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

%I A274844 #19 Feb 16 2025 08:33:36
%S A274844 1,8,100,1664,34336,843776,24046912,779780096,28357004800,
%T A274844 1143189536768,50612287301632,2441525866790912,127479926768287744,
%U A274844 7163315850315825152,431046122080208896000,27655699473265974050816,1884658377677216933085184
%N A274844 The inverse multinomial transform of A001818(n) = ((2*n-1)!!)^2.
%C A274844 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.
%C A274844 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.
%C A274844 In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
%C A274844 We observe that the inverse multinomial transform leaves the value of a(0) undefined.
%C A274844 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.
%C A274844 The IML transform of A001818(n) = ((2*n-1)!!)^2 leads quite unexpectedly to A005411(n), a sequence related to certain Feynman diagrams.
%C A274844 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).
%D A274844 Richard P. Feynman, QED, The strange theory of light and matter, 1985.
%D A274844 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.
%H A274844 M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some Canonical Sequences of Integers</a> Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
%H A274844 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A274844 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H A274844 Eric W. Weisstein MathWorld, <a href="https://mathworld.wolfram.com/LogarithmicTransform.html">Logarithmic Transform</a>.
%H A274844 Wikipedia, <a href="https://en.wikipedia.org/wiki/Feynman_diagram">Feynman diagram</a>
%F A274844 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.
%F A274844 a(n) = A000079(n-1) * A005411(n), n >= 1.
%e A274844 Some a(n) formulas, see A127671:
%e A274844 a(0) = undefined
%e A274844 a(1) = (1/0!) * (1*x(1))
%e A274844 a(2) = (1/1!) * (1*x(2) - x(1)^2)
%e A274844 a(3) = (1/2!) * (1*x(3) - 3*x(2)*x(1) + 2*x(1)^3)
%e A274844 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)
%e A274844 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)
%p A274844 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.
%p A274844 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.
%p A274844 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.
%t A274844 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 *)
%Y A274844 Cf. A274760, A274805, A127671, A001187, A001818, A000079, A005411.
%K A274844 nonn
%O A274844 1,2
%A A274844 _Johannes W. Meijer_, Jul 27 2016