A144498 -id:A144498 - cp's OEIS frontend

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

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

A144301 a(0) = a(1) = 1; thereafter a(n) = (2n-3)a(n-1) + a(n-2).

Original entry on oeis.org

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

Offset: 0

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

A variant of A001515, which is the main entry.

a(n) = number of increasing ordered trees on the vertex set [0,n] (counted by the double factorials A001147) in which n is the label on the leaf that terminates the leftmost path from the root. - David Callan, Aug 24 2011

			G.f. = 1 + x + 2*x^2 + 7*x^3 + 37*x^4 + 266*x^5 + 2431*x^6 + 27007*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..400
E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.
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
W. Mlotkowski and A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.

See A001515 for much more about this sequence.

See A144498 for first differences.

[1] cat [n le 1 select n+1 else (2*n-1)*Self(n) + Self(n-1): n in [0..20]]; // Vincenzo Librandi, Jul 23 2015

a[n_]:= HypergeometricPFQ[{n, 1 - n}, {}, -1/2]; (* Michael Somos, Nov 22 2013 *)
a[n_]:= With[{m= If[n<1, -n, n-1]}, Sum[(m+k)!/((m-k)! k! 2^k), {k,0,m}]]; (* Michael Somos, Nov 22 2013 *)
RecurrenceTable[{a[0]==a[1]==1, a[n]==(2*n-3)*a[n-1] +a[n-2]}, a, {n, 25}] (* Vincenzo Librandi, Jul 23 2015 *)
nxt[{n_,a_,b_}]:={n+1,b,b(2n-1)+a}; NestList[nxt,{1,1,1},30][[All,2]] (* Harvey P. Dale, Nov 29 2022 *)

{a(n) = my(m = if( n<1, -n, n-1)); sum( k=0, m,  (m+k)! / (k! * (m-k)! * 2^k))}; /* Michael Somos, Nov 22 2013 */

def A144301(n): return int(n==0) + sum(binomial(n-1,k)*factorial(n+k-1)/(2^k*factorial(n-1)) for k in range(n))
[A144301(n) for n in range(31)] # G. C. Greubel, Sep 29 2023

a(n) = A001515(n-1) for n>= 1.
E.g.f.: A(x) = exp(1-sqrt(1-2*x)) satisfies A'(x) = A(x)/(1-sqrt(1-2*x)).
Hence a(n+1) = Sum_{k=0..n} ( a(n-k)*binomial(n,k)*(2*k)!/(k!*2^k) ).
A''(x) = (A'(x)/(1-2*x))*(1 + 1/sqrt(1-2*x)).
A''(x) = 2*x*A''(x) + A'(x) + A(x), which is equivalent to the recurrence in the definition.
a(n) = Sum_{k=0..n-1} binomial(n+k-1,2*k)*(2*k)!/(k!*2^k). [See Grosswald, p. 6, Eq. (8).]
a(n) ~ exp(1)*(2n-1)!/(n!*2^n) as n -> oo. [See Grosswald, p. 124]
From Sergei N. Gladkovskii, Oct 06 2012: (Start)
G.f.: 1+x/U(0) where U(k) = 1 - x - x*(2*k+1)/(1 - x - 2*x*(k+1)/U(k+1)); (continued fraction).
G.f.: 1+x*(1-x)/U(0) where U(k) = 1 - 3*x + x^2 - x*4*k - x^2*(2*k+1)*(2*k+2)/U(k+1) ; (continued fraction). (End)
E.g.f.: E(0)/2, 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.: conjecture: 1 + x*(1-x)/(1-3*x+x^2)*Q(0), where Q(k) = 1 - 2*(k+1)*(2*k+1)*x^2/(2*(k+1)*(2*k+1)*x^2 - (1 - 3*x + x^2 - 4*x*k)*(1 - 7*x + x^2 - 4*x*k)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 19 2013
a(1 - n) = a(n) for all n in Z. (a(n+1) + a(n+2))^2 = a(n)*a(n+2) + a(n+1)*a(n+3) for all integer n. - Michael Somos, Nov 22 2013
G.f.: 1 + x/(1-x)*T(0), where T(k) = 1 - x*(k+1)/( x*(k+1) - (1-x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2013

More terms from Vincenzo Librandi, Jul 23 2015

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)

A144495 Row 2 of array in A144502.

