A144301 -id:A144301 - cp's OEIS frontend

A144647 Second differences of A001515 (or A144301).

1, 4, 25, 199, 1936, 22411, 301939, 4649800, 80654599, 1556992441, 33120019516, 769887934729, 19419368959225, 528311452144204, 15421347559288441, 480784227676809991, 15945180393017896024, 560549114426134288675

Offset: 0

N. J. A. Sloane, Jan 26 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A001515, A144301, A144498.

[n le 2 select 4^(n-1) else ( ((2*n-3)*(4*n^2-12*n+13))*Self(n-1) + (4*n^2-8*n+7)*Self(n-2) )/(4*n^2-16*n+19): n in [1..30]]; // G. C. Greubel, Sep 28 2023

A001515 := proc(n) simplify(hypergeom([n+1,-n],[],-1/2)) ; end: A144647 := proc(n) if n =0 then A001515(n) ; else A001515(n+1)-2*A001515(n)+A001515(n-1) ; fi; end: seq(A144647(n),n=0..30) ; # R. J. Mathar, Feb 01 2009

Join[{1},Differences[RecurrenceTable[{a[0]==1,a[1]==2,a[n]== (2n-1)a[n-1]+ a[n-2]},a[n],{n,25}],2]] (* Harvey P. Dale, Jun 18 2011 *)

def A144647_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (-1+4*x+2*(1-x)*sqrt(1-2*x))*exp(1-sqrt(1-2*x))/(sqrt(1-2*x))^3 ).egf_to_ogf().list()
A144647_list(40) # G. C. Greubel, Sep 28 2023

G.f.: (1-x)^2/(x*Q(0)) + 1 - 1/x, where Q(k)= 1 - x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013
G.f.: T(0)*(1-x)/x + 1 - 1/x, where T(k) = 1 - x*(k+1)/( x*(k+1) - (1-x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2013
From G. C. Greubel, Sep 28 2023: (Start)
a(n) = A001515(n+1) - 2*A001515(n) + A001515(n).
a(n) = 2*A001515(n+1) - (2*n+3)*A001515(n).
a(n) = ( ((2*n-1)*(4*n^2 - 4*n + 5))*a(n-1) + (4*n^2 + 3)*a(n-2) )/(4*n^2 - 8*n + 7), with a(0) = 1, a(1) = 4.
E.g.f.: (-1 + 4*x + 2*(1-x)*sqrt(1-2*x))*exp(1-sqrt(1-2*x))/(sqrt(1-2*x))^3. (End)

More terms from R. J. Mathar, Feb 01 2009

A001515 Bessel polynomial y_n(x) evaluated at x=1.

Original entry on oeis.org

1, 2, 7, 37, 266, 2431, 27007, 353522, 5329837, 90960751, 1733584106, 36496226977, 841146804577, 21065166341402, 569600638022431, 16539483668991901, 513293594376771362, 16955228098102446847, 593946277027962411007, 21992967478132711654106, 858319677924203716921141

Offset: 0

N. J. A. Sloane

For some applications it is better to start this sequence with an extra 1 at the beginning: 1, 1, 2, 37, 266, 2431, 27007, 353522, 5329837, ... (again with offset 0). This sequence now has its own entry - see A144301.
Number of partitions of {1,...,k}, n <= k <= 2n, into n blocks with no more than 2 elements per block. Restated, number of ways to use the elements of {1,...,k}, n <= k <= 2n, once each to form a collection of n sets, each having 1 or 2 elements. - Bob Proctor, Apr 18 2005, Jun 26 2006. E.g., for n=2 we get: (k=2): {1,2}; (k=3): {1,23}, {2,13}, {3,12}; (k=4): {12,34}, {13,24}, {14,23}, for a total of a(2) = 7 partitions.
Equivalently, number of sequences of n unlabeled items such that each item occurs just once or twice (cf. A105749). - David Applegate, Dec 08 2008
Numerator of (n+1)-th convergent to 1+tanh(1). - Benoit Cloitre, Dec 20 2002
The following Maple lines show how this sequence and A144505, A144498, A001514, A144513, A144506, A144514, A144507, A144301 are related.
f0:=proc(n) local k; add((n+k)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f0(n),n=0..10)];
# that is this sequence
f1:=proc(n) local k; add((n+k+1)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f1(n),n=0..10)];
# that is A144498
f2:=proc(n) local k; add((n+k+2)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f2(n),n=0..10)];
# that is A144513; divided by 2 gives A001514
f3:=proc(n) local k; add((n+k+3)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f3(n),n=0..10)];
# that is A144514; divided by 6 gives A144506
f4:=proc(n) local k; add((n+k+4)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f4(n),n=0..10)];
# that divided by 24 gives A144507
a(n) is also the numerator of the continued fraction sequence beginning with 2 followed by 3 and the remaining odd numbers: [2,3,5,7,9,11,13,...]. - Gil Broussard, Oct 07 2009
Also, number of scenarios in the Gift Exchange Game when a gift can be stolen at most once. - N. J. A. Sloane, Jan 25 2017

