A001518 -id:A001518 - cp's OEIS frontend

A001498 Triangle a(n,k) (n >= 0, 0 <= k <= n) of coefficients of Bessel polynomials y_n(x) (exponents in increasing order).

1, 1, 1, 1, 3, 3, 1, 6, 15, 15, 1, 10, 45, 105, 105, 1, 15, 105, 420, 945, 945, 1, 21, 210, 1260, 4725, 10395, 10395, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 1, 36, 630, 6930, 51975, 270270, 945945, 2027025, 2027025, 1, 45, 990, 13860, 135135, 945945, 4729725, 16216200, 34459425, 34459425

Offset: 0

N. J. A. Sloane

The row polynomials with exponents in increasing order (e.g., third row: 1+3x+3x^2) are Grosswald's y_{n}(x) polynomials, p. 18, Eq. (7).
Also called Bessel numbers of first kind.
The triangle a(n,k) has factorization [C(n,k)][C(k,n-k)]Diag((2n-1)!!) The triangle a(n-k,k) is A100861, which gives coefficients of scaled Hermite polynomials. - Paul Barry, May 21 2005
Related to k-matchings of the complete graph K_n by a(n,k)=A100861(n+k,k). Related to the Morgan-Voyce polynomials by a(n,k)=(2k-1)!!*A085478(n,k). - Paul Barry, Aug 17 2005
Related to Hermite polynomials by a(n,k)=(-1)^k*A060821(n+k, n-k)/2^n. - Paul Barry, Aug 28 2005
The row polynomials, the Bessel polynomials y(n,x):=Sum_{m=0..n} (a(n,m)*x^m) (called y_{n}(x) in the Grosswald reference) satisfy (x^2)*(d^2/dx^2)y(n,x) + 2*(x+1)*(d/dx)y(n,x) - n*(n+1)*y(n,x) = 0.
a(n-1, m-1), n >= m >= 1, enumerates unordered n-vertex forests composed of m plane (aka ordered) increasing (rooted) trees. Proof from the e.g.f. of the first column Y(z):=1-sqrt(1-2*z) (offset 1) and the Bergeron et al. eq. (8) Y'(z)= phi(Y(z)), Y(0)=0, with out-degree o.g.f. phi(w)=1/(1-w). See their remark on p. 28 on plane recursive trees. For m=1 see the D. Callan comment on A001147 from Oct 26 2006. - Wolfdieter Lang, Sep 14 2007
The asymptotic expansions of the higher order exponential integrals E(x,m,n), see A163931 for information, lead to the Bessel numbers of the first kind in an intriguing way. For the first four values of m these asymptotic expansions lead to the triangles A130534 (m=1), A028421 (m=2), A163932 (m=3) and A163934 (m=4). The o.g.f.s. of the right hand columns of these triangles in their turn lead to the triangles A163936 (m=1), A163937 (m=2), A163938 (m=3) and A163939 (m=4). The row sums of these four triangles lead to A001147, A001147 (minus a(0)), A001879 and A000457 which are the first four right hand columns of A001498. We checked this phenomenon for a few more values of m and found that this pattern persists: m = 5 leads to A001880, m=6 to A001881, m=7 to A038121 and m=8 to A130563 which are the next four right hand columns of A001498. So one by one all columns of the triangle of coefficients of Bessel polynomials appear. - Johannes W. Meijer, Oct 07 2009
a(n,k) also appear as coefficients of (n+1)st degree of the differential operator D:=1/t d/dt, namely D^{n+1}= Sum_{k=0..n} a(n,k) (-1)^{n-k} t^{1-(n+k)} (d^{n+1-k}/dt^{n+1-k}. - Leonid Bedratyuk, Aug 06 2010
a(n-1,k) are the coefficients when expanding (xI)^n in terms of powers of I. Let I(f)(x) := Integral_{a..x} f(t) dt, and (xI)^n := x Integral_{a..x} [ x_{n-1} Integral_{a..x_{n-1}} [ x_{n-2} Integral_{a..x_{n-2}} ... [ x_1 Integral_{a..x_1} f(t) dt ] dx_1 ] .. dx_{n-2} ] dx_{n-1}. Then: (xI)^n = Sum_{k=0..n-1} (-1)^k * a(n-1,k) * x^(n-k) * I^(n+k)(f)(x) where I^(n) denotes iterated integration. - Abdelhay Benmoussa, Apr 11 2025

