A040000 -id:A040000 - cp's OEIS frontend

A112468 Riordan array (1/(1-x), x/(1+x)).

1, 1, 1, 1, 0, 1, 1, 1, -1, 1, 1, 0, 2, -2, 1, 1, 1, -2, 4, -3, 1, 1, 0, 3, -6, 7, -4, 1, 1, 1, -3, 9, -13, 11, -5, 1, 1, 0, 4, -12, 22, -24, 16, -6, 1, 1, 1, -4, 16, -34, 46, -40, 22, -7, 1, 1, 0, 5, -20, 50, -80, 86, -62, 29, -8, 1, 1, 1, -5, 25, -70, 130, -166, 148, -91, 37, -9, 1, 1, 0, 6, -30, 95, -200, 296, -314, 239, -128, 46, -10, 1

Offset: 0

Paul Barry, Sep 06 2005

Row sums are A040000. Diagonal sums are A112469. Inverse is A112467. Row sums of k-th power are 1, k+1, k+1, k+1, .... Note that C(n,k) = Sum_{j=0..n-k} C(n-j-1, n-k-j).
Equals row reversal of triangle A112555 up to sign, where log(A112555) = A112555 - I. Unsigned row sums equals A052953 (Jacobsthal numbers + 1). Central terms of even-indexed rows are a signed version of A072547. Sums of squared terms in rows yields A112556, which equals the first differences of the unsigned central terms. - Paul D. Hanna, Jan 20 2006
Sum_{k=0..n} T(n,k)*x^k = A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively (see the square array in A112739). - Philippe Deléham, Feb 22 2014

			Triangle starts
  1;
  1,  1;
  1,  0,  1;
  1,  1, -1,  1;
  1,  0,  2, -2,  1;
  1,  1, -2,  4, -3,  1;
  1,  0,  3, -6,  7, -4,  1;
Matrix log begins:
  0;
  1,  0;
  1,  0,  0;
  1,  1, -1,  0;
  1,  1,  1, -2,  0;
  1,  1,  1,  1, -3,  0; ...
Production matrix begins
  1,  1,
  0, -1,  1,
  0,  0, -1,  1,
  0,  0,  0, -1,  1,
  0,  0,  0,  0, -1,  1,
  0,  0,  0,  0,  0, -1,  1,
  0,  0,  0,  0,  0,  0, -1,  1.
- _Paul Barry_, Apr 08 2011

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
H. Belbachir and F. Bencherif, On some properties of bivariate Fibonacci and Lucas Polynomials, JIS 11 (2008) 08.2.6.
Hacene Belbachir and Athmane Benmezai, Expansion of Fibonacci and Lucas Polynomials: An Answer to Prodinger's Question, Journal of Integer Sequences, Vol. 15 (2012), #12.7.6.
Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
H. Prodinger, On the expansion of Fibonacci and Lucas Polynomials, JIS 12 (2009) 09.1.6.

Cf. A174294, A174295, A174296, A174297. - Mats Granvik, Mar 15 2010
Cf. A072547 (central terms), A112555 (reversed rows), A112465, A052953, A112556, A112739, A119258.
See A279006 for another version.

T:= function(n,k)
    if k=0 or k=n then return 1;
    else return T(n-1,k-1) - T(n-1,k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 13 2019

a112468 n k = a112468_tabl !! n !! k
a112468_row n = a112468_tabl !! n
a112468_tabl = iterate (\xs -> zipWith (-) ([2] ++ xs) (xs ++ [0])) [1]
-- Reinhard Zumkeller, Jan 03 2014

function T(n,k)
  if k eq 0 or k eq n then return 1;
  else return T(n-1,k-1) - T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 13 2019

T := (n,k,m) -> (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k,k+1)*hypergeom( [1,n+1],[k+2],m)/(k+1)!; A112468 := (n,k) -> T(n,n-k,-1);
seq(print(seq(simplify(A112468(n,k)),k=0..n)),n=0..10); # Peter Luschny, Jul 25 2014

T[n_, 0] = 1; T[n_, n_] = 1; T[n_, k_ ]:= T[n, k] = T[n-1, k-1] - T[n-1, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Jean-François Alcover, Mar 06 2013 *)

{T(n,k)=local(m=1,x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff(polcoeff((1+(m-1)*x)*(1+m*x)/(1+m*x-x*y)/(1-x),n,X),k,Y)} \\ Paul D. Hanna, Jan 20 2006