			The first few Bessel polynomials are (cf. A001497, A001498):
  y_0 = 1
  y_1 = 1 +   x
  y_2 = 1 + 3*x +  3*x^2
  y_3 = 1 + 6*x + 15*x^2 + 15*x^3, etc.
G.f. = 1 + 2*x + 7*x^2 + 37*x^3 + 266*x^4 + 2431*x^5 + 27007*x^6 + 353522*x^7 + ...

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).

Seiichi Manyama, Table of n, a(n) for n = 0..404 (first 101 terms from T. D. Noe)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix I to "Analysis of the gift exchange problem", giving Type D recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix II to "Analysis of the gift exchange problem", giving Type C recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
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.
P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, Boson normal ordering via substitutions and Sheffer-type polynomials, arXiv:quant-ph/0501155, 2005.
Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2019.
Andrew Francis and Michael Hendriksen, Counting spinal phylogenetic networks, arXiv:2502.14223 [q-bio.PE], 2025. See p. 11.
O. Frink and H. L. Krall, A new class of orthogonal polynomials, Trans. Amer. Math. Soc. 65,100-115, 1945. [From _Roger L. Bagula_, Feb 15 2009]
E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From _N. J. A. Sloane_, Sep 15 2012
Wojciech Mlotkowski and Anna Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
Robert 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, Notes 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
Index entries for related partition-counting sequences

Cf. A000108, A000806, A001514, A105749, A143990, A144505, A144506, A144507, A144513, A144514.
See A144301 for other formulas and comments.
Row sums of Bessel triangle A001497 as well as of A001498.
Partial sums: A105748.
First differences: A144498.
Replace "sets" with "lists" in comment: A001517.
The gift scenarios sequences when a gift can be stolen at most s times, for s = 1..9, are this sequence, A144416, A144508, A144509, A149187, A281358, A281359, A281360, A281361.

a001515 = sum . a001497_row -- Reinhard Zumkeller, Nov 24 2014

[(&+[Binomial(n+j, 2*j)*Catalan(j)*Factorial(j+1)/2^j: j in [0..n]]): n in [0..30]]; // G. C. Greubel, Sep 26 2023

A001515 := proc(n) option remember; if n=0 then 1 elif n=1 then 2 else (2*n-1)*A001515(n-1)+A001515(n-2); fi; end;
A001515:=proc(n) local k; add( (n+k)!/((n-k)!*k!*2^k),k=0..n); end;
A001515:= n-> hypergeom( [n+1,-n],[],-1/2);
bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end;

RecurrenceTable[{a[0]==1,a[1]==2,a[n]==(2n-1)a[n-1]+a[n-2]},a[n], {n,25}] (* Harvey P. Dale, Jun 18 2011 *)
Table[Sum[BellY[n+1, k, (2 Range[n+1] - 3)!!], {k, n+1}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

{a(n) = if( n<0, n = -1 - n); sum( k=0, n, (2*n - k)! / (k! * (n-k)!) * 2^(k-n))} /* Michael Somos, Apr 08 2012 */

