A002801 -id:A002801 - cp's OEIS frontend

A192680 Floor-Sqrt transform of Sylvester continuants (A002801).

1, 1, 1, 2, 7, 20, 65, 232, 883, 3581, 15383, 69579, 329769, 1631137, 8391824, 44777342, 247182082, 1408611456, 8270902948, 49953724226, 309870713560, 1971523962002, 12849924998486, 85702472618249, 584305721381119, 4068533850711783, 28907888594859403

Offset: 0

Emanuele Munarini, Jul 07 2011

nonn

FSFromExpSeries[f_,x_,n_] := Map[Floor[Sqrt[#]]&,CoefficientList[Series[f,{x,0,n}],x]Table[k!,{k,0,n}]]
FSFromExpSeries[Exp[x/2]/(1-2x)^(1/4),x,40]

a(n) = floor(sqrt(A002801(n))).

A274804 The exponential transform of sigma(n).

Original entry on oeis.org

1, 1, 4, 14, 69, 367, 2284, 15430, 115146, 924555, 7991892, 73547322, 718621516, 7410375897, 80405501540, 914492881330, 10873902417225, 134808633318271, 1738734267608613, 23282225008741565, 323082222240744379, 4638440974576329923, 68794595993688306903

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The exponential transform [EXP] transforms an input sequence b(n) into the output sequence a(n). The EXP transform is the inverse of the logarithmic transform [LOG], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell's formula. For information about the logarithmic transform see A274805. The EXP transform is related to the multinomial transform, see A274760 and the second formula.
The definition of the EXP transform, see the second formula, shows that n >= 1. To preserve the identity LOG[EXP[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 exponential transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A178867 appear.
We observe that a(0) = 1 and provides no information about any value of b(n), this notwithstanding it is customary to start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the exponential transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A007446 and the first formula. The second program uses the definition of the exponential transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the exponential transform, see A274805.
Some EXP transform pairs are, n >= 1: A000435(n) and A065440(n-1); 1/A000027(n) and A177208(n-1)/A177209(n-1); A000670(n) and A075729(n-1); A000670(n-1) and A014304(n-1); A000045(n) and A256180(n-1); A000290(n) and A033462(n-1); A006125(n) and A197505(n-1); A053549(n) and A198046(n-1); A000311(n) and A006351(n); A030019(n) and A134954(n-1); A038048(n) and A053529(n-1); A193356(n) and A003727(n-1).

			Some a(n) formulas, see A178867:
a(0) = 1
a(1) = x(1)
a(2) = x(1)^2 + x(2)
a(3) = x(1)^3 + 3*x(1)*x(2) + x(3)
a(4) = x(1)^4 + 6*x(1)^2*x(2) + 4*x(1)*x(3) + 3*x(2)^2 + x(4)
a(5) = x(1)^5 + 10*x(1)^3*x(2) + 10*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 5*x(1)*x(4) + 10*x(2)*x(3) + x(5)

Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Alois P. Heinz, Table of n, a(n) for n = 0..531
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.

Cf. A274805, A274760, A178867, A000203, A007446.
Cf. A177208, A177209, A006351, A197505, A144180, A256180, A033462, A198046, A134954, A145460, A188489, A005432, A029725, A124213, A002801.
Cf. A036040, another version.

nmax:=21: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; if n=0 then 1 else add(binomial(n-1, j-1) * b(j) *a(n-j), j=1..n) fi: end: seq(a(n), n=0..nmax); # End first EXP program.
nmax:= 21: with(numtheory): b := proc(n): sigma(n) 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 EXP program.
nmax:=21: with(numtheory): b := proc(n): sigma(n) end: f := series(log(1+add(q(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(0):=1: q(0):=1: a(1):=b(1): q(1):=b(1): for n from 2 to nmax+1 do q(n) := solve(d(n)-b(n), q(n)): a(n):=q(n): od: seq(a(n), n=0..nmax); # End third EXP program.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*DivisorSigma[1, j]*a[n-j], {j, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 22 2017 *)
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 08 2021 *)

a(n) = Sum_{j=1..n} (binomial(n-1,j-1) * b(j) * a(n-j)), n >= 1 and a(0) = 1, with b(n) = A000203(n) = sigma(n).

E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n!) with b(n) = sigma(n) = A000203(n).

