A099022 -id:A099022 - cp's OEIS frontend

A001517 Bessel polynomials y_n(x) (see A001498) evaluated at 2.

1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353, 28875761731, 1098127402131, 46150226651233, 2124008553358849, 106246577894593683, 5739439214861417731, 332993721039856822081, 20651350143685984386753

Offset: 0

N. J. A. Sloane

Numerators of successive convergents to e using continued fraction 1 + 2/(1 + 1/(6 + 1/(10 + 1/(14 + 1/(18 + 1/(22 + 1/26 + ...)))))).

Number of ways to use the elements of {1,...,k}, n <= k <= 2n, once each to form a collection of n lists, each having length 1 or 2. - Bob Proctor, Apr 18 2005, Jun 26 2006

L. Euler, 1737.
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed., Section 0.126, p. 2.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
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).

Robert Israel, Table of n, a(n) for n = 0..334 (first 101 terms from T. D. Noe)
P. Bala, A note on the Catalan transform of a sequence.
J. W. L. Glaisher, On Lambert's proof of the irrationality of Pi and on the irrationality of certain other quantities, Reports of British Assoc. Adv. Sci., 1871, pp. 16-18.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 131.
D. H. Lehmer, Arithmetical periodicities of Bessel functions, Annals of Mathematics, 33 (1932): 143-150. The sequence is on page 149.
D. H. Lehmer, Review of various tables by P. Pederson, Math. Comp., 2 (1946), 68-69.
W. Mlotkowski, A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
J. Riordan, Letter to N. J. A. Sloane, Jul. 1968.
J. Riordan, Letter, Jul 06 1978.
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970.
Index entries for related partition-counting sequences
Index entries for sequences related to Bessel functions or polynomials

Essentially the same as A080893.
a(n) = A099022(n)/n!.
Partial sums: A105747.
Replace "lists" with "sets" in comment: A001515.
Cf. A001515, A001518, A002119, A053556, A053557, A243593.

A:= gfun:-rectoproc({a(n) = (4*n-2)*a(n-1) + a(n-2),a(0)=1,a(1)=3},a(n),remember):
map(A, [$0..20]); # Robert Israel, Jul 22 2015
f:=proc(n) option remember; if n = 0 then 1 elif n=1 then 3 else f(n-2)+(4*n-2)*f(n-1); fi; end;
[seq(f(n), n=0..20)]; # N. J. A. Sloane, May 09 2016
seq(simplify(KummerU(-n, -2*n, 1)), n = 0..16); # Peter Luschny, May 10 2022

Table[(2k)! Hypergeometric1F1[-k, -2k, 1]/k!, {k, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)

PARI
```
a(n)=sum(k=0,n,(n+k)!/k!/(n-k)!)
```

A001517 = lambda n: hypergeometric([-n, n+1], [], -1)
[simplify(A001517(n)) for n in (0..16)] # Peter Luschny, Oct 17 2014

a(n) = Sum_{k=0..n} (n+k)!/(k!*(n-k)!) = (e/Pi)^(1/2) K_{n+1/2}(1/2).
D-finite with recurrence a(n) = (4*n-2)*a(n-1) + a(n-2), n >= 2.
a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n+k)*binomial(n,k)*A000522(n+k). - Vladeta Jovovic, Sep 30 2006
E.g.f. (for offset 1): exp(x*c(x)), where c(x)=(1-sqrt(1-4*x))/(2*x) (cf. A000108). - Vladimir Kruchinin, Aug 10 2010
G.f.: 1/Q(0), where Q(k) = 1 - x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013
a(n) = (1/n!)*Integral_{x>=0} (x*(1 + x))^n*exp(-x) dx. Expansion of exp(x) in powers of y = x*(1 - x): exp(x) = 1 + y + 3*y^2/2! + 19*y^3/3! + 193*y^4/4! + 2721*y^5/5! + .... - Peter Bala, Dec 15 2013
a(n) = exp(1/2) / sqrt(Pi) * BesselK(n+1/2, 1/2). - Vaclav Kotesovec, Mar 15 2014
a(n) ~ 2^(2*n+1/2) * n^n / exp(n-1/2). - Vaclav Kotesovec, Mar 15 2014
a(n) = hypergeom([-n, n+1], [], -1). - Peter Luschny, Oct 17 2014
From G. C. Greubel, Aug 16 2017: (Start)
a(n) = (1/2)_{n} * 4^n * hypergeometric1f1(-n; -2*n; 1).
G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; 4*t/(1-t)^2). (End)
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(n+k,k)*k!. - Ilya Gutkovskiy, Nov 24 2017
a(n) = KummerU(-n, -2*n, 1). - Peter Luschny, May 10 2022

More terms from Vladeta Jovovic, Apr 03 2000

Additional comments from Michael Somos, Jul 15 2002

A105749 Number of ways to use the elements of {1,...,k}, 0 <= k <= 2n, once each to form a sequence of n sets, each having 1 or 2 elements.

