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A132382 Lower triangular array T(n,k) generator for group of arrays related to A001147 and A102625.

%I A132382 #26 Nov 15 2019 23:09:10
%S A132382 1,-1,1,-1,-2,1,-3,-3,-3,1,-15,-12,-6,-4,1,-105,-75,-30,-10,-5,1,-945,
%T A132382 -630,-225,-60,-15,-6,1,-10395,-6615,-2205,-525,-105,-21,-7,1,-135135,
%U A132382 -83160,-26460,-5880,-1050,-168,-28,-8,1,-2027025,-1216215,-374220,-79380,-13230,-1890,-252,-36,-9,1
%N A132382 Lower triangular array T(n,k) generator for group of arrays related to A001147 and A102625.
%C A132382 Let b(n) = LPT[ A001147 ] = -A001147(n-1) for n > 0 and 1 for n=0, where LPT represents the action of the list partition transform described in A133314.
%C A132382 Then T(n,k) = binomial(n,k) * b(n-k) .
%C A132382 Form the matrix of polynomials TB(n,k,t) = T(n,k) * t^(n-k) = binomial(n,k) * b(n-k) * t^(n-k) = binomial(n,k) * Pb(n-k,t),
%C A132382 beginning as
%C A132382      1;
%C A132382     -1,       1;
%C A132382     -1*t,    -2,       1;
%C A132382     -3*t^2,  -3*t,    -3,       1;
%C A132382    -15*t^3, -12*t^2,  -6*t,    -4,    1;
%C A132382   -105*t^4, -75*t^3, -30*t^2, -10*t, -5, 1;
%C A132382 Let Pc(n,t) = LPT(Pb(.,t)).
%C A132382 Then [TB(t)]^(-1) = TC(t) = [ binomial(n,k) * Pc(n-k,t) ] = LPT(TB),
%C A132382 whose first column is
%C A132382 Pc(0,t) = 1
%C A132382 Pc(1,t) = 1
%C A132382 Pc(2,t) = 2 + t
%C A132382 Pc(3,t) = 6 + 6*t + 3*t^2
%C A132382 Pc(4,t) = 24 + 36*t + 30*t^2 + 15*t^3
%C A132382 Pc(5,t) = 120 + 240*t + 270*t^2 + 210*t^3 + 105*t^4.
%C A132382 The coefficients of these polynomials are given by the reverse of A102625 with the highest order coefficients given by A001147 with an additional leading 1.
%C A132382 Note this is not the complete matrix TC. The complete matrix is formed by multiplying along the diagonal of the lower triangular Pascal matrix by these polynomials, embedding trees of coefficients in the matrix.
%C A132382 exp[Pb(.,t)*x] = 1 + [(1-2t*x)^(1/2) - 1] / (t-0) = [1 + a finite diff. of [(1-2t*x)^(1/2)] with step t] = e.g.f. of the first column of TB.
%C A132382 exp[Pc(.,t)*x] = 1 / { 1 + [(1-2t*x)^(1/2) - 1] / t } = 1 / exp[Pb(.,t)*x) = e.g.f. of the first column of TC.
%C A132382 TB(t) and TC(t), being inverse to each other, are the generators of an Abelian group.
%C A132382 TB(0) and TC(0) are generators for a subgroup representing the iterated Laguerre operator described in A132013 and A132014.
%C A132382 Let sb(t,m) and sc(t,m) be the associated sequences under the LPT to TB(t)^m = B(t,m) and TC(t)^m = C(t,m).
%C A132382 Let Esb(t,m) and Esc(t,m) be e.g.f.'s for sb(t,m) and sc(t,m), rB(t,m) and rC(t,m) be the row sums of B(t,m) and C(t,m) and aB(t,m) and aC(t,m) be the alternating row sums.
%C A132382 Then B(t,m) is the inverse of C(t,m), Esb(t,m) is the reciprocal of Esc(t,m) and sb(t,m) and sc(t,m) form a reciprocal pair under the LPT. Similar relations hold among the row sums and the alternating sign row sums and associated quantities.
