A002119 -id:A002119 - 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

A033815 Number of standard permutations of [ a_1..a_n b_1..b_n ] (b_i is not immediately followed by a_i, for all i).

Original entry on oeis.org

1, 1, 14, 426, 24024, 2170680, 287250480, 52370755920, 12585067447680, 3854801333416320, 1465957162768492800, 677696237345719468800, 374281829360322587827200, 243388909697235614324812800, 184070135024053703140543027200, 160192129141963141211280644352000

Offset: 0

N. J. A. Sloane

Also turns up as the solution to Problem #18, p. 326 of Alan Tucker's Applied Combinatorics, 4th ed, Wiley NY 2002 [Tucker's `n' is the `2n' here]. - John L Leonard, Sep 15 2003
Number of acyclic orientations of the Turán graph T(2n,n). - Alois P. Heinz, Jan 13 2016
n-th term of the n-th forward differences of n!. - Alois P. Heinz, Feb 22 2019

R. P. Stanley, Enumerative Combinatorics I, Chap.2, Exercise 10, p. 89.

Reinhard Zumkeller, Table of n, a(n) for n = 0..200
Leo Chao, Paul DesJarlais and John L Leonard, A binomial identity, via derangements, Math. Gaz. 89 (2005), 268-270.
Ira Gessel, Enumerative applications of symmetric functions, Séminaire Lotharingien de Combinatoire, B17a (1987), 17 pp.

Main diagonal of array in A068106 and of A047920.
Cf. A000142, A002119, A116854, A267383.
Column k=2 of A372326.

a033815 n = a116854 (2 * n + 1) (n + 1)
-- Reinhard Zumkeller, Aug 31 2014

A033815 := proc(n) local i; add(binomial(n, i)*(-1)^i*(2*n - i)!, i = 0 .. n) end;
# second Maple program:
A:= proc(n, k) A(n, k):= `if`(k=0, n!, A(n+1, k-1) -A(n, k-1)) end:
a:= n-> A(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 22 2019

a[n_] := (2n)!*Hypergeometric1F1[-n, -2n, -1]; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jun 13 2012, after Vladimir Reshetnikov *)

a(n) = A002119(n)*n!*(-1)^n.
D-finite with recurrence a(n) = 2n*(2n-1)*a(n-1) + n*(n-1)*a(n-2).
a(n) = Sum_{i=0..n} binomial(n, i)*(-1)^i*(2*n-i)!.
From John L Leonard, Sep 15 2003: (Start)
a(n) = Sum_{i=0..n} C(n, i)*(2n-i)!*Sum_{j=0..2n-i} (-1)^j/j!.
a(n) = n!*Sum_{i=0..n} C(n, i)*n!/(n-i)!*Sum_{j=0..n-i} (-1)^j*C(n-i, j)*(n-j)!/i!. (End)
a(n) = Sum_{k=0..n} binomial(n,k)*A000166(n+k). - Vladeta Jovovic, Sep 04 2006
a(n) = A116854(2*n+1,n+1). - Reinhard Zumkeller, Aug 31 2014
a(n) = A267383(2n,n). - Alois P. Heinz, Jan 13 2016
a(n) ~ sqrt(Pi) * 2^(2*n + 1) * n^(2*n + 1/2) / exp(2*n + 1/2). - Vaclav Kotesovec, Feb 18 2017
a(n) = n!*exp(-1/2)*((-1)^n * BesselI(n+1/2,1/2)*Pi^(1/2) + BesselK(n+1/2,1/2)/Pi^(1/2) ). - Mark van Hoeij, Jul 15 2022

A336114 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,1,2,...,2,1); a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 6, 64, 930, 17088, 380870, 9992064, 301738626, 10310669440, 393355695942, 16573741095360, 764401360062626, 38304552622588224, 2072335759298438790, 120390122318741003008, 7474705606285243345410, 493940966313183768532224, 34613731176130328980714886

Offset: 0

Dmitry Efimov, Jul 21 2020

Number of perfect matchings of a chord diagram with 2*n vertices, where neighboring vertices are joined by one chord, and any other pair of vertices is joined by two chords.

