A099597 -id:A099597 - cp's OEIS frontend

A099599 Triangle T read by rows: coefficients of polynomials generating array A099597.

1, 1, 1, 1, 0, 2, 1, 9, -12, 6, 1, -104, 204, -120, 24, 1, 2265, -4840, 3540, -1080, 120, 1, -71064, 164910, -138840, 54360, -10080, 720, 1, 3079825, -7626948, 7134330, -3300360, 808920, -100800, 5040, 1, -176449776, 460982648, -468313104, 244938960, -72266880, 12156480, -1088640, 40320

Offset: 0

Ralf Stephan, Oct 28 2004

Row sums are k (A000027), left edge columns are factorials (A000142). - Peter Bala, Aug 19 2013

			Row polynomials:
  1,
  x + 1,
  2*x^2 + 1,
  6*x^3 - 12*x^2 + 9*x + 1,
  24*x^4 - 120*x^3 + 204*x^2 - 104*x + 1,
  120*x^5 - 1080*x^4 + 3540*x^3 - 4840*x^2 + 2265*x + 1,
  ...

Cf. A000027, A000142, A099597, A132393.

# Define row polynomials R(n, x) recursively:
R := proc(n, x) option remember; if n = 0 then 1 elif n = 1 then 1+x
else (n*x+1)*procname(n-1,x-1) - (n-1)*(x-1)*procname(n-2, x-2) fi end:
Trow := n -> PolynomialTools:-CoefficientList(R(n,x), x);
seq(Trow(n), n = 0..10); # Peter Bala, Aug 19 2013

R[n_, x_] := R[n, x] = (n x + 1) R[n-1, x-1] - (n-1) (x-1) R[n-2, x-2]; R[0, ] = 1; R[1, x] = 1 + x;
Table[CoefficientList[R[n, x], x], {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 13 2019 *)
R[n_,y_] := HypergeometricPFQ[{1,-n,-y},{},1] (* Natalia L. Skirrow, Jul 18 2025 *)

from fractions import Fraction as frac
from math import factorial as fact
from sympy.functions.combinatorial.numbers import stirling
A099599=lambda n,k: sum(map(lambda i: frac(fact(n),fact(n-i))*(-1)**(i-k)*stirling(i,k,kind=1),range(k,n+1))) # Natalia L. Skirrow, Jul 18 2025

The row polynomials satisfy the second order recurrence equation R(n,x) = (n*x+1)*R(n-1,x-1) - (n-1)*(x-1)*R(n-2,x-2), with the initial conditions R(0,x) = 1 and R(1,x) = 1+x. - Peter Bala, Aug 19 2013
From Natalia L. Skirrow, Jul 18 2025: (Start)
T(n,k) = Sum_{i=k..n} (n!/(n-i)!) * A048994(i,k).
T(n,k) = n * ((n-1)*(T(n-2,k)-T(n-1,k)) + T(n-1,k-1)) for k>0 (without this criterion, defines a unique continuation to negative k).
O.g.f.: hypergeom([1,1,-y],[],x/(x-1)) / (1-x).
E.g.f.: e^x*hypergeom([1,-y],[],-x).
Row polynomials are:
R(n,y) = hypergeom([1,-n,-y],[],1).
R(n,y) = n * ((y-n+1)*R(n-1,y) + (n-1)*R(n-2,y)) + 1.
R(n,y) = 1 + n!*y!*I_{y-n}(2) - hypergeom([1],[n+1,y+1],1) where I is the Bessel function. (End)

A099598 Antidiagonal sums of array A099597.

Original entry on oeis.org

1, 2, 4, 8, 19, 50, 162, 576, 2469, 11218, 59984, 331432, 2120663, 13784290, 102616622, 766648128, 6507054537, 54919451490, 523342287196, 4923469621256, 52040350449307, 539901936037202, 6268123599148282

Offset: 0

Ralf Stephan, Oct 28 2004

nonn

a(n)=sum{k=0..n, C(n,k)*(floor(k/2)!)^2}; a(n)=sum{k=0..n, sum{j=0..n-k, C(n-k,j)C(k,j)j!^2}}; - Paul Barry, May 26 2007

A058331 a(n) = 2*n^2 + 1.