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

The following formulas can all be found in (or are easily derived from formulas in) Grosswald's book.
D-finite with recurrence: a(0) = 1, a(1) = 2; thereafter a(n) = (2*n-1)*a(n-1) + a(n-2).
E.g.f.: exp(1-sqrt(1-2*x))/sqrt(1-2*x).
a(n) = Sum_{ k = 0..n } binomial(n+k,2*k)*(2*k)!/(k!*2^k).
Equivalently, a(n) = Sum_{ k = 0..n } (n+k)!/((n-k)!*k!*2^k) = Sum_{ k = n..2n } k!/((2n-k)!*(k-n)!*2^(k-n)).
a(n) = Hypergeometric2F0( [n+1, -n] ; - ; -1/2).
a(n) = A105749(n)/n!.
a(n) ~ exp(1)*(2n)!/(n!*2^n) as n -> oo. [See Grosswald, p. 124]
a(n) = A144301(n+1).
G.f.: 1/(1-x-x/(1-x-2*x/(1-x-3*x/(1-x-4*x/(1-x-5*x/(1-.... (continued fraction). - Paul Barry, Feb 08 2009
From Michael Somos, Apr 08 2012: (Start)
a(-1 - n) = a(n).
(a(n+1) + a(n+2))^2 = a(n)*a(n+2) + a(n+1)*a(n+3) for all integer n. (End)
G.f.: 1/G(0) where G(k) = 1 - x - x*(2*k+1)/(1 - x - 2*x*(k+1)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 05 2012
E.g.f.: E(0)/(2*sqrt(1-2*x)), where E(k) = 1 + 1/(1 - 2*x/(2*x + (k+1)*(1+sqrt(1-2*x))/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 23 2013
G.f.: T(0)/(1-x), where T(k) = 1 - (k+1)*x/((k+1)*x - (1-x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2013
a(n) = (2*BesselI(1/2, 1)+BesselI(3/2, 1))*BesselK(n+1/2, 1). - Jean-François Alcover, Feb 03 2014
a(n) = exp(1)*sqrt(2/Pi)*BesselK(1/2+n,1). - Gerry Martens, Jul 22 2015
From Peter Bala, Apr 14 2017: (Start)
a(n) = (1/n!)*Integral_{x = 0..inf} exp(-x)*x^n*(1 + x/2)^n dx.
E.g.f.: d/dx( exp(x*c(x/2)) ) = 1 + 2*x + 7*x^2/2! + 37*x^3/3! + ..., where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
From G. C. Greubel, Aug 16 2017: (Start)
a(n) = (1/2)_{n} * 2^n * hypergeometric1f1(-n; -2*n; 2).
G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; 2*t/(1-t)^2). (End)

Extensively edited by N. J. A. Sloane, Dec 07 2008

A144498 Column 2 of array in A144502.

Original entry on oeis.org

1, 5, 30, 229, 2165, 24576, 326515, 4976315, 85630914, 1642623355, 34762642871, 804650577600, 20224019536825, 548535471681029, 15969883030969470, 496754110707779461, 16441934503725675485, 576991048929859964160, 21399021201104749243099, 836326710446071005267035

Offset: 0

David Applegate and N. J. A. Sloane, Dec 13 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

First differences of A001515 and A144301.

A144498:= func< n | (&+[Binomial(n,k)*Factorial(n+k+1)/(2^k*Factorial(n)): k in [0..n]]) >;
[A144498(n): n in [0..30]]; // G. C. Greubel, Oct 07 2023

f1:=proc(n) local k; add((n+k+1)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f1(n),n=0..60)];

Table[Sum[(n+k+1)!/((n-k)!*k!*2^k), {k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2013 *)

def A144498(n): return sum(binomial(n,k)*rising_factorial(n+1,k+1)//2^k for k in range(n+1))
[A144498(n) for n in range(31)] # G. C. Greubel, Oct 07 2023

a(n) = A001515(n+1) - A001515(n).
a(n) = A144301(n+2) - A144301(n+1).
E.g.f.: (1 - 2*x + 2*x*sqrt(1-2*x))*exp(1-sqrt(1-2*x))/(1-2*x)^2. - Sergei N. Gladkovskii, Oct 06 2012
G.f.: (1-x)/(x*Q(0)) - 1/x, where Q(k)= 1 - x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013
a(n) ~ 2^(n+3/2) * n^(n+1) / exp(n-1). - Vaclav Kotesovec, Oct 08 2013
G.f.: T(0)/x- 1/x, where T(k) = 1 - (k+1)*x/((k+1)*x - (1-x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2013
(2*n-1)*a(n) = (4*n^2 + 1)*a(n-1) + (2*n+1)*a(n-2). - G. C. Greubel, Oct 07 2023

