A001498 -id:A001498 - cp's OEIS frontend

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

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

A049403 A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).

Original entry on oeis.org

1, 1, 1, 0, 3, 1, 0, 3, 6, 1, 0, 0, 15, 10, 1, 0, 0, 15, 45, 15, 1, 0, 0, 0, 105, 105, 21, 1, 0, 0, 0, 105, 420, 210, 28, 1, 0, 0, 0, 0, 945, 1260, 378, 36, 1, 0, 0, 0, 0, 945, 4725, 3150, 630, 45, 1, 0, 0, 0, 0, 0, 10395, 17325, 6930, 990, 55, 1, 0, 0, 0, 0, 0, 10395, 62370

Offset: 1

Wolfdieter Lang

a(n,1) = A019590(n) = A008279(1,n). a(n,m) =: S1(-1; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A001497(n-1,m-1) (signed Bessel triangle). The monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m, E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

Exponential Riordan array [1+x, x(1+x/2)]. T(n,k) = A001498(k+1, n-k). - Paul Barry, Jan 15 2009

			Triangle a(n,m) (with rows n >= 1 and columns m >= 1) begins as follows:
  1;                 with row polynomial E(1,x) = x;
  1, 1;              with row polynomial E(2,x) = x^2 + x;
  0, 3,  1;          with row polynomial E(3,x) = 3*x^2 + x^3;
  0, 3,  6,   1;     with row polynomial E(4,x) = 3*x^2 + 6*x^3 + x^4;
  0, 0, 15,  10,   1;
  0, 0, 15,  45,  15,   1;
  0, 0,  0, 105, 105,  21,  1;
  0, 0,  0, 105, 420, 210, 28, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First 10 rows of the array and more.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.

Cf. A000085 (row sums), A001497, A001498, A008275, A008279, A008297, A019590, A030528, A039692.

Variations of this array: A096713, A104556, A122848, A130757.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n<2,1,0), 9); # Peter Luschny, Jan 28 2016

t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 11}, {k, n}] // Flatten
(* Second program: *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 13;
M = BellMatrix[If[#<2, 1, 0]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

a(n, m) = n!*A030528(n, m)/(m!*2^(n-m)) for n >= m >= 1.
a(n, m) = (2*m-n+1)*a(n-1, m) + a(n-1, m-1) for n >= m >= 1 with a(n, m) = 0 for n < m, a(n, 0) := 0, and a(1, 1) = 1. [The 0th column does not appear in this array. - Petros Hadjicostas, Oct 28 2019]
E.g.f. for the m-th column: (x*(1 + x/2))^m/m!.
a(n,m) = A122848(n,m). - R. J. Mathar, Jan 14 2011

A144299 Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,0) for n >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 3, 0, 0, 1, 10, 15, 0, 0, 0, 1, 15, 45, 15, 0, 0, 0, 1, 21, 105, 105, 0, 0, 0, 0, 1, 28, 210, 420, 105, 0, 0, 0, 0, 1, 36, 378, 1260, 945, 0, 0, 0, 0, 0, 1, 45, 630, 3150, 4725, 945, 0, 0, 0, 0, 0, 1, 55, 990, 6930, 17325, 10395, 0, 0, 0, 0, 0, 0

Offset: 0

David Applegate and N. J. A. Sloane, Dec 06 2008

T(n,k) is the number of partitions of an n-set into k nonempty subsets, each of size at most 2.
The Grosswald and Choi-Smith references give many further properties and formulas.
Considered as an infinite lower triangular matrix T, lim_{n->infinity} T^n = A118930: (1, 1, 2, 4, 13, 41, 166, 652, ...) as a vector. - Gary W. Adamson, Dec 08 2008