%C A132382 All the group members have the form B(t,m) * C(u,p) = TB(t)^m * TC(u)^p = [ binomial(n,k) * s(n-k) ]
%C A132382 with associated e.g.f. Es(x) = exp[m * Pb(.,t) * x] * exp[p * Pc(.,u) * x] for the first column of the matrix, with terms s(n), so group multiplication is isomorphic to matrix multiplication and to multiplication of the e.g.f.'s for the associated sequences (see examples).
%C A132382 These results can be extended to other groups of integer-valued arrays by replacing the 2 by any natural number in the expression for exp[Pb(.,t)*x].
%C A132382 More generally,
%C A132382 [ G.f. for M = Product_{i=0..j} B[s(i),m(i)] * C[t(i),n(i)] ]
%C A132382 = exp(u*x) * Product_{i=0..j} { exp[m(i) * Pb(.,s(i)) * x] * exp[n(i) * Pc(.,t(i)) * x] }
%C A132382 = exp(u*x) * Product_{i=0..j} { 1 + [ (1 - 2*s(i)*x)^(1/2) - 1 ] / s(i) }^m(i) / { 1 + [ (1 - 2*t(i)*x)^(1/2) - 1 ] / t(i) }^n(i)
%C A132382 = exp(u*x) * H(x)
%C A132382 [ E.g.f. for M ] = I_o[2*(u*x)^(1/2)] * H(x).
%C A132382 M is an integer-valued matrix for m(i) and n(i) positive integers and s(i) and t(i) integers. To invert M, change B to C in Product for M.
%C A132382 H(x) is the e.g.f. for the first column of M and diagonally multiplying the Pascal matrix by the terms of this column generates M. See examples.
%C A132382 The G.f. for M, i.e., the e.g.f. for the row polynomials of M, implies that the row polynomials form an Appell sequence (see Wikipedia and Mathworld). - _Tom Copeland_, Dec 03 2013
%F A132382 [G.f. for TB(n,k,t)] = GTB(u,x,t) = exp(u*x) * { 1 + [ (1 - 2t*x)^(1/2) - 1 ] / t } = exp[(u+Pb(.,t))*x] where TB(n,k,t) = (D_x)^n (D_u)^k /k! GTB(u,x,t) eval. at u=x=0.
%F A132382 [G.f. for TC(n,k,t)] = GTC(u,x,t) = exp(u*x) / { 1 + [ (1 - 2t*x)^(1/2) - 1 ] / t } = exp[(u+Pc(.,t))*x] where TC(n,k,t) = (D_x)^n (D_u)^k /k! GTC(u,x,t) eval. at u=x=0.
%F A132382 [E.g.f. for TB(n,k,t)] = I_o[2*(u*x)^(1/2)] * { 1 + [ (1 - 2t*x)^(1/2) - 1 ] / t } and
%F A132382 [E.g.f. for TC(n,k,t)] = I_o[2*(u*x)^(1/2)] / { 1 + [ (1 - 2t*x)^(1/2) - 1 ] / t }
%F A132382 where I_o is the zeroth modified Bessel function of the first kind, i.e.,
%F A132382 I_o[2*(u*x)^(1/2)] = Sum_{j>=0} (u^j/j!) * (x^j/j!).
%F A132382 So [e.g.f. for TB(n,k)] = I_o[2*(u*x)^(1/2)] * (1 - 2x)^(1/2).
%e A132382 Some group members and associated arrays are
%e A132382 (t,m) :: Array :: Asc. Matrix :: Asc. Sequence :: E.g.f. for sequence
%e A132382 ..............................................................................
%e A132382 (0,1).::.B..::..A132013.::.(1,-1,0,0,0,0,...).....::.s(x).=.1-x
%e A132382 (0,1).::.C..::..A094587.::.(0!,1!,2!,3!,...)......::.1./.s(x)
%e A132382 (0,1).::.rB.::.~A055137.::.(1,0,-1,-2,-3,-4,...)..::.exp(x).*.s(x)
%e A132382 (0,1).::.rC.::....-.....::..A000522...............::.exp(x)./.s(x)
%e A132382 (0,1).::.aB.::....-.....::.(1,-2,3,-4,5,-6,...)...::.exp(-x).*.s(x)
%e A132382 (0,1).::.aC.::..A008290.::..A000166...............::.exp(-x)./.s(x)
%e A132382 ..............................................................................