T(n,k) = if(k==0 || k==n, 1, T(n-1, k-1) - T(n-1, k)); \\ G. C. Greubel, Nov 13 2019

@CachedFunction
def T(n, k):
    if (k<0 or n<0): return 0
    elif (k==0 or k==n): return 1
    else: return T(n-1, k-1) - T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 13 2019

Triangle T(n,k) read by rows: T(n,0)=1, T(n,k) = T(n-1,k-1) - T(n-1,k). - Mats Granvik, Mar 15 2010
Number triangle T(n, k)= Sum_{j=0..n-k} C(n-j-1, n-k-j)*(-1)^(n-k-j).
G.f. of matrix power T^m: (1+(m-1)*x)*(1+m*x)/(1+m*x-x*y)/(1-x). G.f. of matrix log: x*(1-2*x*y+x^2*y)/(1-x*y)^2/(1-x). - Paul D. Hanna, Jan 20 2006
T(n, k) = R(n,n-k,-1) where R(n,k,m) = (1-m)^(-n+k)-m^(k+1)*Pochhammer(n-k,k+1)*hyper2F1([1,n+1],[k+2],m)/(k+1)!. - Peter Luschny, Jul 25 2014

A287804 Number of quinary sequences of length n such that no two consecutive terms have distance 1.

Original entry on oeis.org

1, 5, 17, 59, 205, 713, 2481, 8635, 30057, 104629, 364225, 1267923, 4413861, 15365465, 53490097, 186209299, 648230545, 2256616133, 7855718641, 27347281995, 95201200637, 331413874569, 1153716087665, 4016309864843, 13981555011321, 48672509644725

Offset: 0

David Nacin, Jun 01 2017

			For n=2 the a(2)=17=25-8 sequences contain every combination except these eight: 01,10,12,21,23,32,34,43.

Index entries for linear recurrences with constant coefficients, signature (5,-5,-1).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819.

LinearRecurrence[{5, -5, -1}, {1, 5, 17}, 50]

def a(n):
    if n in [0,1,2]:
        return [1,5,17][n]
    return 5*a(n-1)-5*a(n-2)-a(n-3)

a(n) = 5*a(n-1) - 5a(n-2) - a(n-3), a(0)=1, a(1)=5, a(2)=17.

G.f.: (1 - 3*x^2)/(1 - 5*x + 5*x^2 + x^3).

A287819 Number of nonary sequences of length n such that no two consecutive terms have distance 4.

Original entry on oeis.org

1, 9, 71, 561, 4433, 35031, 276827, 2187585, 17287073, 136608591, 1079529611, 8530826457, 67413620993, 532726379847, 4209793089371, 33267280400913, 262889866978817, 2077449112980255, 16416740845208075, 129730917736941417, 1025179795159015841

Offset: 0

David Nacin, Jun 02 2017

			For n=2 the a(2) = 81 - 10 = 71 sequences contain every combination except these ten: 04,40,15,51,26,62,37,73,48,84.

Index entries for linear recurrences with constant coefficients, signature (8,1,-14).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819.

LinearRecurrence[{8, 1, -14}, {1, 9, 71, 561}, 40]

def a(n):
    if n in [0, 1, 2, 3]:
        return [1, 9, 71, 561][n]
    return 8*a(n-1)+a(n-2)-14*a(n-3)

For n>2, a(n) = 8*a(n-1) + a(n-2) - 14*a(n-3), a(0)=1, a(1)=9, a(2)=71, a(3)=561.

G.f.: (1 + x - 2 x^2 - 2 x^3)/(1 - 8 x - x^2 + 14 x^3).

A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).

Original entry on oeis.org

1, 2, 6, 14, 36, 86, 210, 500, 1194, 2822, 6660, 15638, 36642, 85604, 199626, 464630, 1079892, 2506550, 5811762, 13462484, 31159914, 72071654, 166599972, 384912086, 888906306, 2052031172, 4735527306, 10925175254, 25198866036, 58108609526, 133973643090

