+1) * t^k = (1-t) (1-2t)...(1-(n-1)t) and P(0,t) = 1, exp[P(.,t)*x] = (1+tx)^(1/t) . Compare A094638. T(n,k+1) = (1/k!) (D_t)^k (D_x)^n [ (1+tx)^(1/t) - 1 ] evaluated at t=x=0 . - Tom Copeland (tcjpn(AT)msn.com), Dec 09 2007

PRODUCT((x-i): 1<=i<=n) = SUM(T(n,k)*x^k: 0<=k<=n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 29 2007

3!*C(x,3) = x^3-3*x^2+2*x.

1; 1,-1; 1,-3,2; 1,-6,11,-6; 1,-10,35,-50,24; ...

(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))

See A008275 and A048994, which are the main entries for this triangle of numbers. Cf. A054654, A054655.

Cf. A084938, A145324.

Cf. A094216, A008275, A003422, A000166, A000110, A000204, A000045, A000108.

Adjacent sequences: A008273 A008274 A008275 this_sequence A008277 A008278 A008279

Sequence in context: A088617 A144250 A156367 this_sequence A094638 A143778 A164645

sign,tabl,nice

N. J. A. Sloane (njas(AT)research.att.com).