A206703 -id:A206703 - cp's OEIS frontend

A103194 LAH transform of squares.

0, 1, 6, 39, 292, 2505, 24306, 263431, 3154824, 41368977, 589410910, 9064804551, 149641946796, 2638693215769, 49490245341642, 983607047803815, 20646947498718736, 456392479671188001, 10595402429677269174, 257723100178182605287, 6553958557721713088820

Offset: 0

Vladeta Jovovic, Mar 18 2005

If the e.g.f. of b(n) is E(x) and a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*b(k), then the e.g.f. of a(n) is E(x/(1-x))/(1-x). - Vladeta Jovovic, Apr 16 2005
a(n) is the total number of elements in all partial permutations (injective partial functions) of {1,2,...,n} that are in a cycle.  A fixed point is considered to be in a cycle.  a(n) = Sum_{k=0..n} A206703(n,k)*k. - Geoffrey Critzer, Feb 11 2012
a(n) is the total number of elements in all partial permutations (injective partial functions) of {1,2,...,n} that are undefined, i.e., they do not have an image.- Geoffrey Critzer, Feb 09 2022
a(n) is the total length of all increasing subsequences over all n-permutations.  Cf. A002720. - Geoffrey Critzer, Feb 09 2022

Alois P. Heinz, Table of n, a(n) for n = 0..200
P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 132.
N. J. A. Sloane, Transforms

Cf. A000262, A000290, A001477, A206703.

with(combstruct): SetSeqSetL := [T, {T=Set(S), S=Sequence(U, card >= 1), U=Set(Z, card=1)}, labeled]: seq(k*count(SetSeqSetL, size=k), k=0..18); # Zerinvary Lajos, Jun 06 2007
a := n -> n!*hypergeom([2, 1-n], [1, 1], -1):
seq(simplify(a(n)),n=0..20); # Peter Luschny, Mar 30 2015

nn = 20; a = 1/(1 - x); ay = 1/(1 - y x); D[Range[0, nn]! CoefficientList[ Series[Exp[a x] ay, {x, 0, nn}], x], y] /. y -> 1  (* Geoffrey Critzer, Feb 11 2012 *)

a(n) = Sum_{k=0..n} (n!/k!)*binomial(n-1, k-1)*k^2.
E.g.f.: x/(1-x)^2*exp(x/(1-x)).
Recurrence: (n-1)*a(n) - n*(2*n-1)*a(n-1) + n*(n-1)^2*a(n-2) = 0.
a(n) = n*A000262(n). - Vladeta Jovovic, Mar 20 2005
a(n) ~ n! * exp(-1/2 + 2*sqrt(n))*n^(1/4)/(2*sqrt(Pi)). - Vaclav Kotesovec, Aug 13 2013
a(n) = n!*hypergeom([2, 1-n], [1, 1], -1). - Peter Luschny, Mar 30 2015
a(n) = Sum_{k=1..n} k*(n-k)!*binomial(n,k)^2. - Ridouane Oudra, Jun 17 2025

A088026 Number of "sets of even lists" for even n, cf. A000262.

Original entry on oeis.org

1, 2, 36, 1560, 122640, 15150240, 2695049280, 650948538240, 204637027795200, 81098021561356800, 39516616693678924800, 23204736106751520921600, 16152539421202464036556800, 13145716394493318293898240000, 12363004898960780220305909760000

Offset: 0

Vladeta Jovovic, Nov 02 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A052845, A088009, A206703.

b:= proc(n) option remember; `if`(n=0, 1, add((i->
      b(n-i)*binomial(n-1, i-1)*i!)(2*j), j=1..n/2))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..14);  # Alois P. Heinz, Feb 01 2022

Table[n!*SeriesCoefficient[E^(x^2/(1-x^2)),{x,0,n}],{n,0,40,2}] (* Vaclav Kotesovec, Feb 25 2014 *)

x='x+O('x^66); /* (half) that many terms */
v=Vec(serlaplace(exp(x^2/(1-x^2))));
vector(#v\2,n, v[2*n-1])
/* Joerg Arndt, Jul 29 2012 */

E.g.f.: exp(x^2/(1-x^2)) (even powers only, see PARI code).
E.g.f.: exp(x^2/(1-x^2)) = 4/(2-(x^2/(1-x^2))*G(0))-1 where G(k) = 1 - x^4/(x^4 + 4*(1-x^2)^2*(2*k+1)*(2*k+3)/G(k+1) ) (continued fraction). - Sergei N. Gladkovskii, Dec 10 2012
a(n) ~ 2^(2*n) * n^(2*n-1/4) * exp(sqrt(4*n)-2*n-1/2). - Vaclav Kotesovec, Feb 25 2014
D-finite with recurrence a(n) -2*(2*n-1)^2*a(n-1) +4*(n-1)*(n-2)*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Feb 01 2022
a(n) = A206703(2n,n). - Alois P. Heinz, Feb 19 2022

More terms from Joerg Arndt, Jul 29 2012.