Offset: 0

Johannes W. Meijer, Aug 06 2010; edited Jun 21 2013

a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 or 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the center square the bishop flies into a rage and turns into a raging elephant.
In chaturanga, the old Indian version of chess, one of the pieces was called gaja, elephant in Sanskrit. The Arabs called the game shatranj and the elephant became el fil in Arabic. In Spain chess became chess as we know it today but surprisingly in Spanish a bishop isn't a Christian bishop but a Moorish elephant and it still goes by its original name of el alfil.
On a 3 X 3 chessboard there are 2^9 = 512 ways for an elephant to fly into a rage on the central square (off the center the piece behaves like a normal bishop). The elephant is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the corner squares the 512 elephants lead to 46 different elephant sequences, see the overview of elephant sequences and the crossreferences.
The sequence above corresponds to 16 A[5] vectors with decimal values 71, 77, 101, 197, 263, 269, 293, 323, 326, 329, 332, 353, 356, 389, 449 and 452. These vectors lead for the side squares to A000079 and for the central square to A175655.

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.
David Hooper and Kenneth Whyld, The Oxford Companion to Chess, pp. 74, 366, 1992.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Viswanathan Anand, The Indian Defense, Time, Jun 19 2008.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Johannes W. Meijer, The elephant sequences.
Wikipedia, War Elephant.
Index entries for linear recurrences with constant coefficients, signature (3,1,-6).

Cf. Elephant sequences corner squares [decimal value A[5]]: A040000 [0], A000027 [16], A000045 [1], A094373 [2], A000079 [3], A083329 [42], A027934 [11], A172481 [7], A006138 [69], A000325 [26], A045623 [19], A000129 [21], A095121 [170], A074878 [43], A059570 [15], A175654 [71, this sequence], A026597 [325], A097813 [58], A057711 [27], 2*A094723 [23; n>=-1], A002605 [85], A175660 [171], A123203 [186], A066373 [59], A015518 [341], A134401 [187], A093833 [343].

Cf. A000244, A006130, A006138, A075188, A175655, A180140.

[n le 3 select Factorial(n) else 3*Self(n-1) +Self(n-2) -6*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 08 2021

nmax:=28; m:=1; A[1]:=[0,0,0,0,1,0,0,0,1]: A[2]:=[0,0,0,1,0,1,0,0,0]: A[3]:=[0,0,0,0,1,0,1,0,0]: A[4]:=[0,1,0,0,0,0,0,1,0]: A[5]:=[0,0,1,0,0,0,1,1,1]: A[6]:=[0,1,0,0,0,0,0,1,0]: A[7]:=[0,0,1,0,1,0,0,0,0]: A[8]:=[0,0,0,1,0,1,0,0,0]: A[9]:=[1,0,0,0,1,0,0,0,0]: A:=Matrix([A[1], A[2], A[3], A[4], A[5], A[6], A[7], A[8], A[9]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{3,1,-6}, {1,2,6}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)

a(n)=([0,1,0; 0,0,1; -6,1,3]^n*[1;2;6])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[( (1-x-x^2)/((1-2*x)*(1-x-3*x^2)) ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Dec 08 2021

G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).
a(n) = 3*a(n-1) + a(n-2) - 6*a(n-3) with a(0)=1, a(1)=2 and a(2)=6.
a(n) = ((6+10*A)*A^(-n-1) + (6+10*B)*B^(-n-1))/13 - 2^n with A = (-1+sqrt(13))/6 and B = (-1-sqrt(13))/6.
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n)*2*A000244(n)/(A075118(n) - A006130(n-1)*sqrt(13)).
a(n) = b(n) - b(n-1) - b(n-2), where b(n) = Sum_{k=1..n} Sum_{j=0..k} binomial(j,n-3*k+2*j)*(-6)^(k-j)*binomial(k,j)*3^(3*k-n-j), n>0, b(0)=1, with a(0) = b(0), a(1) = b(1) - b(0). - Vladimir Kruchinin, Aug 20 2010
a(n) = 2*A006138(n) - 2^n = 2*(A006130(n) + A006130(n-1)) - 2^n. - G. C. Greubel, Dec 08 2021
E.g.f.: 2*exp(x/2)*(13*cosh(sqrt(13)*x/2) + 3*sqrt(13)*sinh(sqrt(13)*x/2))/13 - cosh(2*x) - sinh(2*x). - Stefano Spezia, Feb 12 2023

A104350 Partial products of largest prime factors of numbers <= n.

