A158876 -id:A158876 - cp's OEIS frontend

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

1, 1, 10, 478, 68248, 21809656, 13107532816, 13244650672240, 20818058883902848, 48069880140604832128, 156044927762422185270016, 687740710497308621254625536, 4000181720339888446834235653120, 29991260979682976913756629498334208

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

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).

			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

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

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, 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]
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.

Cf. A274844, A274804, A036039, A001818, A001316, A215915, A008279.
Cf. A244430, A213527, A213507, A243953, A158876, A274539, A215915.
Cf. A090998, A008298, A271703, A112302, A135002, A274540.

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 *)

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, ...].

A293847 E.g.f.: exp(Sum_{n>=1} n!*x^n).

Original entry on oeis.org

1, 1, 5, 49, 793, 19361, 672061, 31721425, 1963804529, 154746407233, 15136503333301, 1799712380844401, 255578390749947145, 42713809784784354529, 8296411053128532892013, 1852797862395580239567121, 471358206112272764630500321, 135500644700064476406317390465

Offset: 0

Seiichi Manyama, Oct 17 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..253

Cf. A158876, A159476.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-i)*binomial(n-1, i-1)*i!^2, i=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 02 2021

nmax = 20; CoefficientList[Series[E^Sum[k!*x^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 18 2017 *)

{a(n) = n!*polcoeff(exp(sum(k=1, n, k!*x^k)+x*O(x^n)), n)}

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*k!*a(n-k)/(n-k)! for n > 0.

a(n) ~ n!^2. - Vaclav Kotesovec, Oct 18 2017

A159476 Expansion of e.g.f.: A(x) = exp( Sum_{n>=1} (n-1)!*x^n/n ).

Original entry on oeis.org

1, 1, 2, 8, 62, 862, 19492, 656224, 30739676, 1906807004, 151002453464, 14846381034784, 1772922018732328, 252631570039665832, 42329528274029082608, 8237406877267427867648, 1842215469973381977889808, 469160036709398319115207696, 134976328490030629922214893344

Offset: 0

Paul D. Hanna, Apr 15 2009

nonn

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 62*x^4/4! + 862*x^5/5! + ...
log(A(x)) = x + x^2/2 + 2!*x^3/3 + 3!*x^4/4 + 4!*x^5/5 + 5!*x^6/6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A158876, A293847.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-i)*binomial(n-1, i-1)*(i-1)!^2, i=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 13 2019

a:= CoefficientList[Series[Exp[Sum[(n - 1)!*x^n/n, {n, 1, 500}]], {x, 0, 35}], x]; Table[a[[n]]*(n - 1)!, {n, 1, 30}] (* G. C. Greubel, Jul 09 2018 *)

{a(n)=n!*polcoeff(exp(sum(k=1,n,(k-1)!*x^k/k)+x*O(x^n)),n)}

{a(n)=if(n==0,1,(n-1)!*sum(k=1,n,(k-1)!*a(n-k)/(n-k)!))}

a(n) = (n-1)!*Sum_{k=1..n} (k-1)!*a(n-k)/(n-k)! for n > 0 with a(0)=1.

a(n) ~ (n-1)!^2. - Vaclav Kotesovec, Jul 10 2018

A370084 Expansion of e.g.f. exp( Sum_{k>=1} (2k)! (x/2)^k/k ).

Original entry on oeis.org

1, 1, 7, 199, 15985, 2810401, 916442551, 497173378087, 416055798439009, 508038355486018945, 867346995542879712871, 2001283948891590010467271, 6070787760933663329315253457, 23660956435620796919900079644449

Offset: 0

Seiichi Manyama, Feb 08 2024

nonn

Cf. A158876, A370085.

Cf. A370086.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, (2*k)!*(x/2)^k/k))))

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} (2*k)! * a(n-k) / (2^k * (n-k)!).

A370085 Expansion of e.g.f. exp( Sum_{k>=1} (3k)! (x/6)^k/k ).

Original entry on oeis.org

1, 1, 21, 3421, 2232361, 4047831801, 16492042429501, 131524059703672021, 1862832637043775536721, 43582296764832433769579761, 1592318494850388661944336320101, 86870590496672779178378519636679501, 6822481321458299127004125050236325798521

Offset: 0

Seiichi Manyama, Feb 08 2024

nonn

Cf. A158876, A370084.

Cf. A370087.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, (3*k)!*(x/6)^k/k))))

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} (3*k)! * a(n-k) / (6^k * (n-k)!).

A274539 E.g.f.: exp(sum(bell(n)*z^n/n, n=1..infinity)).

Original entry on oeis.org

1, 1, 3, 17, 155, 2079, 38629, 951187, 29979753, 1175837345, 56066617331, 3187704802281, 212628685506643, 16413606252207007, 1449425836362499605, 144977415195565990619, 16285937949513614300369, 2039447464767566886933057, 282862729890000953318773603

Offset: 0

Johannes W. Meijer, Jun 29 2016

nonn

The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.

Some transform pairs, see the formula section, are: x(n) = A000027(n) and a(n) = A000262(n); x(n) = A000045(n) and a(n) = A244430(n); x(n) = A000079(n) and a(n) = A000165(n); x(n) = A000108(n) and a(n) = A213507(n); x(n) = A000142(n) and a(n) = A158876(n); x(n) = A000203(n) and a(n) = A053529(n).

Cf. A036039, A000110, A000165, A000262, A053529, A158876, A213507, A244430.

a := proc(n): n!*P(n) end: P := proc(n): if n=0 then 1 else P(n):= expand((1/n)*(add(x(n-k) * P(k), k=0..n-1))) fi; end: with(combinat): x := proc(n): bell(n) end: seq(a(n), n=0..18);

nmax = 20; CoefficientList[Series[E^(Sum[BellB[n]*z^n/n, {n, 1, nmax}]), {z, 0, nmax}], z] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 29 2016 *)

a(n) = n! * P(n), with P(n) = (1/n)*(sum(x(n-k) * P(k), k=0..n-1)), n >=1 and P(0) = 1, with x(n) = A000110(n), the Bell numbers.

E.g.f.: exp(sum(x(n)*z^n/n, n=1..infinity)) with x(n) = A000110(n).

cp's OEIS Frontend

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

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A293847 E.g.f.: exp(Sum_{n>=1} n!*x^n).

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A159476 Expansion of e.g.f.: A(x) = exp( Sum_{n>=1} (n-1)!*x^n/n ).

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A370084 Expansion of e.g.f. exp( Sum_{k>=1} (2*k)! * (x/2)^k/k ).

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A370085 Expansion of e.g.f. exp( Sum_{k>=1} (3*k)! * (x/6)^k/k ).

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A274539 E.g.f.: exp(sum(bell(n)*z^n/n, n=1..infinity)).

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A370084 Expansion of e.g.f. exp( Sum_{k>=1} (2k)! (x/2)^k/k ).

A370085 Expansion of e.g.f. exp( Sum_{k>=1} (3k)! (x/6)^k/k ).