A383313 Expansion of e.g.f. exp(-x/2) / (1-2*x)^(1/4).

Original entry on oeis.org

1, 0, 1, 4, 27, 232, 2455, 30852, 449113, 7432624, 137829249, 2830911220, 63796168579, 1565078980536, 41521403685463, 1184510408920468, 36158133322895985, 1176012432875399008, 40599110984252798017, 1482736219224857910756, 57115359439245403771051

Offset: 0

Seiichi Manyama, Apr 23 2025

nonn

Cf. A383314, A383315.

Cf. A002801.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x/2)/(1-2*x)^(1/4)))

a(n) = (-1)^n * n! * Sum_{k=0..n} (1/2)^(n-2*k) * binomial(-1/4,k)/(n-k)!.
a(n) = (n-1) * (2*a(n-1) + a(n-2)) for n > 1.
a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n - 1/4) / (Gamma(1/4) * exp(n + 1/4)). - Vaclav Kotesovec, Apr 23 2025

A002370 a(n) = (2n-1)^2 a(n-1) - 3C(2n-1,3) * a(n-2) for n>1; a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 6, 120, 5250, 395010, 45197460, 7299452160, 1580682203100, 441926274289500, 154940341854097800, 66565404923242024800, 34389901168124209507800, 21034386936107260971255000, 15032296693671903309613950000, 12411582569784462888618434640000

Offset: 0

N. J. A. Sloane

A. C. Aitken, On the number of distinct terms in the expansion of symmetric and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5.
I. M. H. Etherington, Some problems of non-associative combinations, Edinburgh Math. Notes, 32 (1940), 1-6.
T. Muir, The Theory of Determinants in the Historical Order of Development. 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 282.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..225
A. C. Aitken, On the number of distinct terms in the expansion of symmetric and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5. [Annotated scanned copy]
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923, Vol. 2.
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923. [Annotated scans of selected pages] See Vol. 3, page 282.
Wikipedia, Skew-symmetric matrix

Cf. A167028.

a:= proc(n) option remember;
      `if`(n<2, 1, (2*n-1)^2 * a(n-1) -3*binomial(2*n-1, 3) *a(n-2))
    end:
seq(a(n), n=0..20);

a[n_] := Gamma[n+1/2]*HypergeometricPFQ[{1/4, -n}, {}, -4]/Sqrt[Pi]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 17 2014, after Mark van Hoeij *)

x='x+O('x^50);  v=Vec( (1-x)^(-1/4)*exp(x/4) );
vector(#v,n, v[n]*(2*n-2)! ) \\ Joerg Arndt, Jan 21 2011

a(n) = (2*n)! * [x^(2*n)] (1-x^2)^(-1/4)*exp(x^2/4).
a(n) = 2^n*GAMMA(n+1/2)*A002801(n)/Pi^(1/2) = GAMMA(n+1/2)*hypergeom([1/4, -n],[],-4)/Pi^(1/2). - Mark van Hoeij, Oct 26 2011
a(n) ~ (2*n)! * exp(1/4) * GAMMA(3/4) / (Pi * sqrt(2) * n^(3/4)). - Vaclav Kotesovec, Feb 15 2015

More terms from Jon E. Schoenfield, Mar 24 2010

Edited by Alois P. Heinz, Jan 21 2011

A247249 a(n) = (2n-1)a(n-1) + (n-1)*a(n-2) with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 4, 22, 166, 1582, 18232, 246508, 3825244, 67001212, 1307450224, 28126466824, 661290689416, 16869784837288, 464080969569184, 13694525105228368, 431491492805617168, 14458331664269020432, 513376963627111206976, 19255197624159957025888

Offset: 0

Michael Somos, Nov 28 2014

nonn

			G.f. = 1 + x + 4*x^2 + 22*x^3 + 166*x^4 + 1582*x^5 + 18232*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A002801.