			The triangle a(n, k), n >= 0, k = 0..n, begins:
  1
  1  1
  1  3   3
  1  6  15    15
  1 10  45   105    105
  1 15 105   420    945    945
  1 21 210  1260   4725  10395   10395
  1 28 378  3150  17325  62370  135135   135135
  1 36 630  6930  51975 270270  945945  2027025  2027025
  1 45 990 13860 135135 945945 4729725 16216200 34459425 34459425
  ...
And the first few Bessel polynomials are:
  y_0(x) = 1,
  y_1(x) = x + 1,
  y_2(x) = 3*x^2 + 3*x + 1,
  y_3(x) = 15*x^3 + 15*x^2 + 6*x + 1,
  y_4(x) = 105*x^4 + 105*x^3 + 45*x^2 + 10*x + 1,
  y_5(x) = 945*x^5 + 945*x^4 + 420*x^3 + 105*x^2 + 15*x + 1,
  ...
Tree counting: a(2,1)=3 for the unordered forest of m=2 plane increasing trees with n=3 vertices, namely one tree with one vertex (root) and another tree with two vertices (a root and a leaf), labeled increasingly as (1, 23), (2,13) and (3,12). - _Wolfdieter Lang_, Sep 14 2007

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

T. D. Noe, Rows n=0..50 of triangle, flattened
Alexander Alldridge, Joachim Hilgert, and Martin R. Zirnbauer, Chevalley's restriction theorem for reductive symmetric superpairs, arXiv:0812.3530 [math.RT], 2008-2009; J. Alg. 323 (4) (2010) 1159-1185 doi:10.1016/j.jalgebra.2009.11.014, Remark 3.17.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
Juan Antonio Barcelo and Anthony Carbery, On the magnitudes of compact sets in Euclidean spaces, arXiv preprint arXiv:1507.02502 [math.MG], 2015.
François Bergeron, Philippe Flajolet, and Bruno Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, pp. 24-48.
Alexander W. Boldyreff, Decomposition of Rational Fractions into Partial Fractions, Nat. Math. Mag. 17 (6) (1943), 261-267; coefficients (m)N(r).
Alexander Burstein and Toufik Mansour, Words restricted by patterns with at most 2 distinct letters, arXiv:math/0110056 [math.CO], 2001.
Roudy El Haddad, Repeated Integration and Explicit Formula for the n-th Integral of x^m*(ln x)^m', arXiv:2102.11723 [math.GM], 2021.
Andrew Francis and Michael Hendriksen, Counting spinal phylogenetic networks, arXiv:2502.14223 [q-bio.PE], 2025. See p. 9.
Emil Grosswald, Bessel Polynomials: Recurrence Relations, Lecture Notes Math. vol. 698, 1978, p. 18.
Cameron Jakub and Mihai Nica, Depth Degeneracy in Neural Networks: Vanishing Angles in Fully Connected ReLU Networks on Initialization, arXiv:2302.09712 [stat.ML], 2023.
Taekyun Kim, and Dae San Kim, Identities involving Bessel polynomials arising from linear differential equations, arXiv:1602.04106 [math.NT], 2016.
H. L. Krall and Orrin Frink, A new class of orthogonal polynomials, Trans. Amer. Math. Soc. 65, 100-115, 1949.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First ten rows.
B. Leclerc, Powers of staircase Schur functions and symmetric analogues of Bessel polynomials, Discrete Math., 153 (1996), 213-227.
Shi-Mei Ma, Toufik Mansour, and Matthias Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169 [math.CO], 2013.
Shi-Mei Ma, Toufik Mansour, Jean Yeh, and Yeong-Nan Yeh, Normal ordered grammars, arXiv:2404.15119 [math.CO], 2024. See p. 11.
Guillermo Navas-Palencia, On the computation of the cumulative distribution function of the Normal Inverse Gaussian distribution, arXiv:2502.16015 [math.NA], 2025. See p. 25.
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
John Riordan, Notes to N. J. A. Sloane, Jul. 1968
Florian Stober, Average case considerations for MergeInsertion, Master's Thesis, University of Stuttgart, Institute of Formal Methods in Computer Science, 2018.
Florian Stober and Armin Weiß, On the Average Case of MergeInsertion, arXiv:1905.09656 [cs.DS], 2019.
Laszlo A. Székely, Pál L. Erdős, and M. A. Steel, The combinatorics of evolutionary trees, Séminaire Lotharingien de Combinatoire, B28e (1992), 15 pp.
Juan G. Triana, Bessel polynomials by context-free grammars (Polinomios de Bessel mediante gramáticas independientes del contexto), Bistua, Univ. de Pamplona (Colombia, 2024) Vol 22, No. 2. See p. 3.
Jonas Wahl, Traces on diagram algebras II: Centralizer algebras of easy groups and new variations of the Young graph, arXiv:2009.08181 [math.RT], 2020.
Eric Weisstein's World of Mathematics, Modified Spherical Bessel Function of the Second Kind
Index entries for sequences related to Bessel functions or polynomials

