A047974 -id:A047974 - cp's OEIS frontend

A376513 Expansion of e.g.f. exp(x^3 * (1 + x)).

1, 0, 0, 6, 24, 0, 360, 5040, 20160, 60480, 1814400, 19958400, 99792000, 1037836800, 21794572800, 228843014400, 1743565824000, 29640619008000, 542423327846400, 6082255020441600, 70959641905152000, 1429329929803776000, 24977793950613504000

Offset: 0

Seiichi Manyama, Sep 25 2024

Cf. A047974, A376512.

Cf. A373742.

a(n) = n!*sum(k=0, n\3, binomial(k, n-3*k)/k!);

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(k,n-3*k)/k!.
a(n) = (n-1)*(n-2) * (3*a(n-3) + 4*(n-3)*a(n-4)).
a(n) ~ 2^(n/2 - 1) * exp(-27/1024 + 45*2^(-19/2)*n^(1/4) - 9*n^(1/2)/64 + 2^(-3/2)*n^(3/4) - 3*n/4) * n^(3*n/4) * (1 + 264471/(5*2^(37/2)*n^(1/4))). - Vaclav Kotesovec, Sep 26 2024

A377956 a(n) = n! * Sum_{k=0..n} binomial(k+4,n-k) / k!.

Original entry on oeis.org

1, 5, 23, 103, 473, 2261, 11215, 57863, 309713, 1715653, 9831911, 58058375, 353546473, 2210900693, 14215319903, 93610866151, 632159025185, 4362925851653, 30809311250743, 221958273142823, 1632956199823481, 12238229941781845, 93509510960341103, 726913018468699463

Offset: 0

Seiichi Manyama, Nov 12 2024

Cf. A018191, A047974, A377954, A377955.

a(n) = n!*sum(k=0, n, binomial(k+4, n-k)/k!);

E.g.f.: (1 + x)^4 * exp(x + x^2).

a(n) = -(n-6)*a(n-1) + 3*(n-1)*a(n-2) + 2*(n-1)*(n-2)*a(n-3) for n > 2.

A115326 E.g.f.: exp(x/(1-2x))/sqrt(1-4x^2).

Original entry on oeis.org

1, 1, 9, 49, 625, 6561, 109561, 1697809, 35247969, 717436225, 17862589801, 448030761201, 13029739166929, 387070092765409, 12888060720104025, 441427773256896721, 16566268858818121921, 641658452161285040769

Offset: 0

Paul D. Hanna, Jan 20 2006

nonn

Term-by-term square of sequence with e.g.f.: exp(x+m/2*x^2) is given by e.g.f.: exp(x/(1-m*x))/sqrt(1-m^2*x^2) for all m.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Hermite Polynomial
Wikipedia, Hermite polynomials

Cf. A047974.

CoefficientList[Series[E^(x/(1-2*x))/Sqrt[1-4*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 25 2013 *)
RecurrenceTable[{a[0] == a[1] == 1, a[2] == 9, a[n] == (2 n - 1) a[n - 1] + 2 (n - 1) (2 n - 1) a[n - 2] - 8 (n - 2)^2 (n - 1) a[n - 3]}, a, {n, 20}] (* Bruno Berselli, Sep 27 2013 *)
Table[Abs[HermiteH[n, I/2]]^2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 11 2016 *)

a(n)=local(m=2);n!*polcoeff(exp(x/(1-m*x+x*O(x^n)))/sqrt(1-m^2*x^2+x*O(x^n)),n)

Equals the term-by-term square of A047974 which has e.g.f.: exp(x+x^2).
D-finite with recurrence: a(n) = (2*n-1)*a(n-1) + 2*(n-1)*(2*n-1)*a(n-2) - 8*(n-2)^2*(n-1)*a(n-3). - Vaclav Kotesovec, Sep 25 2013
a(n) ~ 2^(n-1)*n^n*exp(sqrt(2*n)-n-1/4) * (1 + 13/(24*sqrt(2*n))). - Vaclav Kotesovec, Sep 25 2013
a(n) = |H_n(i/2)|^2 / 2^n = H_n(i/2) * H_n(-i/2) / 2^n, where H_n(x) is n-th Hermite polynomial, i = sqrt(-1). - Vladimir Reshetnikov, Oct 11 2016

A195186 Number of palindromic double occurrence words of length 2n.

