0,3

Generalization of (signed) Lah number triangle A008297 (amended with a trivial row n=0 and a column k=0 in order to have a Sheffer triangle structure of the Jabotinsky type).

product(s*t-j,j=0..n-1) := fallfac(s*t,n) (falling factorial with n factors) is called generalized factorial of t of order n and scale parameter s in the Charalambides reference p. 301 ch. 8.4.

The s-family of triangles L(s;n,k) (in the Charalambides reference called C(n,k;-s))

is defined for integer s by fallfac(-s*t,n) = ((-1)^n)*risefac(s*t,n) = sum(L(s;n,k)*fallfac(t,k),k=0..n), n>=0. risefac(x,n):=product(x+j,j=0..n-1) for the rising factorials.

For positive s the signless triangles |L(s;n,k)| = L(s;n,k)*(-1)^n satisfies risefac(s*t,n) = sum(|L(s;n,k)|*fallfac(t,k),k=0..n), n>=0.

For negative s see the combinatorial interpretation given in the Charalambides reference, Example 8.8, p. 313: Coupon collector's problem.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, ch. 8.4 p. 301 ff. Table 8.3 (with row n=0 and column k=0 and s=-2).

W. Lang, First ten rows and more.

Recurrence: a(n,k)= 0 if n<k; a(n,0)= 1 if n=0 else 0 and a(n,k)= (-2*k-(n-1))*a(n-1,k)-2*a(n-1,k-1). From the Charalambides reference Theorem 8.19, p. 309, for s=-2.

E.g.f. column k: (1/(1+x)^2 - 1)^k/k!, k>=0. From the Charalambides reference Theorem 8.14, p. 305 for s=-2.(hence a Sheffer triangle of Jabotinsky type).

a(n,k)=sum(((-1)^(k-r))*binomial(k,r)*fallfac(-2*r,n),r=0..k)/k!, n>=k>=0. From the Charalambides reference Theorem 8.15, p. 306 for s=-2.

a(n,k)= sum(S1(n,r)*S2(r,k)*(-2)^r,r=k..n) with the Stirling numbers S1(n,r)= A048993(n,r) and S2(r,k)= A048993(r,k). From the Charalambides reference Theorem 8.13, p.304 for s=-2.

a(n,k)= sum(M_3(n,k,q)*product(fallfac(-2,j)^e(n,m,q,j),j=1..n),q=1..p(n,k)) if n>=k>=1, else 0. Here p(n,k)=A008284(n,m), the number of k parts partitions of n, the M_3 partition numbers are given in A036040 and e(n,m,q,j) is the exponent of j in the q-th k parts partition of n. Rewritten eq. (8.50), Theorem 8.16, p. 307, from the Charalambides reference for s=-2.

[1]; [0,-2]; [0,6,4]; [0,-24,-36,-8]; [0,120,300,144,16]; ...

Recurrence: a(4,2) = -7*a(3,2)-2*a(3,1) = -7*(-36) -2*(-24) = 300.

a(4,2)=300 as sum over the M3 numbers A036040 for the 2 parts partitons of 4: 4*fallfac(-2,1)^1*fallfac(-2,3)^1 + 3*fallfac(-2,2)^2 = 4*(-2)*(-24)+3*6^2 = 300.

Row n=3: [0,-24,-36,-8] for the coefficients in rewriting fallfac(-2*t,3)=((-1)^3)*risefac(2*t,3) = ((-1)^3)*(2*t)*(2*t+1)*(2*t+2) = 0*1 -24*t -36*t*(t-1) -8*t*(t-1)*(t-2).

Column sequences (unsigned) 2*A001710, 4*A136659, 8*A136660, 16*A136661 for k=1..4.

Cf. A136657 without row n=0 and column k=0, divided by 2.

Sequence in context: A055302 A055349 A161174 this_sequence A131595 A124228 A115879

Adjacent sequences: A136653 A136654 A136655 this_sequence A136657 A136658 A136659

sign,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Feb 22 2008, Sep 09 2008