Cf. A001497 (same triangle but rows read in reverse order). Other versions of this same triangle are given in A144331, A144299, A111924 and A100861.
Columns from left edge include A000217, A050534.
Columns 1-6 from right edge are A001147, A001879, A000457, A001880, A001881, A038121.
Bessel polynomials evaluated at certain x are A001515 (x=1, row sums), A000806 (x=-1), A001517 (x=2), A002119 (x=-2), A001518 (x=3), A065923 (x=-3), A065919 (x=4). Cf. A043301, A003215.
Cf. A245066 (central terms). A113025 (y_n(2*x)).

a001498 n k = a001498_tabl !! n !! k
a001498_row n = a001498_tabl !! n
a001498_tabl = map reverse a001497_tabl
-- Reinhard Zumkeller, Jul 11 2014

/* As triangle: */ [[Factorial(n+k)/(2^k*Factorial(n-k)*Factorial(k)): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 15 2016

Bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end; # explicit Bessel polynomials
Bessel := proc(n) option remember; if n <=1 then (1+x)^n else (2*n-1)*x*Bessel(n-1)+Bessel(n-2); fi; end; # recurrence for Bessel polynomials
bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end;
f := proc(n) option remember; if n <=1 then (1+x)^n else (2*n-1)*x*f(n-1)+f(n-2); fi; end;
# Alternative:
T := (n,k) -> pochhammer(n+1,k)*binomial(n,k)/2^k:
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, May 11 2018
T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)
else (n - k + 1)* T(n, k - 1) + T(n - 1, k) fi fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Oct 02 2023

max=50; Flatten[Table[(n+k)!/(2^k*(n-k)!*k!), {n, 0, Sqrt[2 max]//Ceiling}, {k, 0, n}]][[1 ;; max]] (* Jean-François Alcover, Mar 20 2011 *)

{T(n,k)=if(k<0||k>n, 0, binomial(n, k)*(n+k)!/2^k/n!)} /* Michael Somos, Oct 03 2006 */

A001497_ser(N,t='t) = {
  my(x='x+O('x^(N+2)));
  serlaplace(deriv(exp((1-sqrt(1-2*t*x))/t),'x));
};
concat(apply(Vecrev, Vec(A001497_ser(9)))) \\ Gheorghe Coserea, Dec 27 2017

