A047974 -id:A047974 - cp's OEIS frontend

A362773 E.g.f. satisfies A(x) = exp( x * (1+x) * A(x)^2 ).

1, 1, 7, 79, 1377, 32161, 947623, 33746511, 1410518273, 67714577857, 3672410420871, 222082390164559, 14817864737168353, 1081393797641087841, 85691459902207874471, 7327398378967991154511, 672511583942513406768897, 65943097191889528063033729

Offset: 0

Seiichi Manyama, May 02 2023

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A047974, A362771.

Cf. A361065.

nmax = 20; CoefficientList[Series[Sqrt[LambertW[-2*x * (1+x)]/(-2*x * (1+x))], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 10 2023 *)

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

E.g.f.: exp( -LambertW(-2*x * (1+x))/2 ).
a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(k,n-k)/k!.
From Vaclav Kotesovec, Nov 10 2023: (Start)
E.g.f.: sqrt(LambertW(-2*x * (1+x))/(-2*x * (1+x))).
a(n) ~ sqrt(-sqrt(1 + 2*exp(-1)) + 1 + 2*exp(-1)) * 2^(n-1) * n^(n-1) / ((-1 + sqrt(1 + 2*exp(-1)))^n * exp(n-1)). (End)

A067147 Triangle of coefficients for expressing x^n in terms of Hermite polynomials.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 6, 0, 1, 12, 0, 12, 0, 1, 0, 60, 0, 20, 0, 1, 120, 0, 180, 0, 30, 0, 1, 0, 840, 0, 420, 0, 42, 0, 1, 1680, 0, 3360, 0, 840, 0, 56, 0, 1, 0, 15120, 0, 10080, 0, 1512, 0, 72, 0, 1, 30240, 0, 75600, 0, 25200, 0, 2520, 0, 90, 0, 1

Offset: 0

Christian G. Bower, Jan 03 2002

x^n = (1/2^n) * Sum_{k=0..n} a(n,k)*H_k(x).

These polynomials, H_n(x), are an Appell sequence, whose umbral compositional inverse sequence HI_n(x) consists of the same polynomials signed with the e.g.f. e^{-t^2} e^{xt}. Consequently, under umbral composition H_n(HI.(x)) = x^n = HI_n(H.(x)). Other differently scaled families of Hermite polynomials are A066325, A099174, and A060821. See Griffin et al. for a relation to the Catalan numbers and matrix integration. - Tom Copeland, Dec 27 2020

			Triangle begins with:
    1;
    0,   1;
    2,   0,   1;
    0,   6,   0,   1;
   12,   0,  12,   0,   1;
    0,  60,   0,  20,   0,   1;
  120,   0, 180,   0,  30,   0,   1;

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 801. (Table 22.12)

G. C. Greubel, Rows n = 0..100 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Griffin, K. Ono, L. Rolen, and D. Zagier, Jensen polynomials for the Riemann zeta function and other sequences, arXiv:1902.07321 [math.NT], 2019.
Index entries for sequences related to Hermite polynomials

Row sums give A047974. Columns 0-2: A001813, A000407, A001814. Cf. A048854, A060821.

Cf. A060821, A066325, and A099174.

[[Round(Factorial(n)*(1+(-1)^(n+k))/(2*Factorial(k)*Gamma((n-k+2)/2))): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jun 09 2018

T := proc(n, k) (n - k)/2; `if`(%::integer, (n!/k!)/%!, 0) end:
for n from 0 to 11 do seq(T(n, k), k=0..n) od; # Peter Luschny, Jan 05 2021

Table[n!*(1+(-1)^(n+k))/(2*k!*Gamma[(n-k+2)/2]), {n,0,20}, {k,0,n}]// Flatten (* G. C. Greubel, Jun 09 2018 *)

T(n, k) = round(n!*(1+(-1)^(n+k))/(2*k! *gamma((n-k+2)/2)))
for(n=0,20, for(k=0,n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jun 09 2018