Original entry on oeis.org

1, 2, 14, 222, 6384, 291720, 19445040, 1781750880, 214899027840, 33007837322880, 6290830003852800, 1456812592995513600, 402910665227270323200, 131173228963370155161600, 49656810289225281849907200, 21628258853895305337293568000, 10739534026001485514941587456000

Offset: 0

Robert A. Proctor (www.math.unc.edu/Faculty/rap/), Apr 18 2005

Equivalently, number of sequences of n labeled items such that each item occurs just once or twice. - David Applegate, Dec 08 2008

Also, number of assembly trees for a certain star graph, see Vince-Bona, Theorem 4. - N. J. A. Sloane, Oct 08 2012

			a(2) = 14 = |{ ({1},{2}), ({2},{1}), ({1},{2,3}), ({2,3},{1}), ({2},{1,3}), ({1,3},{2}), ({3},{1,2}), ({1,2},{3}), ({1,2},{3,4}), ({3,4},{1,2}), ({1,3},{2,4}), ({2,4},{1,3}), ({1,4},{2,3}), ({2,3},{1,4}) }|.

Alois P. Heinz, Table of n, a(n) for n = 0..100
Samuele Giraudo and Yannic Vargas, Operads of packed words, quotients, and monomial bases, Proc. 37th Conf. Formal Power Series Alg. Comb., Sém. Lotharingien Combin. (2025) Vol. 93B Art. No. 82. See p. 9.
Robert A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
Andrew Vince and Miklos Bona, The Number of Ways to Assemble a Graph, arXiv preprint arXiv:1204.3842 [math.CO], 2012.
Index entries for related partition-counting sequences

Cf. A001515, A003011, A143990.
Replace "sets" by "lists": A099022.
Column n=2 of A181731.

[(&+[Binomial(n,j)*Factorial(n+j)/2^j: j in [0..n]]): n in [0..30]]; // G. C. Greubel, Sep 26 2023

a:= n-> add(binomial(n, k)*(n+k)!/2^k, k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 21 2012

f[n_]:= Sum[Binomial[n,k]*(n+k)!/2^k, {k,0,n}]; Table[f[n], {n,0,20}]

[sum(binomial(n,j)*factorial(n+j)//2^j for j in range(n+1)) for n in range(31)] # G. C. Greubel, Sep 26 2023

a(n) = Sum_{k=0..n} C(n,k) * (n+k)! / 2^k.
a(n) = n! * A001515(n).
A003011(n) = Sum_{k=0..n} C(n, k)*a(k).
a(n) = Gamma(n+1)*Hypergeometric2F0([-n, n+1], [], -1/2). - Peter Luschny, Jul 29 2014
a(n) ~ sqrt(Pi) * 2^(n + 1) * n^(2*n + 1/2) / exp(2*n - 1). - Vaclav Kotesovec, Nov 27 2017
From G. C. Greubel, Sep 26 2023: (Start)
a(n) = n*(2*n-1)*a(n-1) + n*(n-1)*a(n-2).
a(n) = e * sqrt(2/Pi) * n! * BesselK(n+1/2, 1).
a(n) = ((2*n)!/2^n) * Hypergeometric1F1(-n, -2*n, 2).
G.f.: (-2/x) * Integrate_{t=0..oo} exp(-t)/((t+1)^2 - 1 - 2/x) dt.
G.f.: e*( exp(-sqrt(1 + 2/x)) * ExpIntegralEi(-1 + sqrt(1 + 2/x)) - exp(sqrt(1 + 2/x)) * ExpIntegralEi(-1 - sqrt(1 + 2/x)) )/sqrt(x^2 + 2*x).
E.g.f.: ((1-x)/x) * Hypergeometric1F1(1, 3/2, -(1-x)^2/(2*x)).
E.g.f.: (1/(1-x))*Hypergeometric2F0([1, 1/2]; []; 2*x/(1-x)^2). (End)

More terms from Robert G. Wilson v, Apr 23 2005

A076571 Binomial triangle based on factorials.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 6, 8, 11, 16, 24, 30, 38, 49, 65, 120, 144, 174, 212, 261, 326, 720, 840, 984, 1158, 1370, 1631, 1957, 5040, 5760, 6600, 7584, 8742, 10112, 11743, 13700, 40320, 45360, 51120, 57720, 65304, 74046, 84158, 95901, 109601

Offset: 0

Henry Bottomley, Oct 19 2002

			Rows start:
    1;
    1,   2;
    2,   3,   5;
    6,   8,  11,  16;
   24,  30,  38,  49,  65;
  120, 144, 174, 212, 261, 326;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
E. Biondi, L. Divieti, and G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math. 22 1970 22-35. See Table I.
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

Columns include A000142, A001048, A001344, A001345, A001346, A001347.
Right hand columns include A000522, A001339, A001340, A001341, A001342.
Cf. A002627 (row sums), A099022.