a(n, k) = (n+k)!/(2^k*(n-k)!*k!) (see Grosswald and Riordan). - Ralf Stephan, Apr 20 2004
a(n, 0)=1; a(0, k)=0, k > 0; a(n, k) = a(n-1, k) + (n-k+1) * a(n, k-1) = a(n-1, k) + (n+k-1) * a(n-1, k-1). - Len Smiley
a(n, m) = A001497(n, n-m) = A001147(m)*binomial(n+m, 2*m) for n >= m >= 0, otherwise 0.
G.f. for m-th column: (A001147(m)*x^m)/(1-x)^(2*m+1), m >= 0, where A001147(m) = double factorials (from explicit a(n, m) form).
Row polynomials y_n(x) are given by D^(n+1)(exp(t)) evaluated at t = 0, where D is the operator 1/(1-t*x)*d/dt. - Peter Bala, Nov 25 2011
G.f.: conjecture: T(0)/(1-x), where T(k) = 1 - x*y*(k+1)/(x*y*(k+1) - (1-x)^2/T(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 13 2013
Recurrence from Grosswald, p. 18, eq. (5), for the row polynomials: y_n(x) = (2*n-1)*x*y_{n-1} + y_{n-2}(x), y_{-1}(x) = 1 = y_{0} = 1, n >= 1. This becomes, for n >= 0, k = 0..n: a(n, k) = 0 for n < k (zeros not shown in the triangle), a(n, -1) = 0, a(0, 0) = 1 = a(1, 0) and otherwise a(n, k) = (2*n-1)*a(n-1, k-1) + a(n-2, k). Compare with the above given recurrences. - Wolfdieter Lang, May 11 2018
T(n, k) = Pochhammer(n+1,k)*binomial(n,k)/2^k = A113025(n,k)/2^k. - Peter Luschny, May 11 2018
a(n, k) = Sum_{i=0..min(n-1, k)} (n-i)(k-i) * a(n-1, i) where x(n) = x*(x-1)*...*(x-n+1) is the falling factorial, this equality follows directly from the operational formula we wrote in Apr 11 2025.- Abdelhay Benmoussa, May 18 2025

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

Original entry on oeis.org

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

A001514 Bessel polynomial {y_n}'(1).

Original entry on oeis.org

0, 1, 9, 81, 835, 9990, 137466, 2148139, 37662381, 733015845, 15693217705, 366695853876, 9289111077324, 253623142901401, 7425873460633005, 232122372003909045, 7715943399320562331, 271796943164015920914, 10114041937573463433966

Offset: 0

N. J. A. Sloane

nonn

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).

G. C. Greubel, Table of n, a(n) for n = 0..400
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
Index entries for sequences related to Bessel functions or polynomials

Cf. A001515, A001516, A001518, A065920, A144505.

(As in A001497 define:) f := proc(n) option remember; if n <=1 then (1+x)^n else expand((2*n-1)*x*f(n-1)+f(n-2)); fi; end;
[seq( subs(x=1,diff(f(n),x)),n=0..60)];
f2:=proc(n) local k; add((n+k+2)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f2(n),n=0..60)]; # uses a different offset

Table[Sum[(n+k+1)!/((n-k-1)!*k!*2^(k+1)), {k,0,n-1}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)
Join[{0}, Table[n*Pochhammer[1/2, n]*2^n* Hypergeometric1F1[1 - n, -2*n, 2], {n,1,50}]] (* G. C. Greubel, Aug 14 2017 *)

for(n=0,50, print1(sum(k=0,n-1, (n+k+1)!/((n-k-1)!*k!*2^(k+1))), ", ")) \\ G. C. Greubel, Aug 14 2017

a(n) = (1/2) * Sum_{k=0..n} (n+k+2)!/((n-k)!*k!*2^k) (with a different offset).
D-finite with recurrence: (n-1)^2 * a(n) = (2*n-1)*(n^2 - n + 1)*a(n-1) + n^2*a(n-2). - Vaclav Kotesovec, Jul 22 2015
a(n) ~ 2^(n+1/2) * n^(n+1) / exp(n-1). - Vaclav Kotesovec, Jul 22 2015
a(n) = n*2^n*(1/2){n}*hypergeometric1f1(1-n, -2*n, 2), where (a){n} is the Pochhammer symbol. - G. C. Greubel, Aug 14 2017
From G. C. Greubel, Aug 16 2017: (Start)
G.f.: (1/(1-t))*hypergeometric2f0(2, 3/2; -; 2*t/(1-t)^2).
E.g.f.: (1 - 2*x)^(-3/2)*((1 - x)*sqrt(1 - 2*x) + (3*x - 1))*exp((1 - sqrt(1 - 2*x))). (End)

A001880 Coefficients of Bessel polynomials y_n (x).

Original entry on oeis.org

1, 15, 210, 3150, 51975, 945945, 18918900, 413513100, 9820936125, 252070693875, 6957151150950, 205552193096250, 6474894082531875, 216659917377028125, 7675951358500425000, 287080580807915895000, 11303797869311688365625, 467445288360359818884375

Offset: 4

N. J. A. Sloane

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).