{T(n,k) = if(k<0 || nMichael Somos, Jan 15 2020 */

E.g.f. (rel to x): A(x, y) = exp(x*y + x^2).
Sum_{ k>=0 } 2^k*k!*T(m, k)*T(n, k) = T(m+n, 0) = |A067994(m+n)|. - Philippe Deléham, Jul 02 2005
T(n, k) = 0 if n-k is odd; T(n, k) = n!/(k!*((n-k)/2)!) if n-k is even. - Philippe Deléham, Jul 02 2005
T(n, k) = n!/(k!*2^((n-k)/2)*((n-k)/2)!)*2^((n+k)/2)*(1+(-1)^(n+k))/2^(k+1).
T(n, k) = A001498((n+k)/2, (n-k)/2)2^((n+k)/2)(1+(-1)^(n+k))/2^(k+1). - Paul Barry, Aug 28 2005
Exponential Riordan array (e^(x^2),x). - Paul Barry, Sep 12 2006
G.f.: 1/(1-x*y-2*x^2/(1-x*y-4*x^2/(1-x*y-6*x^2/(1-x*y-8*x^2/(1-... (continued fraction). - Paul Barry, Apr 10 2009
The n-th row entries may be obtained from D^n(exp(x*t)) evaluated at x = 0, where D is the operator sqrt(1+4*x)*d/dx. - Peter Bala, Dec 07 2011
As noted in the comments this is an Appell sequence of polynomials, so the lowering and raising operators defined by L H_n(x) = n H_{n-1}(x) and R H_{n}(x) = H_{n+1}(x) are L = D_x, the derivative, and R = D_t log[e^{t^2} e^{xt}] |{t = D_x} = x + 2 D_x, and the polynomials may also be generated by e^{-D^2} x^n = H_n(x). - _Tom Copeland, Dec 27 2020

A293669 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 7, 1, 1, 1, 3, 13, 25, 1, 1, 1, 3, 13, 49, 81, 1, 1, 1, 3, 13, 73, 261, 331, 1, 1, 1, 3, 13, 73, 381, 1531, 1303, 1, 1, 1, 3, 13, 73, 501, 2611, 9073, 5937, 1, 1, 1, 3, 13, 73, 501, 3331, 19993, 63393, 26785, 1, 1, 1, 3, 13, 73, 501, 4051, 27553, 165873, 465769, 133651, 1

Offset: 0

Seiichi Manyama, Oct 14 2017

			Square array begins:
   1,  1,   1,   1,   1, ...
   1,  1,   1,   1,   1, ...
   1,  3,   3,   3,   3, ...
   1,  7,  13,  13,  13, ...
   1, 25,  49,  73,  73, ...
   1, 81, 261, 381, 501, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=1..9 give A000012, A047974, A118589, A193930, A193931, A193932, A193933, A306623, A306624.
Rows n=0-1 give A000012.
Main diagonal gives A000262.
Cf. A293718, A293724.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(n-j, k)*binomial(n-1, j-1)*j!, j=1..min(n, k)))
    end:
seq(seq(A(n, 1+d-n), n=0..d), d=0..12);  # Alois P. Heinz, Nov 11 2020

A[0, ] = 1; A[n /; n >= 0, k_ /; k >= 1] := A[n, k] = (n-1)!*Sum[j*A[n-j, k]/(n-j)!, {j, 1, Min[k, n]}]; A[, ] = 0;
Table[A[n, d-n+1], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 13 2021 *)

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(k,n)} j*A(n-j,k)/(n-j)!.

A088312 Number of sets of lists (cf. A000262) with even number of lists.