Original entry on oeis.org

1, 3, 9, 19, 33, 51, 73, 99, 129, 163, 201, 243, 289, 339, 393, 451, 513, 579, 649, 723, 801, 883, 969, 1059, 1153, 1251, 1353, 1459, 1569, 1683, 1801, 1923, 2049, 2179, 2313, 2451, 2593, 2739, 2889, 3043, 3201, 3363, 3529, 3699, 3873, 4051

Offset: 0

Erich Friedman, Dec 12 2000

Maximal number of regions in the plane that can be formed with n hyperbolas.
Also the number of different 2 X 2 determinants with integer entries from 0 to n.
Number of lattice points in an n-dimensional ball of radius sqrt(2). - David W. Wilson, May 03 2001
Equals A112295(unsigned) * [1, 2, 3, ...]. - Gary W. Adamson, Oct 07 2007
Binomial transform of A166926. - Gary W. Adamson, May 03 2008
a(n) = longest side a of all integer-sided triangles with sides a <= b <= c and inradius n >= 1. Triangle has sides (2n^2 + 1, 2n^2 + 2, 4n^2 + 1).
{a(k): 0 <= k < 3} = divisors of 9. - Reinhard Zumkeller, Jun 17 2009
Number of ways to partition a 3*n X 2 grid into 3 connected equal-area regions. - R. H. Hardin, Oct 31 2009
Let A be the Hessenberg matrix of order n defined by: A[1, j] = 1, A[i, i] := 2, (i > 1), A[i, i - 1] = -1, and A[i, j] = 0 otherwise. Then, for n >= 3, a(n - 1) = coeff(charpoly(A, x), x^(n - 2)). - Milan Janjic, Jan 26 2010
Except for the first term of [A002522] and [A058331] if X = [A058331], Y = [A087113], A = [A002522], we have, for all other terms, Pell's equation: [A058331]^2 - [A002522]*[A087113]^2 = 1; (X^2 - A*Y^2 = 1); e.g., 3^2 -2*2^2 = 1; 9^2 - 5*4^2 = 1; 129^2 - 65*16^2 = 1, and so on. - Vincenzo Librandi, Aug 07 2010
Niven (1961) gives this formula as an example of a formula that does not contain all odd integers, in contrast to 2n + 1 and 2n - 1. - Alonso del Arte, Dec 05 2012
Numbers m such that 2*m-2 is a square. - Vincenzo Librandi, Apr 10 2015
Number of n-tuples from the set {1,0,-1} where at most two elements are nonzero. - Michael Somos, Oct 19 2022
a(n) gives the x-value of the integral solution (x,y) of the Pellian equation x^2 - (n^2 + 1)*y^2 = 1. The y-value is given by 2*n (see Tattersall). - Stefano Spezia, Jul 23 2025

			a(1) = 3 since (0 0 / 0 0), (1 0 / 0 1) and (0 1 / 1 0) have different determinants.
G.f. = 1 + 3*x + 9*x^2 + 19*x^3 + 33*x^4 + 51*x^5 + 73*x^6 + ... - _Michael Somos_, Oct 19 2022

Ivan Niven, Numbers: Rational and Irrational, New York: Random House for Yale University (1961): 17.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 256.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Leo Tavares, Illustration: Triangular Outlines
Reinhard Zumkeller, Enumerations of Divisors.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124.
Second row of array A099597.
See A120062 for sequences related to integer-sided triangles with integer inradius n.
Cf. A112295.
Cf. A087113, A002552.
Cf. A005408, A016813, A086514, A000125, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261.
Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121.
Column 2 of array A188645.
Cf. A001105 and A247375. - Bruno Berselli, Sep 16 2014
Cf. A056106, A251599.
Cf. A000384, A000217, A166926.

