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A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

%I A274760 #23 Mar 07 2025 03:40:40
%S A274760 1,1,10,478,68248,21809656,13107532816,13244650672240,
%T A274760 20818058883902848,48069880140604832128,156044927762422185270016,
%U A274760 687740710497308621254625536,4000181720339888446834235653120,29991260979682976913756629498334208
%N A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.
%C A274760 The multinomial transform [MNL] transforms an input sequence b(n) into the output sequence a(n). Given the fact that the structure of the a(n) formulas, see the examples, lead to the multinomial coefficients A036039 the MNL transform seems to be an appropriate name for this transform. The multinomial transform is related to the exponential transform, see A274804 and the third formula. For the inverse multinomial transform [IML] see A274844.
%C A274760 To preserve the identity IML[MNL[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, otherwise information about b(0) will be lost in transformation.
%C A274760 In the a(n) formulas, see the examples, the multinomial coefficients A036039 appear.
%C A274760 We observe that a(0) = 1 and that this term provides no information about any value of b(n), this notwithstanding we will start the a(n) sequence with a(0) = 1.
%C A274760 The Maple programs can be used to generate the multinomial transform of a sequence. The first program uses the first formula which was found by Paul D. Hanna, see A158876, and Vladimir Kruchinin, see A215915. The second program uses properties of the e.g.f., see the sequences A158876, A213507, A244430 and A274539 and the third formula. The third program uses information about the inverse multinomial transform, see A274844.
%C A274760 Some MNL transform pairs are, n >= 1: A000045(n) and A244430(n-1); A000045(n+1) and A213527(n-1); A000108(n) and A213507(n-1); A000108(n-1) and A243953(n-1); A000142(n) and A158876(n-1); A000203(n) and A053529(n-1); A000110(n) and A274539(n-1); A000041(n) and A215915(n-1); A000035(n-1) and A177145(n-1); A179184(n) and A038205(n-1); A267936(n) and A000266(n-1); A267871(n) and A000090(n-1); A193356(n) and A088009(n-1).
%D A274760 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.
%H A274760 M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some Canonical Sequences of Integers</a>, arXiv:math/0205301 [math.CO], 2002; Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
%H A274760 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 A274760 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H A274760 Eric W. Weisstein MathWorld, <a href="https://mathworld.wolfram.com/ExponentialTransform.html">Exponential Transform</a>.
%F A274760 a(n) = Sum_{k=1..n} ((n-1)!/(n-k)!)*b(k)*a(n-k), n >= 1 and a(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
%F A274760 a(n) = n!*P(n), with P(n) = (1/n)*(Sum_{k=0..n-1} b(n-k)*P(k)), n >= 1 and P(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
%F A274760 E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n) with b(n) = A001818(n) = ((2*n-1)!!)^2.
%F A274760 denom(a(n)/2^n) = A001316(n); numer(a(n)/2^n) = [1, 1, 5, 239, 8531, 2726207, ...].
%e A274760 Some a(n) formulas, see A036039:
%e A274760   a(0) = 1
%e A274760   a(1) = 1*x(1)
%e A274760   a(2) = 1*x(2) + 1*x(1)^2
%e A274760   a(3) = 2*x(3) + 3*x(1)*x(2) + 1*x(1)^3
%e A274760   a(4) = 6*x(4) + 8*x(1)*x(3) + 3*x(2)^2 + 6*x(1)^2*x(2) + 1*x(1)^4
%e A274760   a(5) = 24*x(5) + 30*x(1)*x(4) + 20*x(2)*x(3) + 20*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 10*x(1)^3*x(2) + 1*x(1)^5
%p A274760 nmax:= 13: b := proc(n): (doublefactorial(2*n-1))^2 end: a:= proc(n) option remember: if n=0 then 1 else add(((n-1)!/(n-k)!) * b(k) * a(n-k), k=1..n) fi: end: seq(a(n), n = 0..nmax); # End first MNL program.
%p A274760 nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := exp(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 = 0..nmax); # End second MNL program.
%p A274760 nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(log(1+add(s(n)*x^n/n!, n=1..nmax)), x, nmax+1): d := proc(n): n*coeff(f, x, n) end: a(0) := 1: a(1) := b(1): s(1) := b(1): for n from 2 to nmax do s(n) := solve(d(n)-b(n), s(n)): a(n):=s(n): od: seq(a(n), n=0..nmax); # End third MNL program.
%t A274760 b[n_] := (2*n - 1)!!^2;
%t A274760 a[0] = 1; a[n_] := a[n] = Sum[((n-1)!/(n-k)!)*b[k]*a[n-k], {k, 1, n}];
%t A274760 Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Nov 17 2017 *)
%Y A274760 Cf. A274844, A274804, A036039, A001818, A001316, A215915, A008279.
%Y A274760 Cf. A244430, A213527, A213507, A243953, A158876, A274539, A215915.
%Y A274760 Cf. A090998, A008298, A271703, A112302, A135002, A274540.
%K A274760 nonn
%O A274760 0,3
%A A274760 _Johannes W. Meijer_, Jul 27 2016