A274540 -id:A274540 - 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, ...].

A123168 Continued fraction for c = sqrt(2)*(exp(sqrt(2))-1)/(exp(sqrt(2))+1).

Original entry on oeis.org

0, 1, 6, 5, 14, 9, 22, 13, 30, 17, 38, 21, 46, 25, 54, 29, 62, 33, 70, 37, 78, 41, 86, 45, 94, 49, 102, 53, 110, 57, 118, 61, 126, 65, 134, 69, 142, 73, 150, 77, 158, 81, 166, 85, 174, 89, 182, 93, 190, 97, 198, 101, 206, 105, 214, 109, 222, 113

Offset: 1

Benoit Cloitre, Oct 02 2006

This continued fraction shows exp(sqrt(2)) is irrational (see A274540).

J. Borwein and D. Bailey, Mathematics by experiment, plausible reasoning in the 21st Century, A. K. Peters, p. 77.

G. C. Greubel, Table of n, a(n) for n = 1..1500
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A000217, A000384, A212343.
Odd bisection of A062828 with 0 prepended.
Cf. A274540, A123251.

$MinPrecision = 5 $MachinePrecision; ContinuedFraction[Sqrt[2]* (Exp[Sqrt[2]] - 1)/(Exp[Sqrt[2]] + 1), 100]  (* G. C. Greubel, Aug 17 2018 *) (* or *)
LinearRecurrence[{0, 2, 0, -1}, {0, 1, 6, 5, 14}, 100] (* Georg Fischer, Aug 26 2022 *)

default(realprecision, 1000); contfrac(sqrt(2)*(exp(sqrt(2))-1)/ (exp(sqrt(2))+1)) \\ Michel Marcus, Oct 11 2016

a(2*n) = 4*n-3, a(2*n+1) = 8*n-2.
From Colin Barker, Apr 15 2012: (Start)
a(n) = 2*a(n-2) - a(n-4) for n>5.
G.f.: x^2*(1+6*x+3*x^2+2*x^3)/((1-x)^2*(1+x)^2). (End)
a(n) = (2*n-3)*(3-(-1)^n)/2 for n>1, with a(1) = 0. - Wesley Ivan Hurt, Apr 01 2022

A274541 Decimal expansion of exp(sqrt(2)/2).

Original entry on oeis.org

2, 0, 2, 8, 1, 1, 4, 9, 8, 1, 6, 4, 7, 4, 7, 2, 4, 5, 1, 1, 0, 8, 1, 2, 6, 1, 1, 2, 7, 4, 6, 3, 5, 1, 1, 7, 5, 1, 7, 4, 3, 2, 5, 0, 9, 2, 5, 4, 2, 6, 1, 3, 5, 2, 0, 6, 1, 7, 7, 7, 5, 9, 7, 2, 1, 2, 2, 2, 1, 5, 3, 9, 5, 0, 4, 8, 7, 1, 6, 5, 5, 9, 4, 2, 5, 9, 6

Offset: 1

Johannes W. Meijer, Jun 27 2016

Define P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k), n >= 1 and P(0) = 1 with x(2) = (sqrt(2) + 1) and x(n) = 1 for all other n.
We find that C2 = lim_{n->infinity} P(n) = exp(sqrt(2)/2).
The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.

			c = 2.02811498164747245110812611274635117517432509254...

G. C. Greubel, Table of n, a(n) for n = 1..10000
The Dev Team and Simon Plouffe, The Inverse Symbolic Calculator (ISC).
Index entries for transcendental numbers

Cf. A274540, A274542, A014176, A010503, A036039.

SetDefaultRealField(RealField(100)); Exp[Sqrt[2]/2]; // G. C. Greubel, Aug 19 2018

Digits := 140: evalf(exp(sqrt(2)/2)); # End program 1.
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: x := proc(n): if n=2 then (sqrt(2)+1) else 1 fi: end:
Digits := 140: evalf(P(250)); # End program 2.

First@ RealDigits@ N[Exp[Sqrt[2]/2], 83] (* Michael De Vlieger, Jun 27 2016 *)

my(x=exp(sqrt(2)/2)); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", ")) \\ Felix Fröhlich, Jun 27 2016

c = exp(sqrt(2)/2).

c = lim_{n->infinity} P(n), with P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k), for n >= 1, and P(0) = 1, with x(2) = (1 + sqrt(2)), the silver mean A014176, and x(n) = 1 for all other n.

More digits from Jon E. Schoenfield, Mar 15 2018

A274542 Decimal expansion of exp(sqrt(2)/3).

Original entry on oeis.org

1, 6, 0, 2, 2, 4, 2, 9, 9, 7, 2, 0, 3, 5, 6, 0, 1, 5, 0, 9, 9, 5, 1, 7, 7, 7, 7, 2, 2, 2, 8, 6, 7, 8, 7, 5, 8, 5, 1, 2, 9, 6, 1, 6, 8, 2, 9, 5, 4, 5, 4, 7, 1, 8, 7, 4, 2, 6, 8, 2, 2, 4, 0, 5, 3, 0, 9, 1, 0, 0, 1, 6, 9, 9, 4, 9, 0, 4, 1, 9, 1, 9, 5, 8, 2

Offset: 1

Johannes W. Meijer, Jun 27 2016

Define P(n) = (1/n)*(sum(x(n-k)*P(k), k=0..n-1)), n >= 1 and P(0) =1 with x(3) = (1 + sqrt(2)) and x(n) = 1 for all other n. We find that C2 = limit(P(n), n -> infinity) = exp(sqrt(2)/3).

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

			c = 1.6022429972035601509951777722286787585129616829545471874……

G. C. Greubel, Table of n, a(n) for n = 1..10000
The Dev Team and Simon Plouffe, The Inverse Symbolic Calculator (ISC).
Index entries for transcendental numbers

Cf. A274540, A274541, A014176, A131594, A036039.

SetDefaultRealField(RealField(100)); Exp[Sqrt[2]/3]; // G. C. Greubel, Aug 19 2018

Digits := 85: evalf(exp(sqrt(2)/3)); # End program 1.
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: x := proc(n): if n=3 then (sqrt(2)+1) else 1 fi: end: Digits := 56; evalf(P(120)); # End program 2.

First@ RealDigits@ N[Exp[Sqrt[2]/3], 85] (* Michael De Vlieger, Jun 27 2016 *)

my(x=exp(sqrt(2)/3)); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", ")) \\ Felix Fröhlich, Jun 27 2016

c = exp(sqrt(2)/3)

c = limit(P(n), n -> infinity) with P(n) = (1/n)*(sum(x(n-k)*P(k), k=0..n-1)) for n >= 1, and P(0) =1, with x(3) = (1 + sqrt(2)), the silver mean A014176, and x(n) = 1 for all other n.

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

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A123168 Continued fraction for c = sqrt(2)*(exp(sqrt(2))-1)/(exp(sqrt(2))+1).

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A274541 Decimal expansion of exp(sqrt(2)/2).

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A274542 Decimal expansion of exp(sqrt(2)/3).

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