a058331 = (+ 1) . a001105  -- Reinhard Zumkeller, Dec 13 2014

[2*n^2 + 1 : n in [0..100]]; // Wesley Ivan Hurt, Feb 02 2017

b[g_] := Length[Union[Map[Det, Flatten[ Table[{{i, j}, {k, l}}, {i, 0, g}, {j, 0, g}, {k, 0, g}, {l, 0, g}], 3]]]] Table[b[g], {g, 0, 20}]
2*Range[0, 49]^2 + 1 (* Alonso del Arte, Dec 05 2012 *)

a(n)=2*n^2+1 \\ Charles R Greathouse IV, Jun 16 2011

G.f.: (1 + 3x^2)/(1 - x)^3. - Paul Barry, Apr 06 2003
a(n) = M^n * [1 1 1], leftmost term, where M = the 3 X 3 matrix [1 1 1 / 0 1 4 / 0 0 1]. a(0) = 1, a(1) = 3; a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). E.g., a(4) = 33 since M^4 *[1 1 1] = [33 17 1]. - Gary W. Adamson, Nov 11 2004
a(n) = cosh(2*arccosh(n)). - Artur Jasinski, Feb 10 2010
a(n) = 4*n + a(n-1) - 2 for n > 0, a(0) = 1. - Vincenzo Librandi, Aug 07 2010
a(n) = (((n-1)^2 + n^2))/2 + (n^2 + (n+1)^2)/2. - J. M. Bergot, May 31 2012
a(n) = A251599(3*n) for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n) = sqrt(8*(A000217(n-1)^2 + A000217(n)^2) + 1). - J. M. Bergot, Sep 03 2015
E.g.f.: (2*x^2 + 2*x + 1)*exp(x). - G. C. Greubel, Jul 14 2017
a(n) = A002378(n) + A002061(n). - Bruce J. Nicholson, Aug 06 2017
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(2))*coth(Pi/sqrt(2)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(2))*csch(Pi/sqrt(2)))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(2))*sinh(Pi).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(2))*csch(Pi/sqrt(2)). (End)
From Leo Tavares, May 23 2022: (Start)
a(n) = A000384(n+1) - 3*n.
a(n) = 3*A000217(n) + A000217(n-2). (End)
a(n) = a(-n) for all n in Z and A037235(n) = Sum_{k=0..n-1} a(k). - Michael Somos, Oct 19 2022

Revised description from Noam Katz (noamkj(AT)hotmail.com), Jan 28 2001

A070910 Decimal expansion of BesselI(0,2).

Original entry on oeis.org

2, 2, 7, 9, 5, 8, 5, 3, 0, 2, 3, 3, 6, 0, 6, 7, 2, 6, 7, 4, 3, 7, 2, 0, 4, 4, 4, 0, 8, 1, 1, 5, 3, 3, 3, 5, 3, 2, 8, 5, 8, 4, 1, 1, 0, 2, 7, 8, 5, 4, 5, 9, 0, 5, 4, 0, 7, 0, 8, 3, 9, 7, 5, 1, 6, 6, 4, 3, 0, 5, 3, 4, 3, 2, 3, 2, 6, 7, 6, 3, 4, 2, 7, 2, 9, 5, 1, 7, 0, 8, 8, 5, 5, 6, 4, 8, 5, 8, 9, 8, 9, 8, 4, 5, 9

Offset: 1

Benoit Cloitre, May 20 2002

			2.2795853023360672674372044408115333532858411...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:5:5 at page 20.

Michael Penn, An exponential trigonometric integral., YouTube video, 2020.
Eric Weisstein's World of Mathematics, Factorial Sums.
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind.

Cf. A096789, A070913 (continued fraction), A006040.

Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2))), this sequence (I(0,2)).

