A274844 The inverse multinomial transform of A001818(n) = ((2*n-1)!!)^2.

1, 8, 100, 1664, 34336, 843776, 24046912, 779780096, 28357004800, 1143189536768, 50612287301632, 2441525866790912, 127479926768287744, 7163315850315825152, 431046122080208896000, 27655699473265974050816, 1884658377677216933085184

Offset: 1

Author

Johannes W. Meijer, Jul 27 2016

Keywords

• nonn

Comments

The inverse multinomial transform [IML] transforms an input sequence b(n) into the output sequence a(n). The IML transform inverses the effect of the multinomial transform [MNL], see A274760, and is related to the logarithmic transform, see A274805 and the first formula.

To preserve the identity MNL[IML[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL.

In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.

We observe that the inverse multinomial transform leaves the value of a(0) undefined.

The Maple programs can be used to generate the inverse multinomial transform of a sequence. The first program is derived from a formula given by Alois P. Heinz for the logarithmic transform, see the first formula and A001187. The second program uses the e.g.f. for multivariate row polynomials, see A127671 and the examples. The third program uses information about the inverse of the inverse of the multinomial transform, see A274760.

The IML transform of A001818(n) = ((2*n-1)!!)^2 leads quite unexpectedly to A005411(n), a sequence related to certain Feynman diagrams.

Some IML transform pairs, n >= 1: A000110(n) and 1/A000142(n-1); A137341(n) and A205543(n); A001044(n) and A003319(n+1); A005442(n) and A000204(n); A005443(n) and A001350(n); A007559(n) and A000244(n-1); A186685(n+1) and A131040(n-1); A061711(n) and A141151(n); A000246(n) and A000035(n); A001861(n) and A141044(n-1)/A001710(n-1); A002866(n) and A000225(n); A000262(n) and A000027(n).

Examples

			Some a(n) formulas, see A127671:
a(0) = undefined
a(1) = (1/0!) * (1*x(1))
a(2) = (1/1!) * (1*x(2) - x(1)^2)
a(3) = (1/2!) * (1*x(3) - 3*x(2)*x(1) + 2*x(1)^3)
a(4) = (1/3!) * (1*x(4) - 4*x(3)*x(1) - 3*x(2)^2 + 12*x(2)*x(1)^2 - 6*x(1)^4)
a(5) = (1/4!) * (1* x(5) - 5*x(4)*x(1) - 10*x(3)*x(2) + 20*x(3)*x(1)^2 + 30*x(2)^2*x(1) -60*x(2)*x(1)^3 + 24*x(1)^5)
		

References

• Richard P. Feynman, QED, The strange theory of light and matter, 1985.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Links

• M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
• N. J. A. Sloane, Transforms.
• Eric W. Weisstein MathWorld, Logarithmic Transform.
• Wikipedia, Feynman diagram

Crossrefs

Cf. A274760, A274805, A127671, A001187, A001818, A000079, A005411.

Programs

Maple

nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: c:= proc(n) option remember; b(n) - add(k*binomial(n, k)*b(n-k)*c(k), k=1..n-1)/n end: a := proc(n): c(n)/(n-1)! end: seq(a(n), n=1..nmax); # End first IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := log(1+add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n*coeff(t2, x, n) end: seq(a(n), n=1..nmax); # End second IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(exp(add(t(n)*x^n/n, n=1..nmax)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(1):=b(1): t(1):= b(1): for n from 2 to nmax+1 do t(n) := solve(d(n)-b(n), t(n)): a(n):=t(n): od: seq(a(n), n=1..nmax); # End third IML program.

Mathematica

nMax = 22; b[n_] := ((2*n-1)!!)^2; c[n_] := c[n] = b[n] - Sum[k*Binomial[n, k]*b[n-k]*c[k], {k, 1, n-1}]/n; a[n_] := c[n]/(n-1)!; Table[a[n], {n, 1, nMax}] (* Jean-François Alcover, Feb 27 2017, translated from Maple *)

Formula

a(n) = c(n)/(n-1)! with c(n) = b(n) - Sum_{k=1..n-1}(k*binomial(n, k)*b(n-k)*c(k)), n >= 1 and a(0) = undefined, with b(n) = A001818(n) = ((2*n-1)!!)^2.

a(n) = A000079(n-1) * A005411(n), n >= 1.