			A symmetric 4x4 Toeplitz matrix A with the first row (0,1,2,1) has the form:
0 1 2 1
1 0 1 2
2 1 0 1
1 2 1 0.
Its hafnian equals Hf(A) = a12*a34+a13*a24+a14*a23 = 1*1+2*2+1*1 = 6 = a(2).

Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2020.

Cf. A002119, A336286, A336400.

Join[{1,1},Table[2 HypergeometricU[n,1+2 n,-1],{n,2,16}]] (* Stefano Spezia, Jul 22 2020 *)

a(n) = 2*n*Sum_{k=0..n} (-1)^(n-k)*(n+k-1)!/(k!*(n-k)!), n>=2.
D-finite with recurrence a(n+1) = (4*n+3)*a(n)-(4*n-7)*a(n-1)-a(n-2), n>=4.
D-finite with recurrence a(n+1) = (8*n^2*a(n)+(2*n+1)*a(n-1))/(2*n-1), n>=3.
a(n) = |A002119(n)|-|A002119(n-1)|, n>=2.
a(n) ~ (2*n)!/(sqrt(e)*n!).
a(n) = U(n,1+2*n,-1) for n >= 2, where U(a,b,c) is the confluent hypergeometric function of the second kind. - Stefano Spezia, Jul 22 2020

A336286 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,1,2,...,2,0); a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 5, 57, 859, 16087, 362781, 9593105, 291347603, 9998539791, 382732896853, 16169762600329, 747423640472235, 37523173542935207, 2033249827596197549, 118278700627740322977, 7352204062275501662371, 486343759162888783503775, 34112193002666850227154213

Offset: 0

Dmitry Efimov, Jul 16 2020

Number of perfect matchings of an arc diagram with 2*n vertices, where neighboring vertices are joined by one arc, the vertices 1 and 2*n are not adjacent if n>=2, and all other pairs of vertices are joined by two arcs.

			A symmetric 4 X 4 Toeplitz matrix A with the first row (0,1,2,0) has the form:
  0 1 2 0
  1 0 1 2
  2 1 0 1
  0 2 1 0.
Its hafnian equals Hf(A) = a12*a34 + a13*a24 + a14*a23 = 1*1 + 2*2 + 0*1 = 5 = a(2).

Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2020.

Cf. A002119, A336114, A336400.

[1,1,seq(add((-1)^(n-k-1)*(n+k-1)!*(-3*n+k)/(k!*(n-k)!),k=0..n),n=2..32)] # Georg Fischer, Jun 05 2021

Join[{1,1},RecurrenceTable[{a[n+1] == (4*n+4)*a[n]-(8*n-13)*a[n-1]-2*a[n-2], a[2]==5, a[3]==57, a[4]==859}, a[n], {n,2,32}]] (* Georg Fischer, Jun 05 2021 *)

a(n) = Sum_{k=0..n} (-1)^(n-k)*(n+k-1)!*(3*n-k)/(k!*(n-k)!), n>=2.
D-finite with recurrence a(n+1) = (4n+4)*a(n) - (8n-13)*a(n-1) - 2*a(n-2), n>=4.
D-finite with recurrence a(n+1) = ((32*n^2-12*n+2)*a(n) + (8*n+1)*a(n-1))/(8*n-7), n>=3.
a(n) = |A002119(n)| - 2*|A002119(n-1)|, n>=2.
a(n) ~ (2*n)!/sqrt(e)*n!.

A113025 Triangle of integer coefficients of polynomials P(n,x) of degree n, and falling powers of x, arising in diagonal Padé approximation of exp(x).

Original entry on oeis.org

1, 1, 2, 1, 6, 12, 1, 12, 60, 120, 1, 20, 180, 840, 1680, 1, 30, 420, 3360, 15120, 30240, 1, 42, 840, 10080, 75600, 332640, 665280, 1, 56, 1512, 25200, 277200, 1995840, 8648640, 17297280, 1, 72, 2520, 55440, 831600, 8648640, 60540480, 259459200

