%I A132382
%S A132382 1,1,1,1,2,1,3,3,3,1,15,12,6,4,1,105,75,30,10,5,1,945,630,225,60,15,6,
1,
%T A132382 10395,6615,2205,525,105,21,7,1,135135,83160,26460,5880,1050,168,28,8,
1,
%U A132382 2027025,1216215,374220,79380,13230,1890,252,36,9,1
%V A132382 1,-1,1,-1,-2,1,-3,-3,-3,1,-15,-12,-6,-4,1,-105,-75,-30,-10,-5,1,-945,
-630,-225,-60,
%W A132382 -15,-6,1,-10395,-6615,-2205,-525,-105,-21,-7,1,-135135,-83160,-26460,
-5880,-1050,-168,
%X A132382 -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 columnn 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 interated
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) * Prod(i=0,...,j) { exp[m(i) * Pb(.,s(i)) * x] * exp[n(i)
* Pc(.,t(i)) * x] }
%C A132382 = exp(u*x) * Prod(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.
%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=0,1,...) 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 seq.
%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.::....-.....::.(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.::....-.....::..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 are:
%e A132382 C(1,2)*B(0,1) = B(1,-2)*C(0,-1) = [ Binom(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).
%Y A132382 Sequence in context: A136018 A138022 A113278 this_sequence A048865 A058754
A125087
%Y A132382 Adjacent sequences: A132379 A132380 A132381 this_sequence A132383 A132384
A132385
%K A132382 sign,tabl
%O A132382 0,5
%A A132382 Tom Copeland (tcjpn(AT)msn.com), Nov 11 2007, Nov 12 2007, Nov 19 2007,
Dec 04 2007, Dec 06 2007
%E A132382 More terms from Tom Copeland (tcjpn(AT)msn.com), Dec 05 2007
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