Original entry on oeis.org

1, 2, 6, 12, 60, 180, 1260, 2520, 7560, 37800, 415800, 1247400, 16216200, 113513400, 567567000, 1135134000, 19297278000, 57891834000, 1099944846000, 5499724230000, 38498069610000, 423478765710000, 9740011611330000

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Partial Products of A006530: a(n) = Product_{k=1..n} A006530(k).
a(n) = a(n-1)*A006530(n) for n>1, a(1) = 1;
A020639(a(n)) = A040000(n-1), A006530(a(n)) = A007917(n) for n>1.
A001221(a(n)) = A000720(n), A001222(a(n)) = A001477(n-1).
A007947(a(n)) = A034386(n).
a(n) = A000142(n) / A076928(n). [Corrected by Franklin T. Adams-Watters, Oct 30 2006]
In decimal representation: A104351(n) = number of digits of a(n), A104355(n) = number of trailing zeros of a(n).
A104357(n) = a(n) - 1, A104365(n) = a(n) + 1.

Gérald Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Publ. Inst. Elie Cartan, Vol. 13, Nancy, 1990.

Charles R Greathouse IV, Table of n, a(n) for n = 1..641
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint, arXiv:1202.3670 [math.HO], 2012-2018.
Eric Weisstein's World of Mathematics, Greatest Prime Factor.
Reinhard Zumkeller, Products of largest prime factors of numbers <= n.

Cf. A000142, A002110, A006530, A007947, A020639, A046670, A072486, A076928, A104351, A104355, A104357, A104365.

Cf. A001620, A084945.

a104350 n = a104350_list !! (n-1)
a104350_list = scanl1 (*) a006530_list
-- Reinhard Zumkeller, Apr 10 2014

A104350[n_] := Product[FactorInteger[k][[-1, 1]], {k, 1, n}]; Table[A104350[n], {n, 30}] (* G. C. Greubel, May 09 2017 *)
FoldList[Times,Table[FactorInteger[n][[-1,1]],{n,30}]] (* Harvey P. Dale, May 25 2023 *)

gpf(n)=my(f=factor(n)[,1]); f[#f]
a(n)=prod(i=2,n,gpf(i)) \\ Charles R Greathouse IV, Apr 29 2015

first(n)=my(v=vector(n,i,1)); forfactored(k=2,n, v[k[1]]=v[k[1]-1]*vecmax(k[2][,1])); v \\ Charles R Greathouse IV, May 10 2017

log(a(n)) = c * n * log(n) + c * (1-gamma) * n + O(n * exp(-log(n)^(3/8-eps))), where c is the Golomb-Dickman constant (A084945) and gamma is Euler's constant (A001620) (Tenenbaum, 1990). - Amiram Eldar, May 21 2021

More terms from David Wasserman, Apr 24 2008

A113311 Expansion of (1+x)^2/(1-x).

Original entry on oeis.org

1, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Paul Barry, Oct 25 2005

Row sums of A113310.
Let m=3. We observe that a(n)=Sum_{k=0..floor(n/2)} C(m,n-2*k). Then there is a link with A040000 and A115291: it is the same formula with respectively m=2 and m=4. We can generalize this result with the sequence whose g.f. is given by (1+z)^(m-1)/(1-z). - Richard Choulet, Dec 08 2009
Also continued fraction expansion of (3+sqrt(5))/4. - Bruno Berselli, Sep 23 2011
Also decimal expansion of 121/900. - Vincenzo Librandi, Sep 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A040000, A115291, A171418, A171440-A171443.

CoefficientList[Series[(1+x)^2/(1-x),{x,0,110}],x] (* Harvey P. Dale, Aug 19 2011 *)

a(n)=if(n>1,4,2*n+1) \\ Charles R Greathouse IV, Jun 12 2015

a(n) = Sum_{k=0..n} Sum_{i=0..n-k} (-1)^i*C(i+k-2, i).

E.g.f.: 4*exp(x) - x - 3. - Elmo R. Oliveira, Aug 08 2024

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A040002 Continued fraction for sqrt(5).