Original entry on oeis.org

1, 2, 6, 20, 72, 290, 1198, 5452, 25176, 125874, 637926, 3448708, 18919048, 109412210, 642798510, 3945170012, 24614491704, 159328958690, 1048645656646, 7122719571700, 49185991168968, 349097516604738, 2518145666958126, 18609525157571692, 139704193446510616

Offset: 1

N. J. A. Sloane, Sep 10 2011

nonn

Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, arXiv preprint arXiv:1105.2926 [math.CO], 2011.

A047974 := proc(n) option remember; if n= 1 then 1; elif n=2 then 3; else procname(n-1)+2*(n-1)*procname(n-2) ; end if; end proc:
A195186 := proc(n) if n <= 1 then 1; else A047974(n)-add(procname(n-2*k)*doublefactorial(2*k-1),k=1..floor(n/2)) ; end if; end proc:
seq(A195186(n),n=1..20) ; # R. J. Mathar, Sep 12 2011

b[n_] := Sum[Binomial[k, n - k]*(n!/k!), {k, 0, n}];
a[1] = 1; a[n_] := b[n] - Sum[a[n - 2*k]*(2*k - 1)!!, {k, 1, n/2}];
Array[a, 20] (* Jean-François Alcover, Nov 29 2017, after R. J. Mathar *)

Theorem 3.3 of Burns-Muche gives a recurrence.

A271218 Number of symmetric linked diagrams with n links and no simple link.

Original entry on oeis.org

1, 0, 1, 3, 12, 39, 167, 660, 3083, 13961, 70728, 355457, 1936449, 10587960, 61539129, 361182139, 2224641540, 13880534119, 90090083047, 593246514588, 4038095508691, 27905008440273, 198401618299920, 1432253086621377, 10600146578310209, 79639887325700592, 611739960145556273

Offset: 0

Jonathan Burns, Apr 13 2016

Number of symmetric chord diagrams (where reflection is equivalent) with n chords and no simple chords.

Number of symmetric assembly words that do not contain the subword aa.

			For n=0 the a(0)=1 solution is { ∅ }.
For n=1 there are no solutions since the link in a diagram with one link, 11, is simple.
For n=2 the a(2)=1 solution is { 1212 }.
For n=3 the a(3)=3 solutions are { 123123, 121323, 123231 }.
For n=4 the a(4)=12 solutions are { 12123434, 12132434, 12324341, 12314234, 12341234, 12342341, 12314324, 12341324, 12343412, 12343421, 12324143, 12342143 }.

J. Burns, Counting a Class of Signed Permutations and Chord Diagrams related to DNA Rearrangement, Preprint.

Cf. A047974, A271215.

RecurrenceTable[{a[n]==2a[n-1]+(2n-3)a[n-2]-(2n-5)a[n-3]+2a[n-4]-a[n-5],a[0]==1,a[1]==0,a[2]==1,a[3]==3,a[4]==12},a[n],{n,20}]

lista(nn) = {my(va = vector(nn)); va[1] = 1; va[2] = 0; va[3] = 1; va[4] = 3; va[5] = 12; for (n=5, nn-1, va[n+1] = 2*va[n] + (2*n-3)*va[n-1] - (2*n-5)*va[n-2] + 2*va[n-3] - va[n-4];); va;} \\ Michel Marcus, Jul 28 2020

a(n) = 2*a(n-1) + (2n-3)*a(n-2) - (2n-5)*a(n-3) + 2*a(n-4) - a(n-5).
a(n) = a(n-1) + 2*(n-1)*a(n-2) + a(n-3) + a(n-4) + 2*sum( k=0..n-4, a(k) ).
a(n) ~ 2^(-1/2) * e^(-5/8) * (2n/e)^(n/2) * e^( sqrt(n/2) )  (conjectured).
a(n)/a(n-1) ~ sqrt(2n)  (conjectured).
a(n)/A047974(n) ~ 1/sqrt(e)  (conjectured).