Original entry on oeis.org

1, 0, 1, 6, 37, 260, 2101, 19362, 201097, 2326536, 29668681, 413257790, 6238931821, 101415565836, 1765092183037, 32734873484250, 644215775792401, 13404753632014352, 293976795292186897, 6775966692145553526, 163735077313046119861, 4138498600079573989140

Offset: 0

Vladeta Jovovic, Nov 05 2003

From Peter Bala, Mar 27 2022: (Start)
a(2*n) is odd ; a(2*n+1) is even.
If k is odd then k*(k-1) divides a(k). Consequently, 6 divides a(6*n+3), 10 divides a(10*n+5), 14 divides a(14*n+7), and in general, if k is odd then 2*k divides a(2*k*n + k).
For a positive integer k, a(n+2*k) - a(n) is divisible by k. Thus the sequence obtained by taking a(n) modulo k is purely periodic with period 2*k. Calculation suggests that when k is even the exact period equals k, and when k is odd the exact period equals 2*k. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..444
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
N. J. A. Sloane, LAH transform

Cf. A000262, A001710, A027187, A024429, A024430, A088313, A111884.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Cosh(x/(1-x)) ))); // G. C. Greubel, Dec 13 2022

b:= proc(n, t) option remember; `if`(n=0, t, add(
      b(n-j, 1-t)*binomial(n-1, j-1)*j!, j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016
A088312 := n -> ifelse(n=0, 1, (1/2)*(n - 1)*n!*hypergeom([1 - n/2, 3/2 - n/2], [3/2, 3/2, 2], 1/4)): seq(simplify(A088312(n)), n = 0..21); # Peter Luschny, Dec 14 2022

With[{m=30}, CoefficientList[Series[Cosh[x/(1-x)], {x,0,m}], x] * Range[0,m]!] (* Vaclav Kotesovec, Jul 04 2015 *)
Table[Sum[n!/(2*k)! Binomial[n - 1, 2*k - 1], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Emanuele Munarini, Sep 03 2017 *)

def A088312_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( cosh(x/(1-x)) ).egf_to_ogf().list()
A088312_list(40) # G. C. Greubel, Dec 13 2022

E.g.f.: cosh(x/(1-x)).
a(n) = Sum_{k=1..floor(n/2)} n!/(2*k)!*binomial(n-1,2*k-1).
a(n) ~ 2^(-3/2) * n^(n-1/4) * exp(2*sqrt(n)-n-1/2). - Vaclav Kotesovec, Jul 04 2015
a(n+4) - 2*(2*n+5)*a(n+3) + (6*n^2+24*n+23)*a(n+2) - 2*(n+1)*(n+2)*(2*n+3)*a(n+1) + n*(n+1)^2*(n+2)*a(n) = 0. - Emanuele Munarini, Sep 03 2017
a(n) = (1/2)*(A000262(n) + (-1)^n*A111884(n)). - Peter Bala, Mar 27 2022
a(n) = (1/2)*(n-1)*n!*hypergeom([1 - n/2, 3/2 - n/2], [3/2, 3/2, 2], 1/4) for n >= 1. - Peter Luschny, Dec 14 2022

More terms from Vaclav Kotesovec, Jul 04 2015

a(0)-a(1) prepended by Alois P. Heinz, May 10 2016

A255819 E.g.f.: exp(Sum_{k>=1} k^3 * x^k).

Original entry on oeis.org

1, 1, 17, 211, 3049, 54221, 1131601, 26714647, 700868561, 20208794329, 634445325361, 21512122643771, 782497124407417, 30364699568650981, 1251108918727992689, 54512805637285532671, 2502891521610396838561, 120718449425308259052977, 6099522639316776103853521

Offset: 0

Vaclav Kotesovec, Mar 07 2015

In general, if e.g.f. = exp(Sum_{k>=1} k^m * x^k) and m>0, then a(n) ~ (m+2)^(-1/2) * Gamma(m+2)^(1/(2*m+4)) * exp((m+2)/(m+1) * Gamma(m+2)^(1/(m+2)) * n^((m+1)/(m+2)) + zeta(-m) - n) * n^(n - 1/(2*m+4)).
It appears that the sequence a(n) taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k: if true, then the sequence a(n) taken modulo k would be periodic with period dividing k. - Peter Bala, Nov 14 2017
The above conjecture is true - see the Bala link. - Peter Bala, Jan 20 2018

G. C. Greubel, Table of n, a(n) for n = 0..250
P. Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A082579, A255807, A000262.