RealDigits[ BesselI[0, 2], 10, 110] [[1]] (* Robert G. Wilson v, Jul 09 2004 *)
(* Or *) RealDigits[ Sum[ 1/(n!n!), {n, 0, Infinity}], 10, 110][[1]]

besseli(0,2) \\ Charles R Greathouse IV, Feb 19 2014

Equals Sum_{k>=0} 1/k!^2.
From Peter Bala, Aug 19 2013: (Start)
Continued fraction expansion: 1/(1 - 1/(2 - 1/(5 - 4/(10 - 9/(17 - ... - (n-1)^2/(n^2+1 - ...)))))). See A006040. Cf. A096789.
This continued fraction is the particular case k = 0 of the result BesselI(k,2) = Sum_{n = 0..oo} 1/(n!*(n+k)!) = 1/(k! - k!/((k+2) - (k+1)/((2*k+5) - 2*(k+2)/((3*k+10) - ... - n*(n+k)/(((n+1)*(n+k+1)+1) - ...))))). See the remarks in A099597 for a sketch of the proof. (End)
From Amiram Eldar, May 29 2021: (Start)
Equals (1/e^2) * Sum_{k>=0} binomial(2*k,k)/k! = e^2 * Sum_{k>=0} (-1)^k*binomial(2*k,k)/k!.
Equal (1/(2*Pi)) * Integral_{x=0..2*Pi} exp(2*sin(x)) dx. (End)
Equals BesselJ(0,2*i). - Jianing Song, Sep 18 2021

A006040 a(n) = Sum_{i=0..n} (n!/(n-i)!)^2.

Original entry on oeis.org

1, 2, 9, 82, 1313, 32826, 1181737, 57905114, 3705927297, 300180111058, 30018011105801, 3632179343801922, 523033825507476769, 88392716510763573962, 17324972436109660496553, 3898118798124673611724426, 997918412319916444601453057, 288398421160455852489819933474

Offset: 0

N. J. A. Sloane, Simon Plouffe

R. K. Guy, personal communication.
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..250
D. Deford, Seating rearrangements on arbitrary graphs, 2013. (See Table 1)
D. Deford, Seating rearrangements on arbitrary graphs, involve, Vol. 7 (2014), No. 6, 787-805. (See Table 1)
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
Index entries for sequences related to Bessel functions or polynomials

Main diagonal of array A099597.

Cf. A073701.

a[0]:= 1:
for n from 1 to 30 do a[n]:= n^2*a[n-1] + 1 od:
seq(a[i],i=0..30); # Robert Israel, Dec 15 2014

a = 1; lst = {a}; Do[a = a * n^2 + 1; AppendTo[lst, a], {n, 1, 14}]; lst (* Zerinvary Lajos, Jul 08 2009 *)
Table[Sum[(n!/(n - k)!)^2, {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 15 2017 *)

a(n)=sum(k=0, n, (k!*binomial(n, k))^2 ); \\ Joerg Arndt, Dec 14 2014

def A006040_list(len):
    L = [1]
    for k in range(1,len): L.append(L[-1]*k^2+1)
    return L
A006040_list(18) # Peter Luschny, Dec 15 2014

a(n) = n^2*a(n-1) + 1.
The following formulas will need adjusting, since I have changed the offset. - N. J. A. Sloane, Dec 17 2013
a(n+1) = Nearest integer to BesselI(0, 2)*n!*n!, n >= 1.
a(n+1) = n!^2*Sum_{k = 0..n} 1/k!^2. BesselI(0, 2*sqrt(x))/(1-x) = Sum_{n>=0} a(n+1)*x^n/n!^2. - Vladeta Jovovic, Aug 30 2002
Recurrence: a(1) = 1, a(2) = 2, a(n+1) = (n^2 + 1)*a(n) - (n - 1)^2*a(n-1), n >= 2. The sequence defined by b(n) := (n-1)!^2 satisfies the same recurrence with the initial conditions b(1) = 1, b(2) = 1. It follows that a(n+1) = n!^2*(1 + 1/(1 - 1/(5 - 4/(10 - ... - (n - 1)^2/(n^2 + 1))))). Hence BesselI(0,2) := Sum_{k >= 0} 1/k!^2 = 1 + 1/(1 - 1/(5 - 4/(10 - ... - (n - 1)^2/(n^2 + 1 - ...)))). Cf. A073701. - Peter Bala, Jul 09 2008