A144502 Square array read by antidiagonals upwards: T(n,k) is the number of scenarios for the gift exchange problem in which each gift can be stolen at most once, when there are n gifts in the pool and k gifts (not yet frozen) in peoples' hands.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 7, 7, 5, 1, 37, 37, 30, 16, 1, 266, 266, 229, 155, 65, 1, 2431, 2431, 2165, 1633, 946, 326, 1, 27007, 27007, 24576, 19714, 13219, 6687, 1957, 1, 353522, 353522, 326515, 272501, 198773, 119917, 53822, 13700, 1, 5329837, 5329837, 4976315, 4269271, 3289726, 2199722, 1205857, 486355, 109601, 1

Offset: 0

David Applegate and N. J. A. Sloane, Dec 13 2008

			The array, A(n,k), begins:
    1,    1,     1,      1,       1,        1, ...
    1,    2,     5,     16,      65,      326, ...
    2,    7,    30,    155,     946,     6687, ...
    7,   37,   229,   1633,   13219,   119917, ...
   37,  266,  2165,  19714,  198773,  2199722, ...
  266, 2431, 24576, 272501, 3289726, 42965211, ...
  ...
Antidiagonal triangle, T(n,k), begins as:
      1;
      1,     1;
      2,     2,     1;
      7,     7,     5,     1;
     37,    37,    30,    16,     1;
    266,   266,   229,   155,    65,    1;
   2431,  2431,  2165,  1633,   946,  326,    1;
  27007, 27007, 24576, 19714, 13219, 6687, 1957,   1;

G. C. Greubel, Antidiagonals n = 0..50, 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 [math.CO], 2009.

Rows include A000522, A144495, A144496, A144497.
Columns include A144301, A001515, A144498, A144499, A144500.
Main diagonal is A144501.
Antidiagonal sums give A144503.

A144301:= func< n | (&+[ Binomial(n+k-1,2*k)*Factorial(2*k)/( Factorial(k)*2^k): k in [0..n]]) >;
function A(n,k)
  if n eq 0 then return 1;
  elif k eq 0 then return A144301(n);
  elif k eq 1 then return A144301(n+1);
  else return A(n-1,k+1) + k*A(n,k-1);
  end if;
end function;
A144502:= func< n,k | A(n-k, k) >;
[A144502(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 29 2023

B:=proc(p,r) option remember;
if p=0 then RETURN(1); fi;
if r=0 then RETURN(B(p-1,1)); fi;
B(p-1,r+1)+r*B(p,r-1); end;
seq(seq(B(d-k, k), k=0..d), d=0..9);

t[0, ]= 1; t[n, 0]:= t[n, 0]= t[n-1, 1];
t[n_, k_]:= t[n, k]= t[n-1, k+1] + k*t[n, k-1];
Table[t[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Jan 14 2014, after Maple *)

def A144301(n): return 1 if n<2 else (2*n-3)*A144301(n-1)+A144301(n-2)
@CachedFunction
def A(n,k):
    if n==0: return 1
    elif k==0: return A144301(n)
    elif k==1: return A144301(n+1)
    else: return A(n-1,k+1) + k*A(n,k-1)
def A144502(n,k): return A(n-k,k)
flatten([[A144502(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 29 2023

Let A_n(x) be the e.g.f. for row n. Then A_0(x) = exp(x) and for n >= 1, A_n(x) = (d/dx)A_{n-1}(x)/(1-x).

For n >= 1, the rows A_{n}(x) = P_{n}(x)*exp(x)/(1-x)^(2*n), where P_{n}(x) = (1-x)*(d/dx)( P_{n-1}(x) ) + (2*n-x)*P_{n-1}(x) and P_{0}(x) = 1. - G. C. Greubel, Oct 08 2023

6 more terms from Michel Marcus, Feb 01 2023

A144505 Triangle read by rows: coefficients of polynomials arising from the recurrence A[n](x) = A[n-1]'(x)/(1-x) with A[0] = exp(x).