Offset: 0

Benoit Cloitre, Jan 03 2006

exp(x) is well approximated by P(n,x)/P(n,-x). (P(n,1)/P(n,-1))_{n>=0} is a sequence of convergents to e: i.e., P(n,1) = A001517(n) and P(n,-1) = abs(A002119(n)).
From Roger L. Bagula, Feb 15 2009: (Start)
The row polynomials in rising powers of x are y_n(2*x) = Sum_{k=0..n} binomial(n+k, 2*k)*((2*k)!/k!)*x^k, for n >= 0, with the Bessel polynomials y_n(x) of Krall and Frink, eq. (3), (see also Grosswald, p. 18, eq. (7) and Riordan, p. 77). For the coefficients see A001498. [Edited by Wolfdieter Lang, May 11 2018]
P(n, x) = Sum_{k=0..n} (n+k)!/(k!*(n-k)!)*x^(n-k).
Row sums are A001517. (End)

			P(3,x) = x^3 + 12*x^2 + 60*x + 120.
y_3(2*x) = 1 + 12*x + 60*x^2 + 120*x^3. (Bessel with x -> 2*x).
From _Roger L. Bagula_, Feb 15 2009: (Start)
{1},
{1, 2},
{1, 6, 12},
{1, 12, 60, 120},
{1, 20, 180, 840, 1680},
{1, 30, 420, 3360, 15120, 30240},
{1, 42, 840, 10080, 75600, 332640, 665280},
{1, 56, 1512, 25200, 277200, 1995840, 8648640, 17297280},
{1, 72, 2520, 55440, 831600, 8648640, 60540480, 259459200, 518918400},
{1, 90, 3960, 110880, 2162160, 30270240, 302702400, 2075673600, 8821612800, 17643225600},
{1, 110, 5940, 205920, 5045040, 90810720, 1210809600, 11762150400, 79394515200, 335221286400, 670442572800} (End)

J. Riordan, Combinatorial Identities, Wiley, 1968, p.77, 10. [From Roger L. Bagula, Feb 15 2009]

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
E. Grosswald, Bessel Polynomials: Recurrence Relations, Lecture Notes Math. vol. 698, 1978, p. 18.
H. L. Krall and Orrin Frink, A New Class of Orthogonal Polynomials: The Bessel Polynomials, Trans. Amer. Math. Soc. 65, 100-115, 1949.
Eric Weisstein's World of Mathematics, Padé approximants.
F. Wielonsky, Asymptotics of diagonal Hermite-Pade approximants to exp(x), J. Approx. Theory 90 (1997) 283-298.

Cf. A001498, A001517, A303986 (signed version).

T := (n, k) -> pochhammer(n+1, k)*binomial(n, k):
seq(seq(T(n, k), k=0..n), n=0..9); # Peter Luschny, May 11 2018

L[n_, m_] = (n + m)!/((n - m)!*m!);
Table[Table[L[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%] (* Roger L. Bagula, Feb 15 2009 *)
P[x_, n_] := Sum[ (2*n - k)!/(k!*(n - k)!)*x^(k), {k, 0, n}]; Table[Reverse[CoefficientList[P[x, n], x]], {n,0,10}] // Flatten (* G. C. Greubel, Aug 15 2017 *)

PARI
```
T(n,k)=(n+k)!/k!/(n-k)!
```

From Wolfdieter Lang, May 11 2018: (Start)
T(n, k) = binomial(n+k, 2*k)*(2*k)!/k! = (n+k)!/((n-k)!*k!), n >= 0, k = 0..n. (see the R. L. Baluga comment above).
Recurrence (adapted from A001498, see the Grosswald reference): 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*(2*n-1)*a(n-1, k-1) + a(n-2, k).
(End)
T(n, k) = Pochhammer(n+1, k)*binomial(n, k). # Peter Luschny, May 11 2018

A060475 Triangular array formed from successive differences of factorial numbers, then with factorials removed.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 3, 2, 1, 3, 7, 11, 9, 1, 4, 13, 32, 53, 44, 1, 5, 21, 71, 181, 309, 265, 1, 6, 31, 134, 465, 1214, 2119, 1854, 1, 7, 43, 227, 1001, 3539, 9403, 16687, 14833, 1, 8, 57, 356, 1909, 8544, 30637, 82508, 148329, 133496, 1, 9, 73, 527, 3333, 18089, 81901, 296967, 808393, 1468457, 1334961

Offset: 0

Henry Bottomley, Mar 16 2001

T(n,k) is also the number of partial bijections (of an n-element set) with a fixed domain of size k and without fixed points. Equivalently, T(n,k) is the number of partial derangements with a fixed domain of size k in the symmetric inverse semigroup (monoid), I sub n. - Abdullahi Umar, Sep 14 2008

			Triangle begins
  1,
  1,  0,
  1,  1,  1,
  1,  2,  3,  2,
  1,  3,  7, 11,  9,
  1,  4, 13, 32, 53, 44,
  ...