Original entry on oeis.org

2, 7, 30, 155, 946, 6687, 53822, 486355, 4877250, 53759351, 646098622, 8409146187, 117836551730, 1768850337295, 28318532194206, 481652022466307, 8673291031865602, 164849403644999655, 3297954931572397790, 69274457019123638011, 1524368720086682440242

Offset: 0

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

nonn

Robert Israel, Table of n, a(n) for n = 0..447

Cf. A000522, A001339, A038720, A144496, A144497, A144498.

Cf. A144499, A144500, A144501, A144502, A144503.

A144495:= func< n | (&+[Binomial(n,k)*(k+4)*Factorial(k+1) : k in [0..n]])/2 >;
[A144495(n): n in [0..40]]; // G. C. Greubel, Oct 07 2023

f:= rectoproc({a(n)=((4+3*n)*a(n-1)-(n+3)*(n-1)*a(n-2)+(n-1)*(n-2)*a(n-3))/2,a(0)=2,a(1)=7,a(2)=30},a(n),remember):
map(f, [$0..40]); # Robert Israel, Oct 02 2016

(* First program *)
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];
a[n_] := t[2, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 19 2022 *)
(* Second program *)
a[n_]:= a[n]= If[n==0, 2, (n*(n^2+3*n+1)*a[n-1] -(n+2))/(n^2+n-1)];
Table[a[n], {n,0,40}] (* G. C. Greubel, Oct 07 2023 *)

def A144495(n): return sum(binomial(n,j)*factorial(j+1)*(j+4) for j in range(n+1))/2
[A144495(n) for n in range(41)] # G. C. Greubel, Oct 07 2023

E.g.f.: exp(x)*(2-x)/(1-x)^3.
a(n) = (1/2) * (floor((n+1)*(n+1)!*e) + floor(n*n!*e)). [Gary Detlefs, Jun 06 2010]
a(n) = (1/2) * ( A001339(n) + A001339(n+1) ). [Gary Detlefs, Jun 06 2010]
a(n) = (1/2) * (3 + n + (1 + 3*n + n^2) * A000522(n)). - Gerry Martens, Oct 02 2016
a(n) = ((4+3*n)*a(n-1) - (n+3)*(n-1)*a(n-2) + (n-1)*(n-2)*a(n-3))/2. - Robert Israel, Oct 02 2016
From Peter Bala, May 27 2022: (Start)
a(n) = (1/2)*(A000522(n+2) - A000522(n)).
a(n) = (1/2)*Sum_{k = 0..n} binomial(n,k)*(k+4)*(k+1)!; binomial transform of A038720(n+1).
a(n) = (1/2)*e*Integral_{x >= 1} x^n*(x^2 - 1)*exp(-x).
a(2*n) is even and a(2*n+1) is odd. More generally, a(n+k) == a(n) (mod k) for all n and k. It follows that for each positive integer k, the sequence obtained by reducing a(n) modulo k is periodic, with the exact period dividing k. Various divisibility properties of the sequence follow from this; for example, a(3*n+2) == 0 (mod 3), a(5*n+2) == a(5*n+3) (mod 5), a(7*n+1) == 0 (mod 7) and a(11*n+4) == 0 (mod 11). (End)
a(n) = (n*(n^2 + 3*n + 1)*a(n-1) - (n + 2))/(n^2 + n - 1), with a(0) = 2. - G. C. Greubel, Oct 07 2023

A144496 Row 3 of array in A144502.