Original entry on oeis.org

1, 1, -1, 2, 1, -5, 7, -1, 9, -30, 37, 1, -14, 81, -229, 266, -1, 20, -175, 835, -2165, 2431, 1, -27, 330, -2330, 9990, -24576, 27007, -1, 35, -567, 5495, -34300, 137466, -326515, 353522, 1, -44, 910, -11522, 97405, -561386, 2148139, -4976315, 5329837

Offset: 0

N. J. A. Sloane, Dec 14 2008

			The first few polynomials P[n] (n >= 0) are:
  P[0] = 1;
  P[1] = 1;
  P[2] = -x +2;
  P[3] =  x^2 -5*x +7;
  P[4] = -x^3 + 9*x^2 - 30*x +37;
  P[5] =  x^4 -14*x^3 + 81*x^2 - 229*x +266;
  P[6] = -x^5 +20*x^4 -175*x^3 + 835*x^2 -2165*x +2431;
  P[7] =  x^6 -27*x^5 +330*x^4 -2330*x^3 +9990*x^2 -24576*x +27007;
...
Triangle of coefficients begins:
   1;
   1;
  -1,   2;
   1,  -5,    7;
  -1,   9,  -30,     37;
   1, -14,   81,   -229,    266;
  -1,  20, -175,    835,  -2165,    2431;
   1, -27,  330,  -2330,   9990,  -24576,   27007;
  -1,  35, -567,   5495, -34300,  137466, -326515,   353522;
   1, -44,  910, -11522,  97405, -561386, 2148139, -4976315, 5329837;
...

Seiichi Manyama, Rows n = 0..140, flattened
Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023). See pp. 42, 63.
N. J. A. Sloane, Rows 0 through 25 of the triangle, together with the corresponding polynomials P[n](x).

Columns give A001515 (really A144301), A144498, A001514, A144506, A144507.
Row sums give A001147.
Alternating row sums give A043301.

R:=PowerSeriesRing(Integers(), 50);
f:= func< n,x | x^n*(&+[Binomial(n,j)*Factorial(n+j)*(1-1/x)^(n-j)/(2^j*Factorial(n)) : j in [0..n]]) >;
T:= func< n,k | Coefficient(R!( f(n,x) ), k) >;
[1] cat [T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 02 2023

A[0]:=exp(x);
P[0]:=1;
for n from 1 to 12 do
A[n]:=sort(simplify( diff(A[n-1],x)/(1-x)));
P[n]:=sort(simplify(A[n]*(1-x)^(2*n-1)/exp(x)));
t1:=simplify(x^(degree(P[n],x))*subs(x=1/x,P[n]));
t2:=series(t1,x,2*n+3);
lprint(P[n]);
lprint(seriestolist(t2));
od:

f[n_, x_]:= x^n*Sum[((n+j)!/((n-j)!*j!*2^j))*(1-1/x)^(n-j), {j,0,n}];
t[n_, k_]:= Coefficient[Series[f[n,x], {x,0,30}], x, k];
Join[{1}, Table[t[n,k], {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Oct 02 2023 *)

P. = PowerSeriesRing(QQ, 50)
def f(n,x): return x^n*sum(binomial(n,j)*rising_factorial(n+1,j)*(1-1/x)^(n-j)/2^j for j in range(n+1))
def T(n,k): return P( f(n,x) ).list()[k]
[1] + flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 02 2023

Let A[0](x) = exp(x), A[n](x) = A[n-1]'(x)/(1-x) for n>0 and let P[n](x) = A[n](x)*(1-x)^(2n-1)/exp(x). Row n of triangle gives coefficients of P[n](x) with exponents of x in decreasing order.
From Vladeta Jovovic, Dec 15 2008: (Start)
P[n] = Sum_{k=0..n} ((n+k)!/((n-k)!*k!*2^k)) * (1-x)^(n-k).
E.g.f.: exp((1-x)*(1-sqrt(1-2*y)))/sqrt(1-2*y). (End)

A122850 Exponential Riordan array (1, sqrt(1+2x)-1).