%e A132382 (0,2).::.B..::..A132014.::.(1,-2,2,0,0,0,0...)....::.s(x).=.(1-x)^2
%e A132382 (0,2).::.C..::..A132159.::.(1!,2!,3!,4!,...)......::..1./.s(x).
%e A132382 (0,2).::.rB.::...-......::.(1,-1,-1,1,5,11,19,29,)::.exp(x).*.s(x).
%e A132382 (0,2).::.rC.::...-......::..A001339...............::.exp(x)./.s(x).
%e A132382 (0,2).::.aB.::...-......::.(-1)^n.A002061(n+1)....::.exp(-x).*.s(x).
%e A132382 (0,2).::.aC.::...-......::..A000255...............::.exp(-x)./.s(x).
%e A132382 ..............................................................................
%e A132382 (1,1).::.B..::..T.......::.(1,-A001147(n-1))......::.s(x).=.(1-2x)^(1/2)
%e A132382 (1,1).::.C..::.~A113278.::..A001147...............::.1./.s(x)...
%e A132382 (1,1).::.rB.::...-......::..A055142...............::.exp(x).*.s(x).
%e A132382 (1,1).::.rC.::...-......::..A084262...............::.exp(x)./.s(x).
%e A132382 (1,1).::.aB.::...-......::.(1,-2,2,-4,-4,-56,...).::.exp(-x).*.s(x).
%e A132382 (1,1).::.aC.::...-......::..A053871...............::.exp(-x)./.s(x).
%e A132382 ..............................................................................
%e A132382 (2,1).::.B..::...-......::.(1,-A001813)...........::.s=[1+(1-4x)^(1/2)]/2....
%e A132382 (2,1).::.C..::...-......::..A001761...............::.1./.s(x)..
%e A132382 (2,1).::.rB.::...-......::.(1,0,-3,-20,-183,...)..::.exp(x).*.s(x)..
%e A132382 (2,1).::.rC.::...-......::.(1,2,7,46,485,...).....::.exp(x)./.s(x).
%e A132382 (2,1).::.aB.::...-......::.(1,-2,1,-10,-79,...)...::.exp(-x).*.s(x).
%e A132382 (2,1).::.aC.::...-......::.(1,0,3,20,237,...).....::.exp(-x)./.s(x)
%e A132382 ..............................................................................
%e A132382 (1,2).::.B..::.~A134082.::.(1,-2,0,0,0,0,...).....::.s(x).=.1.-.2x
%e A132382 (1,2).::.C..::....-.....::..A000165...............::.1./.s(x)..
%e A132382 (1,2).::.rB.::....-.....::.(1,-1,-3,-5,-7,-9,...).::.exp(x).*.s(x).
%e A132382 (1,2).::.rC.::....-.....::..A010844...............::.exp(x)./.s(x)..
%e A132382 (1,2).::.aB.::....-.....::.(1,-3,5,-7,9,-11,...)..::.exp(-x).*.s(x).
%e A132382 (1,2).::.aC.::....-.....::..A000354...............::.exp(-x)./.s(x).
%e A132382 ..............................................................................
%e A132382 (The tilde indicates the match is not exact--specifically, there are differences in signs from the true matrices.)
%e A132382 Note the row sums correspond to binomial transforms of s(x) and the alternating row sums, to inverse binomial transforms, or, finite differences.
%e A132382 Some additional examples:
%e A132382 C(1,2)*B(0,1) = B(1,-2)*C(0,-1) = [ binomial(n,k)*A002866(n-k) ] with asc. e.g.f. (1-x) / (1-2x).
%e A132382 B(1,2)*C(0,1) = C(1,-2)*B(0,-1) = 2I - A094587 with asc. e.g.f. (1-2x) / (1-x).
%K A132382 sign,tabl
%O A132382 0,5
%A A132382 _Tom Copeland_, Nov 11 2007, Nov 12 2007, Nov 19 2007, Dec 04 2007, Dec 06 2007
%E A132382 More terms from _Tom Copeland_, Dec 05 2007