Original entry on oeis.org

7, 37, 229, 1633, 13219, 119917, 1205857, 13318249, 160305343, 2088846709, 29297613277, 440110297777, 7050173910619, 119970793032253, 2161243124917849, 41091937905633337, 822320410135133047, 17277401903869659589, 380267691288777510613, 8749454854573455141889

Offset: 0

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

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A144495, A144497, A144498, A144499, A144500, A144501, A144502, A144503.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (7-5*x+x^2)*Exp(x)/(1-x)^5 ))); // G. C. Greubel, Oct 07 2023

CoefficientList[Series[E^x*(7-5*x+x^2)/(1-x)^5, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)

def a(n): # a = A144496
    if (n==0): return 7
    else: return (n*(n^4+10*n^3+33*n^2+44*n+21)*a(n-1) + n^2+6*n+7)/(n^4+6*n^3+9*n^2+4*n+1)
[a(n) for n in range(41)] # G. C. Greubel, Oct 07 2023

E.g.f.: (7-5*x+x^2)*exp(x)/(1-x)^5.
a(n) ~ n! * n^4 * exp(1)/8. - Vaclav Kotesovec, Oct 08 2013
a(n) = (n*(n^4 + 10*n^3 + 33*n^2 + 44*n + 21)*a(n-1) + n^2 + 6*n +
   7)/(n^4 + 6*n^3 + 9*n^2 + 4*n + 1), with a(0) = 7. - G. C. Greubel, Oct 07 2023

A144497 Row 4 of array in A144502.

Original entry on oeis.org

37, 266, 2165, 19714, 198773, 2199722, 26516581, 345921410, 4856217989, 73003575178, 1170146049557, 19921780455746, 359032158501205, 6828661185433514, 136693194501702533, 2872718327660671042, 63240895146440396261, 1455362908778264247050, 34945987212582211588789

Offset: 0

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

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..444

Cf. A144495, A144496, A144498, A144499, A144500, A144501, A144502, A144503.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (37-30*x+9*x^2-x^3)*Exp(x)/(1-x)^7 ))); // G. C. Greubel, Oct 08 2023

a[n_]:= If[n<1, 37, (n*(n^6+21*n^5+172*n^4+705*n^3+1522*n^2+1623*n +653)*a[n-1] -(n^3+12*n^2+41*n+37))/(n^6+15*n^5+82*n^4+207*n^3 +244*n^2+105*n-1)];
Table[a[n], {n,0,40}] (* G. C. Greubel, Oct 08 2023 *)

my(x='x+O('x^25)); Vec(serlaplace(exp(x)*(37-30*x+9*x^2-x^3)/(1-x)^7)) \\ Michel Marcus, Apr 06 2019

def A144497_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (37-30*x+9*x^2-x^3)*exp(x)/(1-x)^7 ).egf_to_ogf().list()
A144497_list(40) # G. C. Greubel, Oct 08 2023

E.g.f.: exp(x)*(37-30*x+9*x^2-x^3)/(1-x)^7.

a(n) = (n*(n^6 + 21*n^5 + 172*n^4 + 705*n^3 + 1522*n^2 + 1623*n + 653)*a(n-1) - (n^3 + 12*n^2 + 41*n + 37))/(n^6 + 15*n^5 + 82*n^4 + 207*n^3 + 244*n^2 + 105*n - 1), with a(0) = 37. - G. C. Greubel, Oct 08 2023

cp's OEIS Frontend

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

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A144301 a(0) = a(1) = 1; thereafter a(n) = (2*n-3)*a(n-1) + a(n-2).

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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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A144495 Row 2 of array in A144502.

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A144496 Row 3 of array in A144502.

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A144301 a(0) = a(1) = 1; thereafter a(n) = (2n-3)a(n-1) + a(n-2).

A144513 a(n) = Sum_{k=0..n} (n+k+2)!/((n-k)!k!2^k).

A144514 a(n) = Sum_{k=0..n} (n+k+3)!/((n-k)!k!2^k).