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 3, -3, 1, 0, -15, 15, -6, 1, 0, 105, -105, 45, -10, 1, 0, -945, 945, -420, 105, -15, 1, 0, 10395, -10395, 4725, -1260, 210, -21, 1, 0, -135135, 135135, -62370, 17325, -3150, 378, -28, 1, 0, 2027025, -2027025, 945945, -270270, 51975, -6930, 630, -36, 1

Offset: 0

Paul Barry, Sep 14 2006

Inverse of number triangle A122848. Entries are Bessel polynomial coefficients. Row sums are A000806.
Also the inverse Bell transform of the sequence "g(n) = 1 if n<2 else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016
Also called Bessel numbers of first kind, and denoted b(n,k). - Abdelhay Benmoussa, Aug 30 2025

			Triangle begins
  1
  0 1
  0 -1 1
  0 3 -3 1
  0 -15 15 -6 1
  0 105 -105 45 -10 1
  0 -945 945 -420 105 -15 1
  0 10395 -10395 4725 -1260 210 -21 1
  0 -135135 135135 -62370 17325 -3150 378 -28 1
  0 2027025 -2027025 945945 -270270 51975 -6930 630 -36 1
  0 -34459425 34459425 -16216200 4729725 -945945 135135 -13860 990 -45 1
  ...

Peter Bala, The white diamond product of power series
Orli Herscovici, Study of the p,q-deformed Touchard polynomials, arXiv:1904.07674 [math.CO], 2019.
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Wikipedia, Bessel polynomials
S. Willerton, The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials, arXiv:1708.03227v1 [math.MG], 2017.

Cf. A000806, A001497, A008277, A039757, A122848, A132062, A144301.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> (-1)^n*doublefactorial(2*n-1), 9); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[Function[n, (-1)^n (2n-1)!!], rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 26 2018, after Peter Luschny *)

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 1 if n<2 else 0, 12).inverse() # Peter Luschny, Jan 19 2016

T(n,k) = (-1)^(n-k)*A132062(n,k). - Philippe Deléham, Nov 06 2011
Triangle equals the matrix product A039757*A008277. Dobinski-type formula for the row polynomials: R(n,x) = x*exp(-x)*Sum_{k = 0..inf} (k-1)*(k-3)*(k-5)*...*(k-(2*n-3))*x^k/k! for n >= 1. Cf. A001497. - Peter Bala, Jun 23 2014
From Peter Bala, Jan 09 2018: (Start)
Alternative Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k = 0..inf} k*(k-2)*(k-4)*...*(k-(2*n-2))*x^k/k!.
Equivalently, R(n,x) = x o (x-2) o (x-4) o...o (x-(2*n-2)), where o denotes the white diamond product of polynomials. See the Bala link for the definition and details.
The white diamond products (x-1) o (x-3) o...o (x-(2*n-3)) give the row polynomials of the array with a factor of x removed.
If d is the first derivative operator f -> d/dx(f(x)) and D is the operator f(x) -> 1/x*d/dx(f(x)) then x^(2*n)*D^n = R(n,x*d), with the understanding that (x*d)^k is to interpreted as the operator f(x) -> x^k*d^k(f(x))/dx^k. (End)
Sum_{k=0..n} (-1)^(n+k) * T(n,k) = A144301(n). - Alois P. Heinz, Aug 31 2022

More terms from Alois P. Heinz, Aug 31 2022

cp's OEIS Frontend

A144647 Second differences of A001515 (or A144301).

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A001515 Bessel polynomial y_n(x) evaluated at x=1.

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A144498 Column 2 of array in A144502.

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A144502 Square array read by antidiagonals upwards: T(n,k) is the number of scenarios for the gift exchange problem in which each gift can be stolen at most once, when there are n gifts in the pool and k gifts (not yet frozen) in peoples' hands.

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A144505 Triangle read by rows: coefficients of polynomials arising from the recurrence A[n](x) = A[n-1]'(x)/(1-x) with A[0] = exp(x).

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A122850 Exponential Riordan array (1, sqrt(1+2x)-1).

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A305536 Expansion of 1/(1 - x/(1 - x - 1x/(1 - x - 2x/(1 - x - 3x/(1 - x - 4x/(1 - ...)))))), a continued fraction.