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A134685
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Irregular triangle read by rows: coefficients C(j,k) of a partition transform for direct Lagrange inversion .
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14
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1, -1, 3, -1, -15, 10, -1, 105, -105, 15, 10, -1, -945, 1260, -280, -210, 21, 35, -1
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OFFSET
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1,3
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COMMENTS
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Let f(t) = u(t) - u(0) = Ev[exp(u.* t) - u(0)] = ln{Ev[(exp(z.* t))/z_0]} = Ev[-ln(1- a.* t)], where the operator Ev denotes umbral evaluation of the umbral variables u., z. or a., e.g., Ev[a.^n + a.^m] = a_n + a_m . The relation between z_n and u_n is given in reference in A127671 and u_n = (n-1)! * a_n .
If u_1 is not equal to 0, then the compositional inverse for these expressions is given by g(t) = sum(j=1,...) P(j,t) where, with u_n denoted by (n') for brevity,
P(1,t) = (1')^(-1) * [ 1 ] * t
P(2,t) = (1')^(-3) * [ -(2') ] * t^2 / 2!
P(3,t) = (1')^(-5) * [ 3 (2')^2 - (1')(3') ] * t^3 / 3!
P(4,t) = (1')^(-7) * [ -15 (2')^3 + 10 (1')(2')(3') - (1')^2 (4') ] * t^4 / 4!
P(5,t) = (1')^(-9) * [ 105 (2')^4 - 105 (1') (2')^2 (3') + 15 (1')^2 (2') (4') + 10 (1')^2 (3')^2 - (1')^3 (5') ] * t^5 / 5!
P(6,t) = (1')^(-11) * [ -945 (2')^5 + 1260 (1') (2')^3 (3') - 280 (1')^2 (2') (3')^2 - 210 (1')^2 (2')^2 (4') + 21 (1')^3 (2')(5') + 35 (1')^3 (3')(4') - (1')^4 (6') ] * t^6 / 6!
...
Substituting ((m-1)') for (m') in each partition and ignoring the (0') factors, the partitions in the brackets of P(n,t) become those of n-1 listed in Abramowitz and Stegun on page 831 and the number of partitions in P(n,t) is given by A000041(n-1).
Combinatorial interpretations are given in the link.
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 831.
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LINKS
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Table of n, a(n) for n=1..19.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
T. Copeland, Short Note on Lagrange Inversion, (posted Sept 2008)
T. Copeland, Lagrange a la Lah
E. Getzler, The semi-classical approximation for modular operads (see pg. 2)
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FORMULA
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The bracketed partitions of P(n,t) are of the form (u_1)^e(1) (u_2)^e(2) ... (u_n)^e(n) with coefficients given by (-1)^(n-1+e(1)) * [2*(n-1)-e(1)]! / [2!^e(2)*e(2)!*3!^e(3)*e(3)! ... n!^e(n)*e(n)! ].
Contribution from Tom Copeland, Sep 05 2011: (Start)
Let h(t) = 1/(df(t)/dt)
= 1/Ev[u.*exp(u.*t)]
= 1/(u_1+(u_2)*t+(u_3)*t^2/2!+(u_4)*t^3/3!+...),
an e.g.f. for the partition polynomials of A133314
(signed A049019) with an index shift.
Then for the partition polynomials of A134685,
n!*P(n,t) = ((t*h(y)*d/dy)^n) y evaluated at y=0,
and the compositional inverse of f(t) is
g(t) = exp(t*h(y)*d/dy) y evaluated at y=0.
Also, dg(t)/dt = h(g(t)). (Cf. A000311 and A134991)(End)
Contribution from Tom Copeland, Oct 30 2011: (Start)
With exp[x* PS(.,t)] = exp[t*g(x)]=exp[x*h(y)d/dy] exp(t*y) eval. at y=0, the raising/creation and lowering/annihilation operators
defined by R PS(n,t)=PS(n+1,t) and L PS(n,t)= n*PS(n-1,t) are
R = t*h(d/dt) = t* 1/[u_1+(u_2)*d/dt+(u_3)*(d/dt)^2/2!+...], and
L =f(d/dt)=(u_1)*d/dt+(u_2)*(d/dt)^2/2!+(u_3)*(d/dt)^3/3!+....
Then P(n,t) = (t^n/n!) dPS(n,z)/dz eval. at z=0. (Cf. A139605, A145271, and link therein to Mathemagical Forests for relation to planted trees on p. 13.) (End)
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EXAMPLE
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Examples and checks:
1) Let u_1 = -1 and u_n = 1 for n>1, then f(t) = exp(u.*
t) - u(0) = exp(t)-2t-1 and g(t) = [e.g.f. of signed A000311];
therefore the row sums of unsigned [C(j,k)] are A000311 =
(0,1,1,4,26,236,2752,...) = (0,-P(1,1),2!*P(2,1),-3!*P(3,1),4!*P(4,1),...) .
2) Let u_1 = -1 and u_n = (n-1)! for n>1, then f(t) =
-ln(1-t)-2t and g(t) = [e.g.f. of signed (0,A032188)] with (0,A032188)
= (0,1,1,5,41,469,6889,...) = (0,-P(1,1),2!*P(2,1),-3!P(3,1),...) .
3) Let u_1 = -1 and u_n = (-1)^n (n-2)! for n>1, then
f(t) = (1+t) ln(1+t) - 2t and g(t) = [e.g.f. of signed (0,A074059)]
with (0,A074059) = (0,1,1,2,7,34,213,...) =
(0,-P(1,1),2!*P(2,1),-3!*P(3,1),...) .
4) Let u_1 = 1, u_2 = -1 and u_n = 0 for n>2, then f(t)
= t(1-t/2) and g(t) = [e.g.f. of (0,A001147)] = 1 - (1-2t)^(1/2)
with (0,A001147) = (0,1,1,3,15,105,945...) =
(0,P(1,1),2!*P(2,1),3!*P(3,1),...) .
5) Let u_1 = 1, u_2 = -2 and u_n = 0 for n>2, then f(t)
= t(1-t) and g(t) = t * [o.g.f. of A000108] = [1 - (1-4t)^(1/2)] / 2
with (0,A000108) = (0,1,1,2,5,14,42,...) =
(0,P(1,1),P(2,1),P(3,1),...) .
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CROSSREFS
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Cf. A145271, (A134991, A019538) = (reduced array, associated g(x)).
Sequence in context: A039815 A147453 A147020 * A130757 A181996 A014621
Adjacent sequences: A134682 A134683 A134684 * A134686 A134687 A134688
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KEYWORD
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sign,tabf,more
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AUTHOR
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Tom Copeland, Jan 26 2008, Sep 13 2008
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STATUS
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approved
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