A105219 a(n) = Sum_{k=0..n} C(n,k)^2(n-k)!k^2.

Original entry on oeis.org

0, 1, 8, 63, 544, 5225, 55656, 653023, 8379008, 116780049, 1757211400, 28394129951, 490371506208, 9013522796473, 175679564492264, 3618800515187775, 78547755741723136, 1791704327280481313, 42846080320725932808, 1071798626271975328639, 27989931083161219661600

Offset: 0

Miklos Kristof, Apr 13 2005

If the e.g.f. of n^2 is E(x) and a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*k^2, then the e.g.f. of a(n) is E(x/(1-x))/(1-x). (Thanks to Vladeta Jovovic for help.)

a(n) is the total number of edges in all matchings of the labeled complete bipartite graph K_n,n. Cf. A144084 for other interpretations. - Geoffrey Critzer, Nov 17 2021

			b(n) = 0,1,4,9,16,25,36,49,64,...
a(3) = C(3,0)^2*3!*b(0) + C(3,1)^2*2!*b(1) + C(3,2)^2*1!*b(2) + C(3,3)^2*0!*b(3) = 1*6*0 + 9*2*1 + 9*1*4 + 1*1*9 = 0 + 18 + 36 + 9 = 63.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.

Cf. A000290, A002720, A144084, A202410, A206703.

for n from 0 to 30 do b[n]:=n^2 od: seq(add(binomial(n,k)^2*(n-k)!*b[k], k=0..n), n=0..30);
seq(`if`(n=0,0,simplify(n!*LaguerreL(n-1,2,-1))),n=0..17); # Peter Luschny, Apr 11 2015

CoefficientList[Series[(x/(1-x)^2+x^2/(1-x)^3)*E^(x/(1-x)), {x, 0, 20}], x]* Table[n!, {n, 0, 20}] (* Vaclav Kotesovec, Oct 17 2012 *)

E.g.f.: (x/(1-x)^2+x^2/(1-x)^3)*exp(x/(1-x)).
a(n) = n^2*A002720(n-1) for n>=1 [Riordan]. - N. J. A. Sloane, Jan 10 2018
a(n) = (n+1)!*(2*L(n,-1)-L(n+1,-1)) where L(n,x) is the n-th Laguerre polynomial. - Peter Luschny, Jan 19 2012
Recurrence: a(n) = 2*(n+2)*a(n-1) - (n^2+4*n-4)*a(n-2) + 2*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+5/4)/sqrt(2)*(1-17/(48*sqrt(n))). - Vaclav Kotesovec, Oct 17 2012
a(n) = n!*L(n-1,2,-1) for n>=1 where L(n,b,x) is the n-th generalized Laguerre polynomial. - Peter Luschny, Apr 11 2015
a(n) = Sum_{k=0...n} A144084(n,k)*k. - Geoffrey Critzer, Nov 17 2021
a(n) = Sum_{k=0..n} (n-k) * A206703(n,k). - Alois P. Heinz, Feb 19 2022
a(n) = Sum_{k=1..n} k*k!*binomial(n,k)^2. - Ridouane Oudra, Jun 15 2025

A260705 Least integer k such that the set of the divisors of k contains exactly n pairs of numbers having the following property: for each pair of two distinct divisors, the reversal of one is equal to the other.

Original entry on oeis.org

84, 168, 336, 1008, 3024, 5544, 11088, 16632, 33264, 49896, 99792, 182952, 365904, 249480, 498960, 1097712, 2162160, 3359664, 1846152, 3027024, 5538456, 6054048, 9081072, 9230760, 14270256, 19891872, 20307672, 25197480, 33297264, 45405360, 55135080, 71351280

Offset: 1

Michel Lagneau, Nov 17 2015

It seems that a(n)==0 (mod 84).

Additional terms with n > 75: a(77) = 15455984544, a(80) = 27719972280, a(83) = 22439977560, a(84) = 18479981520, a(86) = 28559971440. - Lars Blomberg, Jan 04 2016

			a(4)=1008 because the set of the divisors {1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 36, 42, 48, 56, 63, 72, 84, 112, 126, 144, 168, 252, 336, 504, 1008} contains 4 pairs (12, 21), (24, 42), (36, 63) and (48, 84) with the property 21 = reversal(12), 42 = reversal(24), 63 = reversal(36) and 84 = reversal(48).