			Triangle begins:
  n:
  0: 1
  1: 1  0
  2: 1  1   0
  3: 1  3   0    0
  4: 1  6   3    0   0
  5: 1 10  15    0   0  0
  6: 1 15  45   15   0  0  0
  7: 1 21 105  105   0  0  0  0
  8: 1 28 210  420 105  0  0  0  0
  9: 1 36 378 1260 945  0  0  0  0  0
  ...
The row sums give A000085.
For some purposes it is convenient to rotate the triangle by 45 degrees:
  1 0 0 0 0  0  0   0   0    0    0     0 ...
    1 1 0 0  0  0   0   0    0    0     0 ...
      1 3 3  0  0   0   0    0    0     0 ...
        1 6 15 15   0   0    0    0     0 ...
          1 10 45 105 105    0    0     0 ...
             1 15 105 420  945  945     0 ...
                1  21 210 1260 4725 10395 ...
                    1  28  378 3150 17325 ...
                        1   36  630  6930 ...
                             1   45   990 ...
  ...
The latter triangle is important enough that it has its own entry, A144331. Here the column sums give A000085 and the rows sums give A001515.
If the entries in the rotated triangle are denoted by b1(n,k), n >= 0, k <= 2n, then we have the recurrence b1(n, k) = b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2).
Then b1(n,k) is the number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1 or 2.

E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
J. Y. Choi and J. D. H. Smith, On the unimodality and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
Toufik Mansour, Matthias Schork, and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
T. Mansour and M. Shattuck, Partial matchings and pattern avoidance, Appl. Anal. Discrete Math. 7 (2013) 25-50.

Other versions of this same triangle are given in A111924 (which omits the first row), A001498 (which left-adjusts the rows in the bottom view), A001497 and A100861. Row sums give A000085.

Cf. A000085, A000217, A001464, A004526, A118930, A144385, A144643..

a144299 n k = a144299_tabl !! n !! k
a144299_row n = a144299_tabl !! n
a144299_tabl = [1] : [1, 0] : f 1 [1] [1, 0] where
   f i us vs = ws : f (i + 1) vs ws where
               ws = (zipWith (+) (0 : map (i *) us) vs) ++ [0]
-- Reinhard Zumkeller, Jan 01 2014

A144299:= func< n,k | k le Floor(n/2) select Factorial(n)/(Factorial(n-2*k)*Factorial(k)*2^k) else 0 >;
[A144299(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 29 2023

Maple code producing the rotated version:
b1 := proc(n, k)
option remember;
if n = k then 1;
elif k < n then 0;
elif n < 1 then 0;
else b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2);
end if;
end proc;
for n from 0 to 12 do lprint([seq(b1(n,k),k=0..2*n)]); od:

T[n_,0]=0; T[1,1]=1; T[2,1]=1; T[n_, k_]:= T[n-1,k-1] + (n-1)T[n-2,k-1];
Table[T[n,k], {n,12}, {k,n,1,-1}]//Flatten (* Robert G. Wilson v *)
Table[If[k<=Floor[n/2],n!/((n-2 k)! k! 2^k),0], {n, 0, 12},{k,0,n}]//Flatten (* Stefano Spezia, Jun 15 2023 *)

def A144299(n,k): return factorial(n)/(factorial(n-2*k)*factorial(k)*2^k) if k <= (n//2) else 0
flatten([[A144299(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 29 2023

T(n, k) = T(n-1, k-1) + (n-1)*T(n-2, k-1).
E.g.f.: Sum_{k >= 0} Sum_{n = 0..2k} T(n,k) y^k x^n/n! = exp(y(x+x^2/2)). (The coefficient of y^k is the e.g.f. for the k-th row of the rotated triangle shown below.)
T(n, k) = n!/((n - 2*k)!*k!*2^k) for 0 <= k <= floor(n/2) and 0 otherwise. - Stefano Spezia, Jun 15 2023
From G. C. Greubel, Sep 29 2023: (Start)
T(n, 1) = A000217(n-1).
Sum_{k=0..n} T(n,k) = A000085(n).
Sum_{k=0..n} (-1)^k*T(n,k) = A001464(n). (End)

Offset fixed by Reinhard Zumkeller, Jan 01 2014

A043301 a(n) = 2^nSum_{k=0..n} (n+k)!/((n-k)!k!*4^k).

Original entry on oeis.org

1, 3, 13, 77, 591, 5627, 64261, 857901, 13125559, 226566107, 4357258269, 92408688077, 2142828858847, 53940356223483, 1464960933469429, 42699628495507373, 1329548327094606279, 44045893308104036699, 1546924459092019709581, 57412388559637145401293

Offset: 0

Benoit Cloitre, Apr 04 2002

Bruce Berndt, Ramanujan's Notebooks Part II, Springer-Verlag; see Integrals and Asymptotic Expansions, p. 229.
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed., Section 3.737.1, p. 423.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
Wojciech Mlotkowski and Anna Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.

Cf. A043302, A144505.