Original entry on oeis.org

2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

N. J. A. Sloane

Decimal expansion of 11/45. - Natan Arie Consigli, Jan 19 2016

			2.236067977499789696409173668... = 2 + 1/(4 + 1/(4 + 1/(4 + 1/(4 + ...)))). - _Harry J. Smith_, Jun 01 2009

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index of eventually constant sequences
Index of divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A002163 (decimal expansion), A001077/A001076 (convergents), A248235 (Egyptian fraction).
Cf. Continued fraction for sqrt(a^2+1) = (a, 2a, 2a, 2a....): A040000 (contfrac(sqrt(2)) = (1,2,2,...)), A040002, A040006, A040012, A040020, A040030, A040042, A040056, A040072, A040090, A040110 (contfrac(sqrt(122)) = (11,22,22,...)), A040132, A040156, A040182, A040210, A040240, A040272, A040306, A040342, A040380, A040420 (contfrac(sqrt(442)) = (21,42,42,...)), A040462, A040506, A040552, A040600, A040650, A040702, A040756, A040812, A040870, A040930 (contfrac(sqrt(962)) = (31,62,62,...)).
Essentially the same as A010709.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[5],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
PadRight[{2},120,{4}] (* Harvey P. Dale, Jul 06 2019 *)

{ allocatemem(932245000); default(realprecision, 26000); x=contfrac(sqrt(5)); for (n=0, 20000, write("b040002.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

a(0) = 2, a(n) = 4 n>0. - Natan Arie Consigli, Jan 19 2016
From Elmo R. Oliveira, Feb 16 2024: (Start)
G.f.: 2*(1+x)/(1-x).
E.g.f.: 4*exp(x) - 2.
a(n) = 2*A040000(n). (End)

A122542 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 4, 1, 0, 2, 8, 6, 1, 0, 2, 12, 18, 8, 1, 0, 2, 16, 38, 32, 10, 1, 0, 2, 20, 66, 88, 50, 12, 1, 0, 2, 24, 102, 192, 170, 72, 14, 1, 0, 2, 28, 146, 360, 450, 292, 98, 16, 1, 0, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1

Offset: 0

Philippe Deléham, Sep 19 2006, May 28 2007

Riordan array (1, x*(1+x)/(1-x)). Rising and falling diagonals are the tribonacci numbers A000213, A001590.

			Triangle begins:
  1;
  0, 1;
  0, 2,  1;
  0, 2,  4,   1;
  0, 2,  8,   6,   1;
  0, 2, 12,  18,   8,    1;
  0, 2, 16,  38,  32,   10,   1;
  0, 2, 20,  66,  88,   50,  12,   1;
  0, 2, 24, 102, 192,  170,  72,  14,   1;
  0, 2, 28, 146, 360,  450, 292,  98,  16,  1;
  0, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 1.

Other versions: A035607, A113413, A119800, A266213.
Diagonals: A000012, A005843, A001105, A035597 - A035606.
Columns: A000007, A040000, A008575, A005899, A008412-A008416, A008418, A008420, A035706 - A035745.
Sums include: A000007, A001333 (row), A001590 (diagonal), A007483, A057077 (signed row), A078016 (signed diagonal), A086901, A091928, A104934, A122558, A122690.
Cf. A155161, A059283.

a122542 n k = a122542_tabl !! n !! k
a122542_row n = a122542_tabl !! n
a122542_tabl = map fst $ iterate
   (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) $
                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [0, 1])
-- Reinhard Zumkeller, Jul 20 2013, Apr 17 2013

function T(n, k) // T = A122542
  if k eq 0 then return 0^n;
  elif k eq n then return 1;
  else return T(n-1,k) + T(n-1,k-1) + T(n-2,k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 27 2024

CoefficientList[#, y]& /@ CoefficientList[(1-x)/(1 - (1+y)x - y x^2) + O[x]^11, x] // Flatten (* Jean-François Alcover, Sep 09 2018 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, 0, T[n-1,k-1] +T[n-1,k] +T[n-2,k- 1] ]]; (* T = A122542 *)
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 27 2024 *)

def A122542_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)+2*sum(prec(n-i,k-1) for i in (2..n-k+1))
    return [prec(n, k) for k in (0..n)]
for n in (0..10): print(A122542_row(n)) # Peter Luschny, Mar 16 2016

Sum_{k=0..n} x^k*T(n,k) = A000007(n), A001333(n), A104934(n), A122558(n), A122690(n), A091928(n)  for x = 0, 1, 2, 3, 4, 5. - Philippe Deléham, Jan 25 2012
Sum_{k=0..n} 3^(n-k)*T(n,k) = A086901(n).
Sum_{k=0..n} 2^(n-k)*T(n,k) = A007483(n-1), n >= 1. - Philippe Deléham, Oct 08 2006
T(2*n, n) = A123164(n).
T(n, k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1), n > 1. - Philippe Deléham, Jan 25 2012
G.f.: (1-x)/(1-(1+y)*x-y*x^2). - Philippe Deléham, Mar 02 2012
From G. C. Greubel, Oct 27 2024: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = A057077(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A001590(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A078016(n). (End)

A068875 Expansion of (1 + xC)C, where C = (1 - (1 - 4x)^(1/2))/(2x) is the g.f. for Catalan numbers, A000108.