Offset changed by N. J. A. Sloane, Dec 17 2013

A088699 Array read by antidiagonals of coefficients of generating function exp(x)/(1-y-xy).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 21, 34, 21, 6, 1, 1, 7, 31, 73, 73, 31, 7, 1, 1, 8, 43, 136, 209, 136, 43, 8, 1, 1, 9, 57, 229, 501, 501, 229, 57, 9, 1, 1, 10, 73, 358, 1045, 1546, 1045, 358, 73, 10, 1, 1, 11, 91, 529, 1961, 4051, 4051, 1961

Offset: 0

Michael Somos, Oct 08 2003

A(n,m) is the number of ways to pair the elements of two sets (with respectively n and m elements), where each element of either set may be paired with zero or one elements of the other set; number of n X m matrices of zeros and ones with at most one one in each row and column. E.g., A(2,2)=7 because we can pair {A,B} with {C,D} as {AB,CD}, {AC,BD}, {AC,B,D}, {AD,B,C}, {BC,A,D}, {BD,A,C}, or {A,B,C,D}. - Franklin T. Adams-Watters, Feb 06 2006
Compare with A086885. - Peter Bala, Sep 17 2008
A(n,m) is the number of vertex covers and independent vertex sets in the n X m lattice (rook) graph K_n X K_m. - Andrew Howroyd, May 14 2017

			      1       1       1       1       1       1       1       1       1
      1       2       3       4       5       6       7       8       9
      1       3       7      13      21      31      43      57      73
      1       4      13      34      73     136     229     358     529
      1       5      21      73     209     501    1045    1961    3393
      1       6      31     136     501    1546    4051    9276   19081
      1       7      43     229    1045    4051   13327   37633   93289
      1       8      57     358    1961    9276   37633  130922  394353
      1       9      73     529    3393   19081   93289  394353 1441729

Andrew Howroyd, Table of n, a(n) for n = 0..1274
R. J. Mathar, The number of binary nXm matrices with at most k 1's in each row or column, (2014) Table 1.
Eric Weisstein's World of Mathematics, Rook Graph
Eric Weisstein's World of Mathematics, Vertex Cover
Wikipedia, Rook polynomial

Row sums give A081124.
Main diagonal is A002720.
Cf. A099597, A176120.

A088699 := proc(i,j)
    add(binomial(i,k)*binomial(j,k)*k!,k=0..min(i,j)) ;
end proc: # R. J. Mathar, Feb 28 2015

max = 11; se = Series[E^x/(1 - y - x*y), {x, 0, max}, {y, 0, max}] // Normal // Expand; a[i_, j_] := SeriesCoefficient[se, {x, 0, i}, {y, 0, j}]*i!; Flatten[ Table[ a[i - j, j], {i, 0, max}, {j, 0, i}]] (* Jean-François Alcover, May 15 2012 *)

A(i,j)=if(i<0 || j<0,0,i!*polcoeff(exp(x+x*O(x^i))*(1+x)^j,i))

A(i,j)=if(i<0 || j<0,0,i!*polcoeff(exp(x/(1-x)+x*O(x^i))*(1-x)^(i-j-1),i))

A(i,j)=local(M); if(i<0 || j<0,0,M=matrix(j+1,j+1,n,m,if(n==m,1,if(n==m+1,m))); (M^i)[j+1,]*vectorv(j+1,n,1)) /* Michael Somos, Jul 03 2004 */