G. C. Greubel, Rows n=0..100 of triangle, flattened
A. Laradji, and A. Umar, Combinatorial results for the symmetric inverse semigroup, Semigroup Forum 75 (2007), 221-236. - _Abdullahi Umar_, Sep 14 2008
L. Takacs, The Problem of Coincidences, Archive for History of Exact Sciences, Volume 21, No. 3, Sept. 1980. pp 229-244, paragraph 10 (Catalan).

Columns include A000012, A001477, A002061.
Diagonals include A000166, A000255, A000153, A000261, A001909, A001910.
Main diagonal is abs of A002119.
Similar to A076731.
Row sums equal A003470. - Johannes W. Meijer, Jul 27 2011
Cf. A047920, A008290.

[[Factorial(k)*(&+[(-1)^j*Binomial(n-j, k-j)/Factorial(j): j in [0..k]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Mar 04 2019

A060475 := proc(n,k): k! * add(binomial(n-j,k-j)*(-1)^j/j!, j=0..k) end:
seq(seq(A060475(n,k), k=0..n), n=0..7); # Johannes W. Meijer, Jul 27 2011
T := (n,k) -> KummerU(-k, -n, -1):
seq(seq(simplify(T(n, k)), k = 0..n), n = 0..10); # Peter Luschny, Jul 07 2022

t[n_, k_] := k!*Sum[Binomial[n - j, k - j]*(-1)^j/j!, {j, 0, k}]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Aug 08 2011 *)

{T(n,k) = k!*sum(j=0,k, (-1)^j*binomial(n-j, k-j)/j!)};
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Mar 04 2019

[[factorial(k)*sum((-1)^j*binomial(n-j, k-j)/factorial(j) for j in (0..k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 04 2019

T(n,k) = A047920(n,k)/(n-k)! = (n-1)*T(n-1,k-1) + (k-1)*T(n-2,k-2) = (n-k+1)*T(n, k-1) - T(n-1,k-1).
From Abdullahi Umar, Sep 14 2008: (Start)
T(n,k) = k! * Sum_{j=0..k} C(n-j,k-j)*(-1)^j/j!.
C(n,k)*T(n,k) = A144089(n, k). (End)
T(n,k) = A076732(n+1,k+1)/(k+1). - Johannes W. Meijer, Jul 27 2011
E.g.f. as a square array: A(x,y) = exp(-x)/(1 - x - y) = (1 + y + y^2 + y^3 + ...) + (y + 2*y^2 + 3*y^3 + 4*y^4 + ...)*x + (1 + 3*y + 7*y^2 + 13*y^3 + ...)*x^2/2! + (2 + 11*y + 32*y^2 + 71*y^3 + ...)*x^3/3! + .... Observe that (1 - y)*A(x*(1 - y),y) = exp(x*(y - 1))/(1 - x) is the e.g.f. for A008290. - Peter Bala, Sep 25 2013
T(n, k) = KummerU(-k, -n, -1). - Peter Luschny, Jul 07 2022

A131514 Number of ways to design a set of three n-sided dice (using nonnegative integers) such that summing the faces can give any integer from 0 to n^3 - 1.

Original entry on oeis.org

1, 1, 1, 15, 1, 71, 1, 280, 15, 71, 1, 3660, 1, 71, 71, 5775, 1, 3660, 1, 3660, 71, 71, 1, 160440, 15, 71, 280, 3660, 1, 20365, 1, 126126, 71, 71, 71, 415185, 1, 71, 71, 160440, 1, 20365, 1, 3660, 3660, 71, 1, 6387150, 15, 3660, 71, 3660, 1, 160440, 71, 160440

