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A136656 Coefficients for rewriting generalized falling factorials into ordinary falling factorials.

%I A136656 #29 Aug 28 2019 17:15:43
%S A136656 1,0,-2,0,6,4,0,-24,-36,-8,0,120,300,144,16,0,-720,-2640,-2040,-480,
%T A136656 -32,0,5040,25200,27720,10320,1440,64,0,-40320,-262080,-383040,
%U A136656 -199920,-43680,-4032,-128,0,362880,2963520,5503680,3764880,1142400,163968,10752,256,0,-3628800,-36288000,-82978560
%N A136656 Coefficients for rewriting generalized falling factorials into ordinary falling factorials.
%C A136656 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).
%C A136656 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.
%C A136656 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.
%C A136656 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.
%C A136656 For negative s see the combinatorial interpretation given in the Charalambides reference, Example 8.8, p. 313: Coupon collector's problem.
%C A136656 |T(n,k)| = B_{n,k}((j+2)!; j>=0) where B_{n,k} are the partial Bell polynomials. - _Peter Luschny_, May 11 2015
%D A136656 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).
%H A136656 M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janjic/janjic22.html">Some classes of numbers and derivatives</a>, JIS 12 (2009) 09.8.3
%H A136656 W. Lang, <a href="/A136656/a136656.txt">First ten rows and more</a>.
%H A136656 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Marginalia">The Bell Transform</a>
%F A136656 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.
%F A136656 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).
%F A136656 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.
%F A136656 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.
%F A136656 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.
%F A136656 Recurrence for the unsigned case: a(n,k) = Sum_{j=0..n-k} a(n-j-1,k-1)*C(n-1,j)*(j+2)! if k<>0 else k^n. - _Peter Luschny_, Mar 31 2015
%e A136656 Triangle starts:
%e A136656 [1]
%e A136656 [0,   -2]
%e A136656 [0,    6,    4]
%e A136656 [0,  -24,  -36,   -8]
%e A136656 [0,  120,  300,  144,  16]
%e A136656 ...
%e A136656 Recurrence: a(4,2) = -7*a(3,2)-2*a(3,1) = -7*(-36) -2*(-24) = 300.
%e A136656 a(4,2)=300 as sum over the M3 numbers A036040 for the 2 parts partitions of 4: 4*fallfac(-2,1)^1*fallfac(-2,3)^1 + 3*fallfac(-2,2)^2 = 4*(-2)*(-24)+3*6^2 = 300.
%e A136656 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).
%p A136656 # The function BellMatrix is defined in A264428.
%p A136656 BellMatrix(n -> (-1)^(n+1)*(n+2)!, 9); # _Peter Luschny_, Jan 27 2016
%t A136656 fallfac[n_, k_] := Pochhammer[n - k + 1, k]; a[n_, k_] := Sum[(-1)^(k - r)*Binomial[k, r]*fallfac[-2*r, n], {r, 0, k}]/k!; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 09 2013 *)
%t A136656 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A136656 rows = 10;
%t A136656 M = BellMatrix[(-1)^(# + 1)*(# + 2)!&, rows];
%t A136656 Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (*_Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)
%o A136656 (Sage)
%o A136656 @CachedFunction
%o A136656 def T(n, k):  # unsigned case
%o A136656     if k == 0: return 1 if n == 0 else 0
%o A136656     return sum(T(n-j-1,k-1)*(j+1)*(j+2)*gamma(n)/gamma(n-j) for j in (0..n-k))
%o A136656 for n in range(7): [T(n,k) for k in (0..n)] # _Peter Luschny_, Mar 31 2015
%Y A136656 Column sequences (unsigned) 2*A001710, 4*A136659, 8*A136660, 16*A136661 for k=1..4.
%Y A136656 Cf. A136657 without row n=0 and column k=0, divided by 2.
%K A136656 sign,easy,tabl
%O A136656 0,3
%A A136656 _Wolfdieter Lang_, Feb 22 2008, Sep 09 2008