T. D. Noe, Table of n, a(n) for n = 4..100
Selden Crary, Richard Diehl Martinez, Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 1.
J. Riordan, Notes to N. J. A. Sloane, Jul. 1968
Index entries for sequences related to Bessel functions or polynomials

See A001518.
Column 4 of triangle A001497.
Equals the second right hand column of the triangles A094665 and A083061.
Other right hand columns are A001147, A160470, A160471 and A160472.

nn = 25; t = Range[0, nn]! CoefficientList[Series[x (1 + x/2)/(1 - 2 x)^(7/2), {x, 0, nn}], x]; Drop[t, 1] (* T. D. Noe, Aug 10 2012 *)

x='x+O('x^50); Vec(serlaplace(x*(1 + x/2)/(1 - 2*x)^(7/2))) \\ G. C. Greubel, Aug 13 2017

E.g.f.: x*(1 + x/2)/(1 - 2*x)^(7/2); or, if shifted, (1+ 6x+ 3x^2/2!) / (1-2x)^(9/2).
a(n) = (2*n-4)!/(4!*(n-4)!*2^(n-4)).
(n-4)*a(n) = (n-2)*(2*n-5)*a(n-1) for n = 5, 6, .. , with a(4) = 1. - Johannes W. Meijer, May 24 2009
G.f.: x^4*2F0(5/2,3;;2x). - R. J. Mathar, Aug 08 2015

A001516 Bessel polynomial {y_n}''(1).

Original entry on oeis.org

0, 0, 6, 120, 1980, 32970, 584430, 11204676, 233098740, 5254404210, 127921380840, 3350718545460, 94062457204716, 2819367702529560, 89912640142178490, 3040986592542420060, 108752084073199561140, 4101112025363285051526

Offset: 0

N. J. A. Sloane

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).

G. C. Greubel, Table of n, a(n) for n = 0..400
J. Riordan, Letter to N. J. A. Sloane, Jul. 1968
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
Index entries for sequences related to Bessel functions or polynomials

Cf. A001497, A001498, A001514, A001515, A001518, A065944, A144505.

(As in A001497 define:) f := proc(n) option remember; if n <=1 then (1+x)^n else expand((2*n-1)*x*f(n-1)+f(n-2)); fi; end;
[seq( subs(x=1,diff(f(n),x$2)),n=0..60)];

Table[Sum[(n+k+2)!/(2^(k+2)*(n-k-2)!*k!), {k,0,n-2}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)
Join[{0, 0}, Table[n*(n - 1)*Pochhammer[1/2, n]*2^n* Hypergeometric1F1[2 - n, -2*n, 2], {n,2,50}]] (* G. C. Greubel, Aug 14 2017 *)

for(n=0,20, print1(sum(k=0,n-2, (n+k+2)!/(2^(k+2)*(n-k-2)!*k!)), ", ")) \\ G. C. Greubel, Aug 14 2017

G.f.: 6*x^2*(1-x)^(-5)*hypergeom([5/2,3],[],2*x/(x-1)^2). - Mark van Hoeij, Nov 07 2011
D-finite with recurrence: (n-2)*(n-1)*a(n) = (2*n - 1)*(n^2 - n + 2)*a(n-1) + n*(n+1)*a(n-2). - Vaclav Kotesovec, Jul 22 2015
a(n) ~ 2^(n+1/2) * n^(n+2) / exp(n-1). - Vaclav Kotesovec, Jul 22 2015
a(n) = n*(n - 1)*(1/2){n}*2^n* hypergeometric1F1(2 - n, -2*n, 2), where (a){n} is the Pochhammer symbol. - G. C. Greubel, Aug 14 2017
E.g.f.: (-1)*(1 - 2*x)^(-5/2)*((4 - 14*x + 9*x^2)*sqrt(1 - 2*x) + (2*x^3 - 24*x^2 + 18*x - 4))*exp((1 - sqrt(1 - 2*x))). - G. C. Greubel, Aug 16 2017

A001881 Coefficients of Bessel polynomials y_n (x).

Original entry on oeis.org

1, 21, 378, 6930, 135135, 2837835, 64324260, 1571349780, 41247931725, 1159525191825, 34785755754750, 1109981842719750, 37554385678684875, 1343291487737574375, 50661278966102805000, 2009564065655411265000, 83648104232906493905625, 3646073249210806587298125

Offset: 5

N. J. A. Sloane, Simon Plouffe

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).