I:=[3,13]; [1] cat [n le 2 select I[n]  else  (2*n-1)*Self(n-1) + 4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jul 24 2015

f:= gfun:-rectoproc({a(0)=1, a(1)=3, a(n) = (2*n-1)*a(n-1) + 4*a(n-2)}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Jul 23 2015
A043301 := n-> 2^n*hypergeom([n+1, -n], [], -1/4):
seq(simplify(A043301(n)), n=0..19); # Peter Luschny, Nov 10 2016

Table[2^n Sum[(n+k)!/((n-k)!k! 4^k),{k,0,n}],{n,0,20}] (* or *) RecurrenceTable[{a[0]==1,a[1]==3,a[n]==(2n-1)a[n-1]+4a[n-2]}, a[n], {n,20}] (* Harvey P. Dale, Aug 14 2011 *)
CoefficientList[Series[E^(2-2*Sqrt[1-2*x])/Sqrt[1-2*x],{x,0,20}],x]*Range[0,20]! (* Vaclav Kotesovec, Oct 21 2012 *)

x='x+O('x^66); Vec(serlaplace(exp(2-2*sqrt(1-2*x))/sqrt(1-2*x))) \\ Joerg Arndt, May 04 2013

D-finite with recurrence: a(n) = (2*n-1)*a(n-1) + 4*a(n-2), n>1.
a(n) = 2^(n+1)n!(e^2/Pi)*Integral_{t=0..infinity} cos(2t)/(1+t^2)^(n+1)dt.
E.g.f.: 2*(e^2/Pi)*Integral_{t=0..infinity} cos(2t)/(1+t^2-2x)dt.
2^n * y_n(1/2), where y_n(x) are the Bessel polynomials A001498.
G.f.: 1/G(0) where G(k) = 1 - 2*x - x*(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 17 2011
E.g.f.: exp(2-2*sqrt(1-2*x))/sqrt(1-2*x). - Vaclav Kotesovec, Oct 21 2012
a(n) ~ 2^(n+1/2)*n^n/exp(n-2). - Vaclav Kotesovec, Oct 21 2012
G.f.: T(0)/(1-2*x), where T(k) = 1 - x*(k+1)/( x*(k+1) - (1-2*x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 27 2013
a(n) = 2^(n+1)*exp(2)/sqrt(Pi)*BesselK(1/2+n,2). - Gerry Martens, Jul 22 2015
a(n) = 2^n*hypergeom( [n+1, -n], [], -1/4). - Peter Luschny, Nov 10 2016

Edited by Michael Somos, Jul 16 2002

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

A144331 Triangle b(n,k) for n >= 0, 0 <= k <= 2n, read by rows. See A144299 for definition and properties.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 3, 3, 0, 0, 0, 1, 6, 15, 15, 0, 0, 0, 0, 1, 10, 45, 105, 105, 0, 0, 0, 0, 0, 1, 15, 105, 420, 945, 945, 0, 0, 0, 0, 0, 0, 1, 21, 210, 1260, 4725, 10395, 10395, 0, 0, 0, 0, 0, 0, 0, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0, 0, 0, 0, 0, 0

Offset: 0

David Applegate and N. J. A. Sloane, Dec 07 2008

Although this entry is the last of the versions of the underlying triangle to be added to the OEIS, for some applications it is the most important.
Row n has 2n+1 entries.
A001498 has a b-file.

			Triangle begins:
  1
  0 1 1
  0 0 1 3 3
  0 0 0 1 6 15 15
  0 0 0 0 1 10 45 105 105
  0 0 0 0 0  1 15 105 420  945  945
  0 0 0 0 0  0  1  21 210 1260 4725 10395 10395
  ...

