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A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2. 7
1, 1, 10, 478, 68248, 21809656, 13107532816, 13244650672240, 20818058883902848, 48069880140604832128, 156044927762422185270016, 687740710497308621254625536, 4000181720339888446834235653120, 29991260979682976913756629498334208 (list; graph; refs; listen; history; text; internal format)



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.

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.

In the a(n) formulas, see the examples, the multinomial coefficients A036039 appear.

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.

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.

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).


N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.


Table of n, a(n) for n=0..13.

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, Exponential Transform.


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.

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.

E.g.f.: exp(Sum{n >= 1}(b(n)*x^n/n) with b(n) = A001818(n) = ((2*n-1)!!)^2.

denom(a(n)/2^(n)) = A001316(n); numer(a(n)/2^n)) = [1, 1, 5, 239, 8531, 2726207, … ].


Some a(n) formulas, see A036039:

a(0) = 1

a(1) = 1*x(1)

a(2) = 1*x(2) + 1*x(1)^2

a(3) = 2*x(3) + 3*x(1)*x(2) + 1*x(1)^3

a(4) = 6*x(4) + 8*x(1)*x(3) + 3*x(2)^2 + 6*x(1)^2*x(2) + 1*x(1)^4

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


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.

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.

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.


b[n_] := (2*n - 1)!!^2;

a[0] = 1; a[n_] := a[n] = Sum[((n-1)!/(n-k)!)*b[k]*a[n-k], {k, 1, n}];

Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 17 2017 *)


Cf. A274844, A274804, A036039, A001818, A001316, A215915, A008279.

Cf. A244430, A213527, A213507, A243953, A158876, A274539, A215915.

Cf. A163930, A008298, A271703, A112302, A135002, A274540.

Sequence in context: A323205 A257133 A159533 * A173157 A035320 A200458

Adjacent sequences: A274757 A274758 A274759 * A274761 A274762 A274763




Johannes W. Meijer, Jul 27 2016



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