nmax=20; CoefficientList[Series[Exp[Sum[k^3*x^k,{k,1,nmax}]],{x,0,nmax}],x] * Range[0,nmax]!
nn = 20; Range[0, nn]! * CoefficientList[Series[Product[Exp[k^3*x^k], {k, 1, nn}], {x, 0, nn}], x] (* Vaclav Kotesovec, Mar 21 2016 *)

E.g.f.: exp(x*(1 + 4*x + x^2)/(1-x)^4).
a(n) ~ 2^(3/10) * 3^(1/10) * 5^(-1/2) * n^(n-1/10) * exp(1/120 + 5 * 2^(-7/5) * 3^(1/5) * n^(4/5) - n).
a(n) = y(n)*n! where y(0)=1 and y(n)=(Sum_{k=0..n-1} (n-k)^4*y(k))/n for n>=1. - Benedict W. J. Irwin, Jun 02 2016
E.g.f.: Product_{k>=1} 1/(1 - x^k)^(J_4(k)/k), where J_4(k) is the Jordan function (A059377). - Ilya Gutkovskiy, May 25 2019

A002747 Logarithmic numbers.

Original entry on oeis.org

1, -2, 9, -28, 185, -846, 7777, -47384, 559953, -4264570, 61594841, -562923252, 9608795209, -102452031878, 2017846993905, -24588487650736, 548854382342177, -7524077221125234, 187708198761024553, -2859149344027588940, 78837443479630312281, -1320926996940746090302

Offset: 1

N. J. A. Sloane

sign

abs(a(n)) is also the number of distinct routes starting from a point A and ending at a point B, without traversing any edge more than once, when there are n bi-directional edges connecting A and B. E.g., if there are 3 edges p, q and r from A to B, then the 9 routes starting from A and ending at B are p, q, r, pqr, prq, rpq, rqp, qpr and qrp. - Nikita Kiran, Sep 02 2022

Reducing the sequence modulo the odd integer 2*k + 1 results in a purely periodic sequence with period dividing 4*k + 2, For example, reduced modulo 5 the sequence becomes the purely periodic sequence [1, 3, 4, 2, 0, 4, 2, 1, 3, 0, 1, 3, 4, 2, 0, 4, 2, 1, 3, 0, ...] with period 10. - Peter Bala, Sep 12 2022

J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
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 = 1..200
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k
J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
Simon Plouffe, Simple inverter lookup on 1.8134302039235
Simon Plouffe, Smart inverter lookup on 1.8134302039235
Index entries for sequences related to logarithmic numbers

Cf. A000166, A000522, A087208.

a:= proc(n) a(n):= n*`if`(n<2, n, (n-1)*a(n-2)-(-1)^n) end:
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 10 2013

egf = x/Exp[x]/(1-x^2); a[n_] := SeriesCoefficient[egf, {x, 0, n}]*n!; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Jan 17 2014, after Vladeta Jovovic *)
a[n_] := (Exp[-1] Gamma[1 + n, -1] - (-1)^n Exp[1] Gamma[1 + n, 1])/2;
Table[a[n], {n, 1, 22}] (* Peter Luschny, Dec 18 2017 *)

a(n) = (-1)^(n+1)*sum(k=0, n, binomial(n, k)*k!*(1-(-1)^k)/2); \\ Michel Marcus, Jan 13 2022

E.g.f.: x/exp(x)/(1-x^2). - Vladeta Jovovic, Feb 09 2003
a(n) = n*((n-1)*a(n-2)-(-1)^n). - Matthew Vandermast, Jun 30 2003
From Gerald McGarvey, Jun 06 2004: (Start)
For n odd, a(n) = n! * Sum_{i=0..n-1, i even} 1/i!.
For n even, a(n) = n! * Sum_{i=1..n-1, i odd} 1/i!.
For n odd, lim_{n->infinity} a(n)/n! = cosh(1).
For n even, lim_{n->infinity} a(n)/n! = sinh(1).
For n even, lim_{n->infinity} n*a(n)*a(n-1)/n!^2 = cosh(1)*sinh(1).
For signed values, Sum_{n>=1} a(n)/n!^2 = 0.
For unsigned values, Sum_{n>=1} a(n)/n!^2 = cosh(1)*sinh(1). (End)
a(n) = (-1)^(n-1)*Sum_{k=0..n} C(n, k)*k!*(1-(-1)^k)/2. - Paul Barry, Sep 14 2004
a(n) = (-1)^(n+1)*n*A087208(n-1). - R. J. Mathar, Jul 24 2015
a(n) = (exp(-1)*Gamma(1+n,-1) - (-1)^n*exp(1)*Gamma(1+n,1))/2 = (A000166(n) - (-1)^n*A000522(n))/2. - Peter Luschny, Dec 18 2017