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Cf. A144299, A278990.
Row sums give A001515, column sums A000085.
Other versions of this triangle are given in A001497, A001498, A111924 and A100861.
See A144385 for a generalization.

a144331 n k = a144331_tabf !! n !! k
a144331_row n = a144331_tabf !! n
a144331_tabf = iterate (\xs ->
  zipWith (+) ([0] ++ xs ++ [0]) $ zipWith (*) (0:[0..]) ([0,0] ++ xs)) [1]
-- Reinhard Zumkeller, Nov 24 2014

A144331:= func< n,k | k le n-1 select 0 else Factorial(k)/(2^(k-n)*Factorial(k-n)*Factorial(2*n-k)) >;
[A144331(n,k): k in [0..2*n], n in [0..12]]; // G. C. Greubel, Oct 04 2023

Flatten[Table[PadLeft[Table[(n+k)!/(2^k*k!*(n-k)!), {k,0,n}], 2*n+1, 0], {n,0,12}]] (* Jean-François Alcover, Oct 14 2011 *)

def A144331(n, k): return 0 if kA144331(n,k) for k in range(2*n+1)] for n in range(13)]) # G. C. Greubel, Oct 04 2023

E.g.f.: Sum_{n >= 0} Sum_{k = 0..2n} b(n,k) y^n * x^k/k! = exp(x*y*(1 + x/2)).
b(n, k) = 2^(n-k)*k!/((2*n-k)!*(k-n)!).
Sum_{k=0..2*n} b(n, k) = A001515(n).
Sum_{n >= 0} b(n, k) = A000085(k).
From G. C. Greubel, Oct 04 2023: (Start)
T(n, k) = 0 for 0 <= k <= n-1, otherwise T(n, k) = k!/(2^(k-n)*(k-n)!*(2*n-k)!) for n <= k <= 2*n.
Sum_{k=0..2*n} (-1)^k * T(n, k) = A278990(n). (End)

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

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

A240440 Number of ways to place 3 points on a triangular grid of side n so that they are not vertices of an equilateral triangle of any orientation.

Original entry on oeis.org

0, 0, 15, 105, 420, 1260, 3150, 6930, 13860, 25740, 45045, 75075, 120120, 185640, 278460, 406980, 581400, 813960, 1119195, 1514205, 2018940, 2656500, 3453450, 4440150, 5651100, 7125300, 8906625, 11044215, 13592880, 16613520, 20173560, 24347400, 29216880

Offset: 1

Heinrich Ludwig, Apr 08 2014

a(n) = 15 * A000579(n+3).

a(n) = A001498(n,3), the fourth column of coefficients of Bessel polynomials. - Ran Pan, Dec 03 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. (2010) Vol. 13, Issue 4, Art. No. 10.4.4. See p=5 in the last equation on page 3.
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000579, A001498, A240441, A240442, A240439.

If one of the initial zeros is omitted, this is a row of the array in A129533.

[(n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48 : n in [1..50]]; // Wesley Ivan Hurt, Dec 03 2015

A240440:=n->(n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48; seq(A240440(n), n=1..50); # Wesley Ivan Hurt, Apr 08 2014

Table[(n+3)(n+2)(n+1)n(n-1)(n-2)/48, {n, 50}] (* Wesley Ivan Hurt, Apr 08 2014 *)
CoefficientList[Series[15 x^2/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 19 2014 *)

Vec(15*x^3/(1-x)^7 + O(x^100)) \\ Colin Barker, Apr 18 2014

vector(100,n,(n^2-1)*(n^2-4)*(n+3)*n/48) \\ Derek Orr, Dec 24 2015

a(n) = (n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48.
G.f.: 15*x^3 / (1-x)^7. - Colin Barker, Apr 18 2014
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7) for n>7. - Wesley Ivan Hurt, Dec 03 2015

A365673 Array A(n, k) read by ascending antidiagonals. Polygonal number weighted generalized Catalan sequences.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 15, 8, 1, 1, 1, 5, 34, 105, 16, 1, 1, 1, 6, 61, 496, 945, 32, 1, 1, 1, 7, 96, 1385, 11056, 10395, 64, 1, 1, 1, 8, 139, 2976, 50521, 349504, 135135, 128, 1, 1, 1, 9, 190, 5473, 151416, 2702765, 14873104, 2027025, 256, 1

Offset: 0

Peter Luschny, Sep 30 2023

Using polygonal numbers as weights, a recursion for triangles is defined, whose main diagonals represents a family of sequences, which include, among others, the powers of 2, the double factorial of odd numbers, the reduced tangent numbers, and the Euler numbers.
Apart from the edge cases k = 0 and k = n the recursion is T(n, k) = w(n, k) * T(n, k - 1) + T(n - 1, k). T(n, 0) = 1 and T(n, n) = T(n, n-1) if n > 0.
The weights w(n, k) identical to 1 yield the recursion of the Catalan triangle A009766 (with main diagonal the Catalan numbers). Here the polygonal numbers are used as weights in the form w(n, k) = p(s, n - k + 1), where the parameter s is the number of sides of the polygon and p(s, n) = ((s-2) * n^2 - (s-4) * n) / 2, see A317302.