A076571:= func< n,k| (&+[Binomial(k,j)*Factorial(n-j): j in [0..k]]) >;
[A076571(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 05 2023

A076571[n_, k_]:= n!*Hypergeometric1F1[-k,-n,1];
Table[A076571[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 05 2023 *)

def A076571(n,k): return sum(binomial(k,j)*factorial(n-j) for j in range(k+1))
flatten([[A076571(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 05 2023

T(n, k) = Sum_{j=0..k} binomial(k, j)*(n-j)!.
T(n, k) = T(n, k-1) + T(n-1, k-1) with T(n, 0) = n!.
T(n, n) = A000522(n).
Sum_{k=0..n} T(n, k) = A002627(n+1).
From G. C. Greubel, Oct 05 2023: (Start)
T(n, k) = n! * Hypergeometric1F1([-k], [-n], 1).
T(2*n, n) = A099022(n). (End)

A328826 Triangle read by rows: binomial(n,k)(2n-k)!, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 2, 1, 24, 12, 2, 720, 360, 72, 6, 40320, 20160, 4320, 480, 24, 3628800, 1814400, 403200, 50400, 3600, 120, 479001600, 239500800, 54432000, 7257600, 604800, 30240, 720, 87178291200, 43589145600, 10059033600, 1397088000, 127008000, 7620480, 282240, 5040, 20922789888000

Offset: 0

R. J. Mathar, Oct 28 2019

Vertex-labeled disconnected Goldstone diagrams with n vertices and k single-particle potentials.

			The triangle starts
    1;
    2     1;
   24    12     2;
  720   360    72     6;
40320 20160  4320   480    24;

P. J. Rossky, M. Karplus, The enumeration of Goldstone diagrams in many-body perturbation theory, J. Chem. Phys. 64 (1976) 1569, equation (9) and Table 1.

Cf. A099022 (row sums), A000142 (diagonal), A010050 (column k=0), A002674 (k=1).

A328826 := proc(n,k)
        binomial(n,k)*(2*n-k)! ;
end proc:

Table[Binomial[n,k](2n-k)!,{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Feb 03 2022 *)

T(n,k)= binomial(n,k)*(2*n-k)!.

T(n,k) = A328921(n,k) + A328922(n,k). - R. J. Mathar, Nov 02 2019

A082765 Trinomial transform of the factorial numbers (A000142).

Original entry on oeis.org

1, 4, 45, 1282, 70177, 6239016, 817234189, 147950506390, 35370826189857, 10791515504716012, 4091225768720823181, 1886585105032464025674, 1039774852573506696192385, 674970732343624159361034832

Offset: 0

Emanuele Munarini, May 21 2003

Number of ways to use the elements of {1,..,k}, 0<=k<=2n, once each to form a sequence of n (possibly empty) lists, each of length at most 2. - Bob Proctor, Apr 18 2005

Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math.CO/0606404, Jan 05, 2007
Index entries for related partition-counting sequences

a(n) = Sum[C(n, k)*A099022(k), 0<=k<=n]
Replace "sequence" by "collection" in comment: A105747.
Replace "lists" by "sets" in comment: A003011.

a(n) = Sum[ Trinomial[n, k] k!, {k, 0, 2n} ] where Trinomial[n, k] = trinomial coefficients (A027907)

Integral_{x=0..infinity} (x^2+x+1)^n*exp(-x) dx - Gerald McGarvey, Oct 14 2006

cp's OEIS Frontend

A001517 Bessel polynomials y_n(x) (see A001498) evaluated at 2.

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A105749 Number of ways to use the elements of {1,...,k}, 0 <= k <= 2n, once each to form a sequence of n sets, each having 1 or 2 elements.

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A076571 Binomial triangle based on factorials.

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A328826 Triangle read by rows: binomial(n,k)*(2*n-k)!, n>=0, 0<=k<=n.

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Maple

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A082765 Trinomial transform of the factorial numbers (A000142).

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A328826 Triangle read by rows: binomial(n,k)(2n-k)!, n>=0, 0<=k<=n.