More terms from Michel Marcus, Jul 28 2020

A291632 Column 1 of A122832.

Original entry on oeis.org

0, 1, 2, 9, 28, 125, 486, 2317, 10424, 53433, 267850, 1470161, 8032212, 46925749, 275437358, 1702883925, 10630404976, 69192858737, 455957606034, 3110617216153, 21512638153100, 153234193139181, 1107087138215542, 8206182165264029, 61703155328534568

Offset: 0

Vaclav Kotesovec, Aug 28 2017

nonn

Cf. A122832, A047974.

Table[n!*Sum[Binomial[i, n - 1 - i]/i!, {i, 0, n - 1}], {n, 0, 30}]
CoefficientList[Series[x*E^(x*(x+1)), {x, 0, 20}], x] * Range[0, 20]!

E.g.f.: x*exp(x*(x+1)).
Recurrence: (n-1)*a(n) = n*a(n-1) + 2*(n-1)*n*a(n-2).
a(n) ~ 2^(n/2 - 1) * n^((n+1)/2) * exp(sqrt(n/2) - n/2 - 1/8).

A354015 Expansion of e.g.f. 1/(1 - x)^(1 - log(1-x)).

Original entry on oeis.org

1, 1, 4, 18, 106, 750, 6188, 58184, 613156, 7149780, 91319712, 1267089912, 18969355656, 304646227704, 5222700792528, 95169251327040, 1836450816902928, 37403582826055824, 801728489886598848, 18037821249349491360, 424970923585819603872, 10462258547232790348512

Offset: 0

Seiichi Manyama, May 14 2022

nonn

Cf. A000776, A047974, A189423, A353995, A354013.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x)^(1-log(1-x))))

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, (1+2*sum(k=1, j-1, 1/k))*v[i-j+1]/(i-j)!)); v;

a(n) = sum(k=0, n, k!*sum(j=0, k\2, 1/(j!*(k-2*j)!))*abs(stirling(n, k, 1)));

E.g.f.: exp( -log(1-x) * (1 - log(1-x)) ).
a(0) = 1; a(n) = Sum_{k=1..n} A000776(k-1) * binomial(n-1,k-1) * a(n-k) = (n-1)! * Sum_{k=1..n} (1 + 2*Sum_{j=1..k-1} 1/j) * a(n-k)/(n-k)!.
a(n) = Sum_{k=0..n} A047974(k) * |Stirling1(n,k)|.

A361036 a(n) = n! * [x^n] (1 + x)^n * exp(x*(1 + x)^n).

Original entry on oeis.org

1, 2, 11, 124, 2225, 56546, 1928707, 85029596, 4687436609, 314255427490, 25077179715131, 2343489559096412, 253185531592066801, 31279831940279656514, 4376923336721600128115, 687815536092999747916156, 120491486068612766739548417, 23378730923206887237941740226

Offset: 0

Peter Bala, Mar 13 2023

We conjecture that a(n+k) == a(n) (mod k) for all n and k. If true, then for each k, the sequence a(n) taken modulo k is a periodic sequence and the period divides k. For example, modulo 7 the sequence becomes [1, 2, 4, 5, 6, 0, 4, 1, 2, 4, 5, 6, 0, 4, 1, 2, 4, 5, 6, 0, 4, ...], apparently a periodic sequence of period 7.

More generally, let F(x) and G(x) denote power series with integer coefficients with F(0) = G(0) = 1. Define b(n) = n! * [x^n] exp(x*G(x)^n)*F(x)^n. Then we conjecture that b(n+k) == b(n) (mod k) for all n and k.