Original entry on oeis.org

1, 2, 4, 10, 28, 84, 264, 858, 2860, 9724, 33592, 117572, 416024, 1485800, 5348880, 19389690, 70715340, 259289580, 955277400, 3534526380, 13128240840, 48932534040, 182965127280, 686119227300, 2579808294648, 9723892802904, 36734706144304, 139067101832008

Offset: 0

N. J. A. Sloane, Jun 06 2002

A Catalan transform of A040000 under the mapping g(x) -> g(x*c(x)), where c(x) is the g.f. of A000108. Sequence A040000 can be retrieved using the mapping g(x) -> g(x*(1-x)). A040000(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k) * (-1)^k * a(n-k). a(n) and A040000 may be described as a Catalan pair. - Paul Barry, Nov 14 2004
a(n) is the number of Dyck (n+1)-paths all of whose nonterminal descents to ground level are of odd length. For example, a(2) counts UUUDDD, UUDUDD, UDUUDD, UDUDUD. - David Callan, Jul 25 2005
From Gary W. Adamson, Jul 11 2011: (Start)
a(n) is the sum of the top row terms in M^n, where M is the following infinite square production matrix:
  1, 1, 0, 0, 0, 0, ...
  0, 1, 1, 0, 0, 0, ...
  1, 1, 1, 1, 0, 0, ...
  0, 1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, 1, ...
  ...
For example, the top row of M^3 = (2, 4, 3, 1) with sum = 10 = a(3). (End)
For n >= 1, a(n) is the number of binary trees with n+1 internal node in which one of the subtrees of the root is empty. Cf. A002057. [Sedgewick and Flajolet] - Geoffrey Critzer, Jan 05 2013
Empirical: a(n) is the number of entries of absolute value 1 that appear among all partitions in the canonical basis of the Temperley-Lieb algebra of order n. - John M. Campbell, Oct 17 2017
For n >= 1, a(n) is the number of Dyck paths of size n+2, whose corresponding unit interval graph has P3-hull number equal to 2. This result is due to Alrik Sandberg. - Per W. Alexandersson, Jan 09 2024

			G.f. = 1 + 2*x + 4*x^2 + 10*x^3 + 28*x^4 + 84*x^5 + 264*x^6 + 858*x^7 + ...
For example, the canonical basis of the Temperley-Lieb algebra of order 3 is {{{-3, 1}, {-2, -1}, {2, 3}}, {{-3, 3}, {-2, 2}, {-1, 1}}, {{-3, 3}, {-2, -1}, {1, 2}}, {{-3, -2}, {-1, 1}, {2, 3}}, {{-3, -2}, {-1, 3}, {1, 2}}}, and we see that the total number of entries of absolute value 1 that appear among the partitions in this basis is a(3) = 10.

R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, 1996, p. 225.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995), 743-751. [Annotated scanned copy]
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, 2017.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

A002420 and A262543 are essentially the same sequence as this.

Cf. A000108, A000984, A002057, A040000, A068875, A071721.

[1] cat [2*Binomial( 2*n, n)/(n+1): n in [1..30]]; // Vincenzo Librandi, Oct 17 2017

Z:=(1-sqrt(1-4*x))/2/x: G:=(2-(1+x)*Z)/Z: Gser:=series(-G, x=0, 30): (1,seq(coeff(Gser, x, n), n=2..26)); # Zerinvary Lajos, Dec 23 2006
Z:=-(1-z-sqrt(1-z))/sqrt(1-z): Zser:=series(Z, z=0, 32): (1,seq(coeff(Zser*4^n, z, n), n=2..26)); # Zerinvary Lajos, Jan 01 2007
A068875List := proc(m) local A, P, n; A := [1, 2]; P := [2];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A068875List(26); # Peter Luschny, Mar 24 2022

nn=30;t=(1-(1-4x )^(1/2))/(2x);Prepend[Table[Coefficient[Series[1+x (y t -y+1)^2,{x,0,nn}],x ^n y],{n,2,nn}],1]  (* Geoffrey Critzer, Jan 05 2013 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 2 Binomial[ 2 n, n]/(n + 1)]; (* Michael Somos, Jun 18 2014 *)
a[ n_] := SeriesCoefficient[ -1 + 4 / (1 + Sqrt[ 1 - 4 x]), {x, 0, n}]; (* Michael Somos, Jun 18 2014 *)
Table[If[n==0, 1, 2 CatalanNumber[n]], {n,0,25}] (* Peter Luschny, Feb 27 2017 *)