E.g.f.: exp(x)/(1-y-xy)=Sum_{i, j} A(i, j) y^j x^i/i!.
A(i, j) = A(i-1, j)+j*A(i-1, j-1)+(i==0) = A(j, i).
T(n, k) = sum{j=0..k, C(n, k-j)*k!/j!} = sum{j=0..k, (k-j)!*C(k, j)C(n, k-j)}. - Paul Barry, Nov 14 2005
A(i,j) = sum_k C(i,k)*C(j,k)*k!. E.g.f.: sum_{i,j} a(i,j)*x^i/i!*y^j/j! = e^{x+y+xy}. - Franklin T. Adams-Watters, Feb 06 2006
The LDU factorization of this array, formatted as a square array, is P * D * transpose(P), where P is Pascal's triangle A007318 and D = diag(0!, 1!, 2!, ... ). Compare with A099597. - Peter Bala, Nov 06 2007
A(i,j) = (-1)^-i HypergeometricU(-i, 1 - i + j, -1). - Eric W. Weisstein, May 10 2017

A228229 Recurrence a(n) = n(n + 1)a(n-1) + 1 with a(0) = 1.

Original entry on oeis.org

1, 3, 19, 229, 4581, 137431, 5772103, 323237769, 23273119369, 2094580743211, 230403881753211, 30413312391423853, 4744476733062121069, 863494765417306034559, 181333900737634267257391, 43520136177032224141773841, 11837477040152764966562484753

Offset: 0

Peter Bala, Aug 19 2013

Main subdiagonal (and main superdiagonal) of A099597. Cf. A006040 and A228230.

Seiichi Manyama, Table of n, a(n) for n = 0..252
E. W. Weisstein, Modified Bessel Function of the First Kind

Cf. A006040, A096789, A099597, A228230, A368837, A368838.

A228229 :=proc(n) option remember
    if n = 0 then 1
    else n*(n+1)*procname(n-1) + 1
    end if:
end proc:
seq(A228229(n), n = 0..20);

RecurrenceTable[{a[n] == n*(n + 1)*a[n-1] + 1, a[0] == 1},a,{n,0,20}] (* Vaclav Kotesovec, May 06 2015 *)

a(n) = n!*(n + 1)!*sum {k = 0..n} 1/(k!*(k + 1)!).
Generating function: 1/(1 - x)*1/sqrt(x)*BesselI(1, 2*sqrt(x)) = sum {n >= 0} a(n)*x^n/(n!*(n + 1)!).
Defining recurrence equation: a(n) = n*(n + 1)*a(n-1) + 1 with a(0) = 1.
Alternative recurrence equation: a(0) = 1, a(1) = 3, and for n >= 2, a(n) = (n*(n + 1) + 1)*a(n-1) - n*(n - 1)*a(n-2).
The sequence b(n) := n!*(n + 1)! satisfies the same recurrence with the initial conditions b(0) = 1, b(1) = 2. It follows that we have the finite continued fraction expansion a(n) = n!*(n + 1)!*(1/(1 - 1/(3 - 2/(7 - 6/(13 - … - n*(n - 1)/(n^2 + n + 1)))))). Taking the limit yields the continued fraction expansion for the modified Bessel function value BesselI(1,2) = sum {k = 0..inf} 1/(k!*(k + 1)!) = 1/(1 - 1/(3 - 2/(7 - 6/(13 - ...- n*(n - 1)/(n^2 + n + 1 - ...))))) = 1.590636... (see A096789).
a(n) ~ BesselI(1,2) * n!*(n+1)!. - Vaclav Kotesovec, May 06 2015

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A099599 Triangle T read by rows: coefficients of polynomials generating array A099597.

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A099598 Antidiagonal sums of array A099597.

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A058331 a(n) = 2*n^2 + 1.

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A070910 Decimal expansion of BesselI(0,2).

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A006040 a(n) = Sum_{i=0..n} (n!/(n-i)!)^2.

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A088699 Array read by antidiagonals of coefficients of generating function exp(x)/(1-y-xy).

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PARI

PARI

PARI

Formula

A228229 Recurrence a(n) = n*(n + 1)*a(n-1) + 1 with a(0) = 1.

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A228229 Recurrence a(n) = n(n + 1)a(n-1) + 1 with a(0) = 1.