I:=[1,4]; [1] cat [n le 2 select I[n] else (2*n-1)*Self(n-1) + (n-1)*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 03 2018

a[ n_] := If[ n < 0, 0, HypergeometricPFQ[ {3/4, -n}, {}, 4] / (-2)^n];
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[-x/2] (1 - 2 x)^(-3/4), {x, 0, n}]];
a[ n_] := If[ n < 0, 0, n! / (-2)^n Sum[ (-4)^k Binomial[ k - 1/4, k] / (n - k)!, {k, 0, n}]];
a[ n_] := If[ n < 0, 0, RecurrenceTable[ {a[k] == (2 k - 1) a[k + 1] + (k - 1) a[k], a[0] == a[1] == 1}, a, {k, n, n}]];
nxt[{n_,a_,b_}]:={n+1,b,(2n+1)b+a n}; NestList[nxt,{1,1,1},20][[All,2]] (* Harvey P. Dale, Jan 16 2023 *)

{a(n) = if( n<0, 0, n! / (-2)^n * sum(k=0, n, (-4)^k * binomial(k - 1/4, k) / (n-k)!))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( exp(-x / 2 + A) * (1 - 2*x + A)^(-3/4), n))};

a(n) = n! / (-2)^n * Sum_{k=0..n} (-4)^k * binomial(k - 1/4, k) / (n-k)!.
E.g.f.: exp(-x/2) * (1-2*x)^(-3/4).
E.g.f. A(x) satisfies 0 = A(x)*(x+1) + A'(x)*(2*x-1).
a(n) ~ (n/e)^(n+1/4) * 2^n * Gamma(1/4) / sqrt(Pi).
0 = +a(n)*(+a(n+1) + 3*a(n+2) - a(n+3)) + a(n+1)*(+a(n+1) + 5*a(n+2) - 2*a(n+3)) + a(n+2)*(+2*a(n+2)) for all integer n>=0.

A152148 Riordan array [exp(-x/2)(1-2x)^(-1/4),x].

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 4, 3, 0, 1, 27, 16, 6, 0, 1, 232, 135, 40, 10, 0, 1, 2455, 1392, 405, 80, 15, 0, 1, 30852, 17185, 4872, 945, 140, 21, 0, 1, 449113, 246816, 68740, 12992, 1890, 224, 28, 0, 1, 7432624, 4042017, 1110672, 206220

Offset: 0

Paul Barry, Nov 26 2008

Row sums are A002801. The Riordan array [exp(-x/m)(1-mx)^(-1/m^2),x] has production matrix with

e.g.f. exp(x*t)(t+1/(1-mx)).

			Triangle begins
1,
0, 1,
1, 0, 1,
4, 3, 0, 1,
27, 16, 6, 0, 1,
232, 135, 40, 10, 0, 1,
2455, 1392, 405, 80, 15, 0, 1
The production matrix of this array begins
0, 1,
1, 0, 1,
4, 2, 0, 1,
24, 12, 3, 0, 1,
192, 96, 24, 4, 0, 1;
1920, 960, 240, 40, 5, 1
and has e.g.f. exp(x*t)(t+1/(1-2x)).

A381505 Expansion of e.g.f. exp(2x/3) / (1-3x)^(1/9).

Original entry on oeis.org

1, 1, 2, 10, 88, 1064, 16144, 293968, 6241280, 151328512, 4124855296, 124843943936, 4153947277312, 150699794606080, 5919989155033088, 250339939417452544, 11338037538551824384, 547552961327680913408, 28087260712728645468160, 1525087432592278987866112

Offset: 0

Seiichi Manyama, Apr 23 2025

nonn

Cf. A002801, A381506.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*x/3)/(1-3*x)^(1/9)))

a(n) = n! * Sum_{k=0..n} (-3)^k * (2/3)^(n-k) * binomial(-1/9,k)/(n-k)!.
a(n) = (3*n-2)*a(n-1) - 2*(n-1)*a(n-2) for n > 1.
a(n) ~ sqrt(2*Pi) * 3^n * n^(n - 7/18) / (Gamma(1/9) * exp(n - 2/9)). - Vaclav Kotesovec, Apr 23 2025

A381506 Expansion of e.g.f. exp(3x/4) / (1-4x)^(1/16).