{a(n) = if( n<1, n==0, 2 * binomial( 2*n, n) / (n + 1))}; /* Michael Somos, Aug 17 2005 */

{a(n) = if( n<0, 0, polcoeff( -1 + 4 / (1 + sqrt(1 - 4*x + x * O(x^n))), n))}; /* Michael Somos, Aug 17 2005 */

Apart from initial term, twice Catalan numbers.
G.f.: (1 - x - sqrt(1 - 4*x)) / x. - Michael Somos, Apr 13 2012
From Paul Barry, Nov 14 2004: (Start)
G.f.: (1 + x*c(x))/(1 - x*c(x)), where c(x) is the g.f. of A000108.
a(n) = C(n)*(2-0^n), where C(n) = A000108(n).
a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(2*n, n-k) *((2*k + 1)/(n + k + 1)) * binomial(k, j) * (-1)^(j-k) * (2 - 0^j). (End)
Assuming offset 1, then series reversion of g.f. A(x) is -A(-x). - Michael Somos, Aug 17 2005
Assuming offset 2, then A(x) satisfies A(x - x^2) = x^2 - x^4 and so A(x) = C(x)^2 - C(x)^4, A(A(x)) = C(x)^4 - C(x)^8, A(A(A(x))) = C(x)^8 - C(x)^16, etc., where C(x) = (1-sqrt(1-4*x))/2 = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + ... . - Paul D. Hanna, May 16 2008
Apart from initial term, INVERTi transform of A000984(n+1) = binomial(2*n+2,n+1), also, for n >= 1, a(n) = (1/Pi)*Integral_{x=0..4} x^(n-1)*sqrt(x*(4 - x)). - Groux Roland, Mar 15 2011
D-finite with recurrence (n+2)*a(n) - 2*(2*n+1)*a(n-1) = 0, n > 1. - R. J. Mathar, Nov 14 2011
For n > 0, a(n) = C(2*n+2, n+1) mod 4*C(2*n, n - 1). - Robert G. Wilson v, May 02 2012
For n > 0, a(n) = 2^(2*n + 1) * Gamma(n + 1/2)/(sqrt(Pi) * (n + 1)!). - Vaclav Kotesovec, Sep 16 2013
G.f.: 1 + 2*x/(Q(0) - x), where Q(k) = 2*x + (k + 1)/(2*k + 1) - 2*x*(k + 1)/(2*k + 1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2013
G.f.: 3 - 4*x - 2*S(0), where S(k) = 2*k + 1 - x*(2*k + 3)/(1 - x*(2*k + 1)/S(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 23 2013
0 = a(n)*(16*a(n+1) - 10*a(n+2)) + a(n+1)*(2*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Jun 18 2014
If A(x)^t = 1 + 2*t*x + Sum_{n >= 2} t*P(n,t)*x^n, then we conjecture that all the zeros of the polynomial P(n,t) lie on the vertical line Re(t) = -n/2 in the complex plane. - Peter Bala, Oct 05 2015
a(n+1) = a(n) + (1/2)*(Sum_{k=0..n} a(k)*a(n-k)) if n > 0. - Michael Somos, Apr 22 2022
b(n) = a(n+1) - a(n) for all n in Z if b(0) = 2, a(-1) = -1, a(0) = 0, a(-1) = 3, a(-2) = -1 where b = A071721. - Michael Somos, Apr 23 2022

A108561 Triangle read by rows: T(n,0)=1, T(n,n)=(-1)^n, T(n+1,k)=T(n,k-1)+T(n,k) for 0 < k < n.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 1, -1, 1, 2, 2, 0, 1, 1, 3, 4, 2, 1, -1, 1, 4, 7, 6, 3, 0, 1, 1, 5, 11, 13, 9, 3, 1, -1, 1, 6, 16, 24, 22, 12, 4, 0, 1, 1, 7, 22, 40, 46, 34, 16, 4, 1, -1, 1, 8, 29, 62, 86, 80, 50, 20, 5, 0, 1, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, -1, 1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6, 0, 1, 1, 11, 56, 174, 367