G. C. Greubel, Table of n, a(n) for n = 5..400 (terms 5..100 from T. D. Noe)
Selden Crary, Richard Diehl Martinez, Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 1.
J. Riordan, Notes to N. J. A. Sloane, Jul. 1968
Index entries for sequences related to Bessel functions or polynomials

See A001518.
(1/4) the coefficient of x^2 of polynomials in A098503.
Column 5 of triangle A001497.
Third column (m=2) of Laguerre-Sonin a=1/2 triangle A130757.

[Factorial(2*n-5)/(120*Factorial(n-5)*2^(n-5) ): n in [5..30]]; // Vincenzo Librandi, Aug 14 2017

With[{nn = 50}, CoefficientList[Series[x*(1 + 3*x/2)/(1 - 2*x)^(9/2), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 13 2017 *)

x='x+O('x^50); Vec(serlaplace(x*(1 + 3*x/2)/(1 - 2*x)^(9/2))) \\ G. C. Greubel, Aug 13 2017

a(n) = (2n-5)!/( 5!*(n-5)!*2^(n-5) ).
a(n) = binomial(n-3,2)*(2*n-5)!!/5!!, n >= 5, with (2*n-5)!! = A001147(n-2).
E.g.f.: x*(1 + 3*x/2)/(1 - 2*x)^(9/2), with offset 1. - G. C. Greubel, Aug 13 2017
G.f.: t^5 * hypergeometric2F0(3, 7/2; -; 2*t) = t^5 + 21*t^6 + .... - G. C. Greubel, Aug 16 2017

A065919 Bessel polynomial y_n(4).

Original entry on oeis.org

1, 5, 61, 1225, 34361, 1238221, 54516085, 2836074641, 170218994545, 11577727703701, 880077524475821, 73938089783672665, 6803184337622361001, 680392371852019772765, 73489179344355757819621, 8525425196317119926848801, 1057226213522667226687070945

Offset: 0

N. J. A. Sloane, Dec 08 2001

Main diagonal of A143411. - Peter Bala, Aug 14 2008

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Harry J. Smith, Table of n, a(n) for n = 0..100
W. Mlotkowski and A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
Index entries for sequences related to Bessel functions or polynomials

Cf. A001515, A001517, A001518.
Cf. A143411 (main diagonal), A143412.
Polynomial coefficients are in A001498.

A065919:= func< n | (&+[Binomial(n,k)*Factorial(n+k)*2^k/Factorial(n): k in [0..n]]) >;
[A065919(n): n in [0..30]]; // G. C. Greubel, Oct 05 2023

seq(simplify(2^n*KummerU(-n,-2*n,1/2)), n=0..16); # Peter Luschny, May 10 2022

Table[Sum[(n+k)!*2^k/((n-k)!*k!), {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)

for (n=0, 100, if (n>1, a=4*(2*n - 1)*a1 + a2; a2=a1; a1=a, if (n, a=a1=5, a=a2=1)); write("b065919.txt", n, " ", a) ) \\ Harry J. Smith, Nov 04 2009

a(n) = sum(k=0,n, (n+k)!*2^k/((n-k)!*k!) ); \\ Joerg Arndt, May 17 2013

def A065919(n): return sum(binomial(n,k)*factorial(n+k)*2^k/factorial(n) for k in range(n+1))
[A065919(n) for n in range(31)] # G. C. Greubel, Oct 05 2023