Offset: 1

H.B. Wassenaar (towr(AT)ai.rug.nl), Aug 14 2007

nonn

Also the number of ways to factor (x^(n^3)-1)/(x-1) into p(x)*q(x)*r(x), such that p(x),q(x),r(x) are polynomials with exactly n terms and all coefficients +1 (and all exponents nonnegative). (Krasner and Ranulac, 1937)

a(n) depends only on the prime signature of n. Hence a(n) will be 1 for all primes, 15 for all squares of primes, 71 for all products of distinct primes, and so on. - William P. Orrick, Jan 26 2023

			a(4)=15 because we can choose any of the following 15 configurations for our three dice:
  [ {0, 1,  2,  3}, {0, 4,  8, 12}, {0, 16, 32, 48} ],
  [ {0, 1,  2,  3}, {0, 4, 16, 20}, {0,  8, 32, 40} ],
  [ {0, 1,  2,  3}, {0, 4, 32, 36}, {0,  8, 16, 24} ],
  [ {0, 1,  4,  5}, {0, 2,  8, 10}, {0, 16, 32, 48} ],
  [ {0, 1,  4,  5}, {0, 2, 16, 18}, {0,  8, 32, 40} ],
  [ {0, 1,  4,  5}, {0, 2, 32, 34}, {0,  8, 16, 24} ],
  [ {0, 1,  8,  9}, {0, 2,  4,  6}, {0, 16, 32, 48} ],
  [ {0, 1,  8,  9}, {0, 2, 16, 18}, {0,  4, 32, 36} ],
  [ {0, 1,  8,  9}, {0, 2, 32, 34}, {0,  4, 16, 20} ],
  [ {0, 1, 16, 17}, {0, 2,  4,  6}, {0,  8, 32, 40} ],
  [ {0, 1, 16, 17}, {0, 2,  8, 10}, {0,  4, 32, 36} ],
  [ {0, 1, 16, 17}, {0, 2, 32, 34}, {0,  4,  8, 12} ],
  [ {0, 1, 32, 33}, {0, 2,  4,  6}, {0,  8, 16, 24} ],
  [ {0, 1, 32, 33}, {0, 2,  8, 10}, {0,  4, 16, 20} ],
  [ {0, 1, 32, 33}, {0, 2, 16, 18}, {0,  4,  8, 12} ].

William P. Orrick, Table of n, a(n) for n = 1..10000
M. Krasner and B. Ranulac, Sur une propriété des polynomes de la division du cercle, Comptes Rendus Académie des Sciences Paris, 240:397-399, 1937.
Matthew C. Lettington and Karl Michael Schmidt, Divisor Functions and the Number of Sum Systems, arXiv:1910.02455 [math.NT], 2019.

Cf. A273013, A002119.

@cached_function
def R3(i,j,k):
    if i > 1 and j==1 and k==1:
        return(1)
    elif j > 1 or k > 1:
        divList = divisors(i)[:-1]
        return(sum(G3(d,j,k) for d in divList) + sum(B3(d,j,k) for d in divList))
@cached_function
def G3(i,j,k):
    if i==1 and j > 1 and k==1:
        return(1)
    elif i > 1 or k > 1:
        divList = divisors(j)[:-1]
        return(sum(R3(i,d,k) for d in divList) + sum(B3(i,d,k) for d in divList))
@cached_function
def B3(i,j,k):
    if i==1 and j==1 and k > 1:
        return(1)
    elif i > 1 or j > 1:
        divList = divisors(k)[:-1]
        return(sum(R3(i,j,d) for d in divList) + sum(G3(i,j,d) for d in divList))
def a3(n):
    if n == 1:
        return(1)
    else:
        return(R3(n,n,n) / 2) # William P. Orrick, Jan 26 2023

Recurrence: a(1) = 1. For n > 1, a(n) = r(n,n,n) / 2 where r(i,1,1) = g(1,j,1) = b(1,1,k) = 1 for all i, j, k > 1, r(i,j,k) = Sum_{d|i,dWilliam P. Orrick, Jan 26 2023

Terms a(16) and beyond from William P. Orrick, Jan 26 2023

cp's OEIS Frontend

A272648 a(n) = A002119(n) mod 7.

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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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A033815 Number of standard permutations of [ a_1..a_n b_1..b_n ] (b_i is not immediately followed by a_i, for all i).

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A336114 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,1,2,...,2,1); a(0)=a(1)=1.

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A336286 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,1,2,...,2,0); a(0)=a(1)=1.

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