More terms from Jeffrey Shallit

More terms from Vladeta Jovovic, Feb 09 2003

A122832 Exponential Riordan array (e^(x(1+x)),x).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 7, 9, 3, 1, 25, 28, 18, 4, 1, 81, 125, 70, 30, 5, 1, 331, 486, 375, 140, 45, 6, 1, 1303, 2317, 1701, 875, 245, 63, 7, 1, 5937, 10424, 9268, 4536, 1750, 392, 84, 8, 1, 26785, 53433, 46908, 27804, 10206, 3150, 588, 108, 9, 1

Offset: 0

Paul Barry, Sep 12 2006

Row sums are A000898. Inverse is A122833. Product of A007318 and A067147.

			Triangle begins:
   1;
   1,   1;
   3,   2,  1;
   7,   9,  3,  1;
  25,  28, 18,  4, 1;
  81, 125, 70, 30, 5, 1;
  ...
From _Peter Bala_, May 14 2012: (Start)
T(3,1) = 9. The 9 ways to select a subset of {1,2,3} of size 1 and arrange the remaining elements into a set of lists (denoted by square brackets) of length 1 or 2 are:
{1}[2,3], {1}[3,2], {1}[2][3],
{2}[1,3], {2}[3,1], {2}[1][3],
{3}[1,2], {3}[2,1], {3}[1][2]. (End)

Michel Marcus, Rows n=0..50 of triangle, flattened

A000898 (row sums), A047974 (column 0), A291632 (column 1), A122833 (inverse array).

Cf. A059344, A111062.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[E^(#(1+#))&, #&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

T(n,k) = (n!/k!)*sum(i=0, n-k, binomial(i,n-k-i)/i!); \\ Michel Marcus, Aug 28 2017

Number triangle T(n,k) = (n!/k!)*Sum_{i = 0..n-k} C(i,n-k-i)/i!.
From Peter Bala, May 14 2012: (Start)
Array is exp(S + S^2) where S is A132440 the infinitesimal generator for Pascal's triangle.
T(n,k) = binomial(n,k)*A047974(n-k).
So T(n,k) gives the number of ways to choose a subset of {1,2,...,n} of size k and then arrange the remaining n-k elements into a set of lists of length 1 or 2. (End)
From Peter Bala, Oct 24 2023: (Start)
n-th row polynomial: R(n,x) = exp(D + D^2) (x^n) = exp(D^2) (1 + x)^n, where D denotes the derivative operator d/dx. Cf. A111062.
The sequence of polynomials defined by R(n,x-1) = exp(D^2) (x^n) begins [1, 1, 2 + x^2, 6*x + x^3, 12 + 12*x^2 + x^4, ...] and is related to the Hermite polynomials. See A059344. (End)

More terms from Michel Marcus, Aug 28 2017

A361567 Expansion of e.g.f. exp(x^2/2 * (1+x)^2).

Original entry on oeis.org

1, 0, 1, 6, 15, 60, 555, 3150, 17745, 158760, 1399545, 10914750, 102920895, 1104323220, 11249313075, 119330961750, 1426411411425, 17429852840400, 213417453474225, 2791671804271350, 38524272522310575, 537569719902715500, 7732658753799054075

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A047974, A361568, A361569.

Cf. A335344, A361278.