y_n(x) = Sum_{k=0..n} (n+k)!*(x/2)^k/((n-k)!*k!).
From Peter Bala, Aug 14 2008: (Start)
Recurrence relation: a(0) = 1, a(1) = 5, a(n) = 4*(2*n-1)*a(n-1) + a(n-2) for n >= 2. Sequence A143412(n) satisfies the same recurrence relation.
1/sqrt(e) = 1 - 2*Sum_{n = 0..inf} (-1)^n/(a(n)*a(n+1)) = 1 - 2*( 1/(1*5) - 1/(5*61) + 1/(61*1225) - ... ). (End)
G.f.: 1/Q(0), where Q(k)= 1 - x - 4*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013
a(n) = exp(1/4)/sqrt(2*Pi)*BesselK(n+1/2,1/4). - Gerry Martens, Jul 22 2015
a(n) ~ 2^(3*n+1/2) * n^n / exp(n-1/4). - Vaclav Kotesovec, Jul 22 2015
From Peter Bala, Apr 12 2017: (Start)
a(n) = 1/n!*Integral_{x = 0..inf} x^n*(1 + 2*x)^n dx.
E.g.f.: d/dx( exp(x*c(2*x)) ) = 1 + 5*x + 61*x^2/2! + 1225*x^3/3! + ..., where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
G.f.: (1/(1-x))*hypergeometric2f0(1,1/2; - ; 8*x/(1-x)^2). - G. C. Greubel, Aug 16 2017
a(n) = 2^n*KummerU(-n, -2*n, 1/2). - Peter Luschny, May 10 2022

A065944 Bessel polynomial {y_n}''(-1).

Original entry on oeis.org

0, 0, 6, -60, 720, -9870, 153510, -2679264, 51934680, -1107917910, 25807660560, -651977992380, 17758547202396, -518856566089680, 16188283372489410, -537210169663283760, 18894951642157260480, -702160022681408982114

Offset: 0

N. J. A. Sloane, Dec 08 2001

sign

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Cf. A001518, A001516.

f:=Factorial;; Concatenation([0,0], List([2..20], n-> Sum([0..n-2], k-> (-1)^k*f(n+k+2)/(2^(k+2)*f(n-k-2)*f(k)) ))); # G. C. Greubel, Jul 10 2019

f:=Factorial; [0,0] cat [(&+[((-1)^k*f(n+k+2)/(2^(k+2)*f(n-k-2) *f(k))): k in [0..n-2]]): n in [2..20]]; // G. C. Greubel, Jul 10 2019

Table[Sum[(n+k+2)!*(-1)^k/(2^(k+2)*(n-k-2)!*k!), {k,0,n-2}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)
Join[{0, 0}, Table[4*n*(n-1)*Pochhammer[1/2, n]*(-2)^(n-2)* Hypergeometric1F1[2-n, -2*n, -2], {n, 2,20}]] (* G. C. Greubel, Aug 14 2017 *)

for(n=0,20, print1(sum(k=0,n-2, (n+k+2)!*(-1)^k/(2^(k+2)*(n-k-2)!*k!)), ", ")) \\ G. C. Greubel, Aug 14 2017

f=factorial; [0,0]+[sum((-1)^k*f(n+k+2)/(2^(k+2)*f(n-k-2)*f(k)) for k in (0..n-2)) for n in (2..20)] # G. C. Greubel, Jul 10 2019

Recurrence: (n-2)*(n-1)*a(n) = -(n-2)*(n+1)*(2*n-1)*a(n-1) + n*(n+1)*a(n-2). - Vaclav Kotesovec, Jul 22 2015
a(n) ~ (-1)^n * 2^(n+1/2) * n^(n+2) / exp(n+1). - Vaclav Kotesovec, Jul 22 2015
From G. C. Greubel, Aug 14 2017: (Start)
a(n) = 2*n*(n-1)*(1/2){n}*(-2)^(n - 1)* hypergeometric1f1(2 - n, -2*n, -2), where (a){n} is the Pochhammer symbol.
E.g.f.: (1 + 2*x)^(-5/2)*(x*(x + 2)*sqrt(1 + 2*x) + (2*x^3 - 2*x)) * exp(-1 + sqrt(1 + 2*x)). (End)
G.f.: (6*x^2/(1-x)^5)*hypergeometric2f0(3,5/2; - ; -2*x/(1-x)^2). - G. C. Greubel, Aug 16 2017

cp's OEIS Frontend

A001498 Triangle a(n,k) (n >= 0, 0 <= k <= n) of coefficients of Bessel polynomials y_n(x) (exponents in increasing order).

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

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A001514 Bessel polynomial {y_n}'(1).

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A001880 Coefficients of Bessel polynomials y_n (x).

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A001516 Bessel polynomial {y_n}''(1).

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A001881 Coefficients of Bessel polynomials y_n (x).

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