			Array A(n, k) starts:                            (polygon|diagonal|triangle)
[0] 1, 1, 1,   1,     1,       1,         1, ...  A258837  A000012
[1] 1, 1, 2,   4,     8,      16,        32, ...  A080956  A011782
[2] 1, 1, 3,  15,   105,     945,     10395, ...  A001477  A001147  A001498
[3] 1, 1, 4,  34,   496,   11056,    349504, ...  A000217  A002105  A365674
[4] 1, 1, 5,  61,  1385,   50521,   2702765, ...  A000290  A000364  A060058
[5] 1, 1, 6,  96,  2976,  151416,  11449296, ...  A000326  A126151  A366138
[6] 1, 1, 7, 139,  5473,  357721,  34988647, ...  A000384  A126156  A365672
[7] 1, 1, 8, 190,  9080,  725320,  87067520, ...  A000566  A366150  A366149
[8] 1, 1, 9, 249, 14001, 1322001, 188106489, ...  A000567
           A054556                         A366137

Wikipedia, Polygonal number.

Poly weights: A258837, A080956, A001477, A000217, A000290, A000326, A000384.
Rows: A000012, A011782, A001147, A002105, A000364, A126151, A126156, A366150.
Triangles: A001498, A365674, A060058, A366138, A365672, A366149.
Cf. A009766, A366137 (central diagonal), A317302 (table of polygonal numbers).
Cf. A112934, A303943, A305532, A305533.

poly := (s, n) -> ((s - 2) * n^2 - (s - 4) * n) / 2:
T := proc(s, n, k) option remember; if k = 0 then 1 else if k = n then T(s, n, k-1) else poly(s, n - k + 1) * T(s, n, k - 1) + T(s, n - 1, k) fi fi end:
for n from 0 to 8 do A := (n, k) -> T(n, k, k): seq(A(n, k), k = 0..9) od;
# Alternative, using continued fractions:
A := proc(p, L) local CF, poly, k, m, P, ser;
   poly := (s, n) -> ((s - 2)*n^2 - (s - 4)*n)/2;
   CF := 1 + x;
   for k from 1 to L do
       m := L - k + 1;
       P := poly(p, m);
       CF := 1/(1 - P*x*CF)
   od;
   ser := series(CF, x, L);
   seq(coeff(ser, x, m), m = 0..L-1)
end:
for p from 0 to 8 do lprint(A(p, 8)) od;

poly[s_, n_] := ((s - 2) * n^2 - (s - 4) * n) / 2;
T[s_, n_, k_] := T[s, n, k] = If[k == 0, 1, If[k == n, T[s, n, k - 1], poly[s, n - k + 1] * T[s, n, k - 1] + T[s, n - 1, k]]];
A[n_, k_] := T[n, k, k];
Table[A[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 27 2023, from first Maple program *)

A(p, n) = {
       my(CF = 1 + x,
           poly(s, n) = ((s - 2)*n^2 - (s - 4)*n)/2,
           m, P
       );
       for(k = 1, n,
           m = n - k + 1;
           P = poly(p, m);
           CF = 1/(1 - P*x*CF)
        );
        Vec(CF + O(x^(n)))
}
for(p = 0, 8, print(A(p, 8)))
\\  Michel Marcus and Peter Luschny, Oct 02 2023

from functools import cache
@cache
def T(s, n, k):
    if k == 0: return 1
    if k == n: return T(s, n, k - 1)
    p = (n - k + 1) * ((s - 2) * (n - k + 1) - (s - 4)) // 2
    return p * T(s, n, k - 1) + T(s, n - 1, k)
def A(n, k): return T(n, k, k)
for n in range(9): print([A(n, k) for k in range(9)])

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

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A049403 A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).

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A144299 Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,0) for n >= 0.

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A043301 a(n) = 2^n*Sum_{k=0..n} (n+k)!/((n-k)!*k!*4^k).

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

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A144331 Triangle b(n,k) for n >= 0, 0 <= k <= 2n, read by rows. See A144299 for definition and properties.

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A043301 a(n) = 2^nSum_{k=0..n} (n+k)!/((n-k)!k!*4^k).