0,7

This is a binomial convolution triangle (Sheffer triangle) of the Appell type: (exp(-x)/(1-x),x), i.e. the e.g.f. of column k is (exp(-x)/(1-x))*(x^k/k!). See the e.g.f. given by V. Jovovic below. W. Lang, Jan 21 2008.

The formula T(n,k)=binomial(n,k)*A000166(n-k), with the derangements numbers (subfactorials) A000166 (see also the Charalambides reference) shows the Appell type of this triangle. W. Lang, Jan 21 2008.

T(n,k) is the number of permutations of {1,2,...,n} having k pairs of consecutive right-to-left minima (0 is considered a right-to-left minimum for each permutation). Example: T(4,2)=6 because we have 1243, 1423, 4123, 1324, 3124, and 2134; for example, 1324 has right-to-left minima in positions 0-1,3-4 and 2134 has right-to-left minima in positions 0,2-3-4, the consecutive ones being joined by -. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 29 2008

S. K. Das and N. Deo, Rencontres graphs: a family of bipartite graphs, Fib. Quart., Vol. 25, No. 3, August 1987, 250-262.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 194.

I. Kaplansky, Symbolic solution of certain problems in permutations, Bull. Amer. Math. Soc., 50 (1944), 906-914.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p.173, Table 5.2 (without row n=0 and column k=0).

T. D. Noe, Rows n=0..50 of triangle, flattened

T(n, k) = T(n-1, k)*n+C(n, k)*(-1)^(n-k) = T(n, k-1)/k+C(n, k)*(-1)^(n-k)/(n-k+1) = T(n-1, k-1)*n/k = T(n-k, 0)*C(n, k) [with T(0, 0) = 1]; so T(n, n) = 1, T(n, n-1) = 0, T(n, n-2) = n*(n-1)/2 for n >= 0.

Sum[ T[n, k], {k, 0, n}] ==Sum[ k T[n, k], {k, 0, n}] == n! for all n>0, n, k integers. - wouter.meeussen(AT)pandora.be, May 29 2001

O.g.f. for k-th column is 1/k!*Sum_{i>=k} i!*x^i/(1+x)^(i+1). O.g.f. for k-th row is k!*Sum_{i=0..k} (-1)^i/i!*(1-x)^i. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 12 2002

E.g.f.: exp((y-1)*x)/(1-x). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 18 2002

Sum(k=0..n, T(n, k)*x^k ) is the permanent of the n X n matrix with x's on the diagonal and 1's elsewhere; for x = 0, 1, 2, 3, 4, 5, 6 see A000166, A000142, A000522, A010842, A053486, A053487, A080954. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 12 2003

T(n, k)= Sum(j=0..n, A008290(n, j)*k^(n-j)) is the permanent of the n X n matrix with 1's on the diagonal and k's elsewhere; for k = 0, 1, 2 see A000012 A000142 A000354 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 13 2003

T(n,k)=sum{j=0..n, (-1)^(j-k)*binomial(j,k)*n!/j!}; - Paul Barry (pbarry(AT)wit.ie), May 25 2006

T(n,k)=(n!/k!)*sum(((-1)^j)/j!,j=0..n-k), n>=k>=0. From the Appell type of the triangle and the subfactorial formula.

T(n,0)=n*sum((j/(j+1))*T(n-1,j),j=0..n-1), T(0,0)=1. From the z-sequence of this Sheffer triangle z(j)=j/(j+1) with e.g.f. (1-exp(x)*(1-x))/x. See the W. Lang link under A006232 for Sheffer a- and z-sequences. W. Lang. Jan 21 2008.

T(n,k)= (n/k)*T(n-1,k-1) for k>=1. See above. From the a-sequence of this Sheffer triangle a(0)=1, a(n)=0, n>=1 with e.g.f. 1. See the W. Lang link under A006232 for Sheffer a- and z-sequences. W. Lang. Jan 21 2008.

exp((y-1)*x)/(1-x) = 1+y*x+1/2!*(1+y^2)*x^2+1/3!*(2+3*y+y^3)*x^3+1/4!*(9+8*y+6*y^2+y^4)*x^4+1/5!*(44+45*y+20*y^2+10*y^3+y^5)*x^5+...

Triangle begins:

1

0 1

1 0 1

2 3 0 1

9 8 6 0 1

44 45 20 10 0 1

265 264 135 40 15 0 1

1854 1855 924 315 70 21 0 1

14833 14832 7420 2464 630 112 28 0 1

133496 133497 66744 22260 5544 1134 168 36 0 1

a[0] = 1; a[1] = 0; a[n_] := Round[n!/E] /; n >= 1 size = 8; Table[Binomial[n, k]a[n - k], {n, 0, size}, {k, 0, n}] // TableForm (from Harlan J. Brothers, Mar 19 2007)

(PARI) {T(n, k)= if(k<0|k>n, 0, n!/k!*sum(i=0, n-k, (-1)^i/i!))}

Columns give A000166, A000240, A000387, A000449, A000475. Cf. A055137, A008291.

T(n, k)=binomial(n, k)*A000166(n-k)

Cf. A080955.

Cf. A000012 A000142 A000354.

Adjacent sequences: A008287 A008288 A008289 this_sequence A008291 A008292 A008293

Sequence in context: A055137 A128888 A004443 this_sequence A059066 A059067 A065861

nonn,tabl,nice

njas

Comments and more terms from Michael Somos, Apr 26 2000 and Christian G. Bower (bowerc(AT)usa.net), Apr 26 2000