Vaclav Kotesovec, Table of n, a(n) for n = 0..270

Cf. A047974, A278070, A361281.

seq( n!*add(add(binomial(n,i+j)*binomial(j*n,i)/j!, j = 0..n-i), i = 0..n), n = 0..20);

Table[n! * Sum[Sum[Binomial[n, i + j]*Binomial[j*n, i]/j!, {j, 0, n - i}], {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 27 2023 *)

a(n) = n!*Sum_{i = 0..n} Sum_{j = 0..n-i} binomial(n,i+j)*binomial(j*n,i)/j!.

a(n) ~ n! * exp(r*(1+r)^n) * (1+r)^(n/2 + 1) / (sqrt(2*Pi*n*(3 + n*r)) * r^(n+1)), where r = 2*LambertW(n/2)/n - (n + 2*LambertW(n/2)) * (n - 4*LambertW(n/2)^3) / (n^3 * (3 + 2*LambertW(n/2))). - Vaclav Kotesovec, Mar 28 2023

A361570 Expansion of e.g.f. exp( (x * (1+x))^2 ).

Original entry on oeis.org

1, 0, 2, 12, 36, 240, 2280, 15120, 122640, 1330560, 13335840, 136382400, 1657212480, 20860519680, 262278656640, 3585207225600, 52249374777600, 772773281280000, 11907924610982400, 193962388523904000, 3253343368231756800, 56051640629816832000

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A047974, A361571.

Cf. A052887, A361278.

With[{nn=30},CoefficientList[Series[Exp[(x(1+x))^2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 18 2024 *)

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, i, j*binomial(2, j-2)*v[i-j+1]/(i-j)!)); v;

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(2*k,n-2*k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=2..n} k * binomial(2,k-2) * a(n-k)/(n-k)!.
From Vaclav Kotesovec, Mar 25 2023: (Start)
a(n) ~ 2^(n/2 - 1) * exp(1/64 - 3*n^(1/4)/2^(13/2) - sqrt(n)/16 + n^(3/4)/sqrt(2) - 3*n/4) * n^(3*n/4).
a(n) = 2*(n-1)*a(n-2) + 6*(n-2)*(n-1)*a(n-3) + 4*(n-3)*(n-2)*(n-1)*a(n-4). (End)

A382134 Number of completely asymmetric matchings (not containing centered or coupled arcs) of [2n].

Original entry on oeis.org

1, 0, 0, 8, 48, 384, 4480, 59520, 897792, 15368192, 293769216, 6198589440, 143130972160, 3590253477888, 97214510235648, 2826205634330624, 87801981951344640, 2902989352269250560, 101776549707306237952, 3771425415371470405632, 147285455218020210180096

Offset: 0

R. J. Mathar, Mar 17 2025

nonn

Sergi Elizalde and Emeric Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7, theorem 5.1

Cf. A000079, A001205, A047974, A053871.

g:= exp(-x-x^2)/sqrt(1-2*x) ;
seq( coeftayl(g,x=0,n)*n!,n=0..10) ;

E.g.f: exp(-x-x^2)/sqrt(1-2*x).
a(n) = 2^n * A001205(n).
D-finite with recurrence a(n) +2*(-n+1)*a(n-1) -4*(n-1)*(n-2)*a(n-3)=0.

cp's OEIS Frontend

A376513 Expansion of e.g.f. exp(x^3 * (1 + x)).

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A377956 a(n) = n! * Sum_{k=0..n} binomial(k+4,n-k) / k!.

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A115326 E.g.f.: exp(x/(1-2*x))/sqrt(1-4*x^2).

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A195186 Number of palindromic double occurrence words of length 2n.

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A271218 Number of symmetric linked diagrams with n links and no simple link.

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A291632 Column 1 of A122832.

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A354015 Expansion of e.g.f. 1/(1 - x)^(1 - log(1-x)).

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A361036 a(n) = n! * [x^n] (1 + x)^n * exp(x*(1 + x)^n).

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A115326 E.g.f.: exp(x/(1-2x))/sqrt(1-4x^2).