Offset: 0

Reinhard Zumkeller, Jun 10 2005

Sum_{k=0..n} T(n,k) = A078008(n);
Sum_{k=0..n} abs(T(n,k)) = A052953(n-1) for n > 0;
T(n,1) = n - 2 for n > 1;
T(n,2) = A000124(n-3) for n > 2;
T(n,3) = A003600(n-4) for n > 4;
T(n,n-6) = A001753(n-6) for n > 6;
T(n,n-5) = A001752(n-5) for n > 5;
T(n,n-4) = A002623(n-4) for n > 4;
T(n,n-3) = A002620(n-1) for n > 3;
T(n,n-2) = A008619(n-2) for n > 2;
T(n,n-1) = n mod 2 for n > 0;
T(2*n,n) = A072547(n+1).
Sum_{k=0..n} T(n,k)*x^k = A232015(n), A078008(n), A000012(n), A040000(n), A001045(n+2), A140725(n+1) for x = 2, 1, 0, -1, -2, -3 respectively. - Philippe Deléham, Nov 17 2013, Nov 19 2013
(1,a^n) Pascal triangle with a = -1. - Philippe Deléham, Dec 27 2013
T(n,k) = A112465(n,n-k). - Reinhard Zumkeller, Jan 03 2014

			From _Philippe Deléham_, Nov 17 2013: (Start)
Triangle begins:
  1;
  1, -1;
  1,  0,  1;
  1,  1,  1, -1;
  1,  2,  2,  0,  1;
  1,  3,  4,  2,  1, -1;
  1,  4,  7,  6,  3,  0,  1; (End)

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
C. Merino, S. D. Noble, M. Ramirez-Ibanez, R. Villarroel-Flores, On the structure of the h-vector of a paving matroid, Eur. J. Comb. 33, No. 8, 1787-1799 (2012).
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318 (a=1), A008949(a=2), A164844(a=10).
Similar to the triangles A035317, A059259, A080242, A112555.
Cf. A228196
Cf. A072547 (central terms).

Flat(List([0..13],n->List([0..n],k->Sum([0..k],i->Binomial(n,i)*(-2)^(k-i))))); # Muniru A Asiru, Feb 19 2018

a108561 n k = a108561_tabl !! n !! k
a108561_row n = a108561_tabl !! n
a108561_tabl = map reverse a112465_tabl
-- Reinhard Zumkeller, Jan 03 2014

A108561 := (n, k) -> add(binomial(n, i)*(-2)^(k-i), i = 0..k):
seq(seq(A108561(n,k), k = 0..n), n = 0..12); # Peter Bala, Feb 18 2018

Clear[t]; t[n_, 0] = 1; t[n_, n_] := t[n, n] = (-1)^Mod[n, 2]; t[n_, k_] := t[n, k] = t[n-1, k] + t[n-1, k-1]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013 *)

def A108561_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return -prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^k*prec(n, k) for k in (1..n-1)]+[(-1)^(n+1)]
for n in (1..12): print(A108561_row(n)) # Peter Luschny, Mar 16 2016

G.f.: (1-y*x)/(1-x-(y+y^2)*x). - Philippe Deléham, Nov 17 2013
T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0)=T(1,0)=1, T(1,1)=-1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 17 2013
From Peter Bala, Feb 18 2018: (Start)
T(n,k) = Sum_{i = 0..k} binomial(n,i)*(-2)^(k-i), 0 <= k <= n.
The n-th row polynomial is the n-th degree Taylor polynomial of the rational function (1 + x)^n/(1 + 2*x) about 0. For example, for n = 4, (1 + x)^4/(1 + 2*x) = 1 + 2*x + 2*x^2 + x^4 + O(x^5). (End)

Definition corrected by Philippe Deléham, Dec 26 2013

cp's OEIS Frontend

A112468 Riordan array (1/(1-x), x/(1+x)).

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A287804 Number of quinary sequences of length n such that no two consecutive terms have distance 1.

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A287819 Number of nonary sequences of length n such that no two consecutive terms have distance 4.

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A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).

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A104350 Partial products of largest prime factors of numbers <= n.

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A113311 Expansion of (1+x)^2/(1-x).

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A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).

A068875 Expansion of (1 + xC)C, where C = (1 - (1 - 4x)^(1/2))/(2x) is the g.f. for Catalan numbers, A000108.