Original entry on oeis.org

1, 1, 2, 12, 138, 2202, 44172, 1064664, 29947644, 962720316, 34812065304, 1398413067984, 61779789904248, 2976866834860728, 155364530441352912, 8730749828092965408, 525584335643810008848, 33743905825099188235536, 2301524700814009677800736

Offset: 0

Seiichi Manyama, Apr 23 2025

nonn

Cf. A002801, A381505.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(3*x/4)/(1-4*x)^(1/16)))

a(n) = n! * Sum_{k=0..n} (-4)^k * (3/4)^(n-k) * binomial(-1/16,k)/(n-k)!.
a(n) = (4*n-3)*a(n-1) - 3*(n-1)*a(n-2) for n > 1.
a(n) ~ sqrt(Pi) * n^(n - 7/16) * 2^(2*n + 1/2) / (Gamma(1/16) * exp(n - 3/16)). - Vaclav Kotesovec, Apr 23 2025

A383316 Expansion of e.g.f. exp(x/2) / (1-4*x)^(1/8).

Original entry on oeis.org

1, 1, 3, 23, 281, 4593, 93643, 2285959, 64981809, 2107824353, 76819828499, 3107456481399, 138145505435977, 6694550810809297, 351219409831557339, 19832058937696108007, 1199219012904515868257, 77314609952787255980481, 5293934640303567123132451

Offset: 0

Seiichi Manyama, Apr 23 2025

nonn

Cf. A002801, A383317.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/2)/(1-4*x)^(1/8)))

a(n) = n! * Sum_{k=0..n} (-2)^k * (1/2)^(n-2*k) * binomial(-1/8,k)/(n-k)!.
a(n) = (4*n-3)*a(n-1) - 2*(n-1)*a(n-2) for n > 1.
a(n) ~ sqrt(Pi) * 2^(2*n + 1/2) * n^(n - 3/8) / (Gamma(1/8) * exp(n - 1/8)). - Vaclav Kotesovec, Apr 23 2025

A383317 Expansion of e.g.f. exp(x/2) / (1-6*x)^(1/12).

Original entry on oeis.org

1, 1, 4, 46, 838, 20398, 619768, 22564252, 957247708, 46363595644, 2524152072304, 152582368541224, 10139721673875976, 734706716925462184, 57646381491830349472, 4869084744694710293392, 440492624600086270972432, 42494068518463022190243088, 4354423933547086885775444032

Offset: 0

Seiichi Manyama, Apr 23 2025

nonn

Cf. A002801, A383316.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/2)/(1-6*x)^(1/12)))

a(n) = n! * Sum_{k=0..n} (-3)^k * (1/2)^(n-2*k) * binomial(-1/12,k)/(n-k)!.
a(n) = (6*n-5)*a(n-1) - 3*(n-1)*a(n-2) for n > 1.
From Vaclav Kotesovec, Apr 23 2025: (Start)
a(n) ~ (sqrt(3) - 1) * 2^(n-1) * 3^n * n^(n - 5/12) * Gamma(11/12) / (sqrt(Pi) * exp(n - 1/12)).
Equivalently, a(n) ~ Pi * (2 - sqrt(3))^(1/4) * 2^(n + 1/2) * 3^(n - 3/8) * n^(n - 5/12) / (Gamma(1/3) * Gamma(1/4) * exp(n - 1/12)). (End)

cp's OEIS Frontend

A192680 Floor-Sqrt transform of Sylvester continuants (A002801).

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A274804 The exponential transform of sigma(n).

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A383313 Expansion of e.g.f. exp(-x/2) / (1-2*x)^(1/4).

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A002370 a(n) = (2*n-1)^2 * a(n-1) - 3*C(2*n-1,3) * a(n-2) for n>1; a(0) = a(1) = 1.

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A247249 a(n) = (2*n-1)*a(n-1) + (n-1)*a(n-2) with a(0) = a(1) = 1.

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Magma

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A152148 Riordan array [exp(-x/2)(1-2x)^(-1/4),x].

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A381505 Expansion of e.g.f. exp(2*x/3) / (1-3*x)^(1/9).

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A381506 Expansion of e.g.f. exp(3*x/4) / (1-4*x)^(1/16).

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PARI

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A383316 Expansion of e.g.f. exp(x/2) / (1-4*x)^(1/8).

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A002370 a(n) = (2n-1)^2 a(n-1) - 3C(2n-1,3) * a(n-2) for n>1; a(0) = a(1) = 1.

A247249 a(n) = (2n-1)a(n-1) + (n-1)*a(n-2) with a(0) = a(1) = 1.

A381505 Expansion of e.g.f. exp(2x/3) / (1-3x)^(1/9).

A381506 Expansion of e.g.f. exp(3x/4) / (1-4x)^(1/16).