Lars Blomberg, Table of n, a(n) for n = 1..75.

Cf. A000005, A027750, A083815, A206703, A260704.

with(numtheory):nn:=10^8:
for n from 1 to 16 do:
ii:=0:
for m from 1 to nn while(ii=0) do:
it:=0:d:=divisors(m):d0:=nops(d):
  for i from 1 to d0 do:
   dd:=d[i]:y:=convert(dd,base,10):n1:=length(dd):
   s:=sum('y[j]*10^(n1-j)', 'j'=1..n1):
    for k from i+1 to d0 do:
     if s=d[k]
     then
     it:=it+1:
     else fi:
    od:
    od:
    if it=n
    then
    ii:=1:printf("%d %d \n",n,m):
    else fi:
od:
od:

nbr(vd) = {nb = 0; for (j=1, #vd, da = vd[j]; rda = eval(concat(Vecrev(Str(da)))); rrda = eval(concat(Vecrev(Str(rda)))); if ((da != rda) && vecsearch(vd,rda) && (da == rrda), nb++);); nb/2;}
a(n) = {k=1; while (nbrp(divisors(k)) != n, k++); k;} \\ Michel Marcus, Dec 27 2015

a(14)-a(15) corrected by Lars Blomberg, Dec 27 2015

a(7), a(19), a(20) corrected and a(21)-a(32) added by Lars Blomberg, Jan 04 2016

A331725 E.g.f.: exp(x/(1 - x)) / (1 + x).

Original entry on oeis.org

1, 0, 3, 4, 57, 216, 2755, 18348, 247569, 2368432, 35256771, 436248660, 7235178313, 108919083144, 2010150360387, 35421547781116, 723689454172065, 14543895730321248, 326843345169621379, 7354350135365751972, 180610925178770615001, 4488323611011676811320

Offset: 0

Ilya Gutkovskiy, Jan 25 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A000262, A002720, A009940, A133942, A182386, A206703.

A331725 := proc(n)
    add((-1)^k*binomial(n,k)*k!*A000262(n-k),k=0..n) ;
end proc:
seq(A331725(n),n=0..42) ; # R. J. Mathar, Aug 20 2021

nmax = 21; CoefficientList[Series[Exp[x/(1 - x)]/(1 + x), {x, 0, nmax}], x] Range[0, nmax]!
A000262[n_] := If[n == 0, 1, n! Sum[Binomial[n - 1, k]/(k + 1)!, {k, 0, n - 1}]]; a[n_] := Sum[(-1)^k Binomial[n, k] k! A000262[n - k], {k, 0, n}]; Table[a[n], {n, 0, 21}]
a[n_] := (-1)^n n! (1 - Sum[(-1)^j*LaguerreL[j, 1, -1]/(j+1), {j,0,n-1}]);
Table[a[n], {n, 0, 21}] (* Peter Luschny, Feb 20 2022 *)

seq(n)={Vec(serlaplace(exp(x/(1 - x) + O(x*x^n)) / (1 + x)))} \\ Andrew Howroyd, Jan 25 2020

def gen_a():
    F, L, S, N = 1, 1, 1, 1
    while True:
        yield F * S
        L = gen_laguerre(N - 1, 1, -1) / N
        S += L if F < 0 else -L
        F *= -N; N += 1
a = gen_a(); print([next(a) for  in range(21)]) # _Peter Luschny, Feb 20 2022

a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * k! * A000262(n-k).
a(n) ~ n^(n - 1/4) / (2^(3/2) * exp(1/2 - 2*sqrt(n) + n)). - Vaclav Kotesovec, Jan 26 2020
D-finite with recurrence a(n) +(-n+1)*a(n-1) -(n-1)*(n+1)*a(n-2) +(n-1)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Aug 20 2021
a(n) = Sum_{k=0..n} (-1)^k * A206703(n,k). - Alois P. Heinz, Feb 19 2022
a(n) = (-1)^n*n!*(1 - Sum_{j=0..n-1}((-1)^j*LaguerreL(j, 1, -1)/(j + 1))). - Peter Luschny, Feb 20 2022

cp's OEIS Frontend

A103194 LAH transform of squares.

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A088026 Number of "sets of even lists" for even n, cf. A000262.

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A105219 a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*k^2.

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A260705 Least integer k such that the set of the divisors of k contains exactly n pairs of numbers having the following property: for each pair of two distinct divisors, the reversal of one is equal to the other.

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Comments

Examples

Links

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Maple

PARI

Extensions

A331725 E.g.f.: exp(x/(1 - x)) / (1 + x).

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Formula

A105219 a(n) = Sum_{k=0..n} C(n,k)^2(n-k)!k^2.