Table[n! * Sum[Binomial[2*k,n-2*k]/(2^k * k!), {k,0,n/2}], {n,0,20}] (* Vaclav Kotesovec, Mar 25 2023 *)

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/2*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)/(2^k * k!).
a(0) = 1; a(n) = ((n-1)!/2) * Sum_{k=2..n} k * binomial(2,k-2) * a(n-k)/(n-k)!.
From Vaclav Kotesovec, Mar 25 2023: (Start)
a(n) = (n-1)*a(n-2) + 3*(n-2)*(n-1)*a(n-3) + 2*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 2^(n/4 - 1) * exp(1/128 - 3*2^(-29/4)*n^(1/4) - sqrt(n/2)/16 + 2^(-3/4)*n^(3/4) - 3*n/4) * n^(3*n/4). (End)

A375173 Expansion of e.g.f. exp( (1/(1 - 4*x)^(1/2) - 1)/2 ).

Original entry on oeis.org

1, 1, 7, 79, 1225, 24121, 575311, 16105447, 517380529, 18752175505, 756760712311, 33645775575391, 1633792107752377, 86022043957561609, 4880923725657950335, 296882100064302393271, 19269430292162925519841, 1329278651404123963041697

Offset: 0

Seiichi Manyama, Aug 02 2024

For k >= 2, the difference a(n+k) - a(n) is divisible by k. It follows that for each k, the sequence formed by taking a(n) modulo k is periodic with period dividing k. For example, modulo 10 the sequence becomes [1, 1, 7, 9, 5, 1, 1, 7, 9, 5, ...], a purely periodic sequence of period 5. Cf. A047974. - Peter Bala, Feb 11 2025

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A080893, A375175.

Cf. A004211.

Table[4^n * Sum[Abs[StirlingS1[n, k]] * BellB[k, 1/2] / 2^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 02 2024 *)

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

a(n) = Sum_{k=0..n} 4^(n-k) * |Stirling1(n,k)| * A004211(k) = 4^n * Sum_{k=0..n} (1/2)^k * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.
From Vaclav Kotesovec, Aug 02 2024: (Start)
a(n) = 6*(2*n - 3)*a(n-1) - (48*n^2 - 192*n + 191)*a(n-2) + 32*(n-3)*(n-2)*(2*n - 5)*a(n-3).
a(n) ~ 2^(2*n - 1/6) * n^(n - 1/3) / (sqrt(3) * exp(n - 3*2^(-4/3)*n^(1/3) + 1/2)) * (1 - 31/(72*2^(2/3)*n^(1/3)) - 4607/(20736*2^(1/3)*n^(2/3))). (End)
a(n) = (1/exp(1/2)) * (-4)^n * n! * Sum_{k>=0} binomial(-k/2,n)/(2^k * k!). - Seiichi Manyama, Jan 18 2025

A377954 a(n) = n! * Sum_{k=0..n} binomial(k+2,n-k) / k!.

Original entry on oeis.org

1, 3, 9, 31, 117, 471, 2053, 9339, 45321, 227467, 1203681, 6556023, 37316029, 217944351, 1321360797, 8201728531, 52577120913, 344433580179, 2321103364921, 15960060854607, 112534486969221, 808555930139623, 5942117054417589, 44446333314841131

Offset: 0

Seiichi Manyama, Nov 12 2024

Cf. A018191, A047974, A377955, A377956.

a(n) = n!*sum(k=0, n, binomial(k+2, n-k)/k!);

E.g.f.: (1 + x)^2 * exp(x + x^2).
a(n) = -(n-4)*a(n-1) + 3*(n-1)*a(n-2) + 2*(n-1)*(n-2)*a(n-3) for n > 2.
a(n) = ((n^2-7*n+3)*a(n-1) + 2*(n-1)*(n^2-3*n-1)*a(n-2))/(n^2-5*n+3) for n > 1.
a(n) ~ n^(n/2 + 1) * 2^(n/2 - 3/2) / exp(1/8 - sqrt(n/2) + n/2) * (1 + 157/(48*sqrt(2*n))). - Vaclav Kotesovec, Nov 12 2024

cp's OEIS Frontend

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