A174295 -id:A174295 - cp's OEIS frontend

A174297 First column of A174295.

1, -1, -1, 0, -1, 1, -3, 6, -15, 36, -91, 232, -603, 1585, -4213, 11298, -30537, 83097, -227475, 625992, -1730787, 4805595, -13393689, 37458330, -105089229, 295673994, -834086421, 2358641376, -6684761125, 18985057351, -54022715451

Offset: 0

Mats Granvik, Mar 15 2010

sign

First 6 terms as in Mobius function A008683. Signed version of A099323 with an additional leading 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. 099323, A112467, A112468, A174294, A174295, A174296.

a:= func< n | n lt 2 select (-1)^n else (&+[(-1)^(k+1)*Binomial(n-2, k)*Catalan(k): k in [0..n-2]]) >;
[a(n): n in [0..30]]; // G. C. Greubel, Nov 25 2021

a[n_]:= a[n]= If[n<2, (-1)^n, Sum[(-1)^(j+1)*Binomial[n-2, j]*CatalanNumber[j], {j, 0, n-2}]]; Table[a[n], {n,0,40}] (* G. C. Greubel, Nov 25 2021 *)

[1,-1]+[sum( (-1)^(j+1)*binomial(n-2,j)*catalan_number(j) for j in (0..n-2) ) for n in (2..40)] # G. C. Greubel, Nov 25 2021

a(n) = -(-3)^(n-3/2)*hypergeometric2F1([3/2, n-1],[2],4) for n > 2. - Mark van Hoeij, Jul 02 2010

a(n) = (-1)^n if n < 2 otherwise Sum_{j=0..n-2} (-1)^(j-1)*binomial(n-2, j)*Catalan(j). - G. C. Greubel, Nov 25 2021

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

Original entry on oeis.org

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

A112467 Riordan array ((1-2x)/(1-x), x/(1-x)).

Original entry on oeis.org

1, -1, 1, -1, 0, 1, -1, -1, 1, 1, -1, -2, 0, 2, 1, -1, -3, -2, 2, 3, 1, -1, -4, -5, 0, 5, 4, 1, -1, -5, -9, -5, 5, 9, 5, 1, -1, -6, -14, -14, 0, 14, 14, 6, 1, -1, -7, -20, -28, -14, 14, 28, 20, 7, 1, -1, -8, -27, -48, -42, 0, 42, 48, 27, 8, 1, -1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1, -1, -10, -44, -110, -165, -132, 0, 132, 165, 110

Offset: 0

Paul Barry, Sep 06 2005

Row sums are A000007. Diagonal sums are -F(n-2). Inverse is A112468. T(2n,n)=0.
(-1,1)-Pascal triangle. - Philippe Deléham, Aug 07 2006
Apart from initial term, same as A008482. - Philippe Deléham, Nov 07 2006
Each column equals the cumulative sum of the previous column. - Mats Granvik, Mar 15 2010
Reading along antidiagonals generates in essence rows of A192174. - Paul Curtz, Oct 02 2011
Triangle T(n,k), read by rows, given by (-1,2,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2011

			Triangle starts:
    1;
   -1,  1;
   -1,  0,   1;
   -1, -1,   1,   1;
   -1, -2,   0,   2,   1;
   -1, -3,  -2,   2,   3,   1;
   -1, -4,  -5,   0,   5,   4,  1;
   -1, -5,  -9,  -5,   5,   9,  5,  1;
   -1, -6, -14, -14,   0,  14, 14,  6,  1;
   -1, -7, -20, -28, -14,  14, 28, 20,  7,  1;
   -1, -8, -27, -48, -42,   0, 42, 48, 27,  8, 1;
   -1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1;
  ...
From _Paul Barry_, Apr 08 2011: (Start)
Production matrix begins:
   1,  1,
  -2, -1,  1,
   2,  0, -1,  1,
  -2,  0,  0, -1,  1,
   2,  0,  0,  0, -1,  1,
  -2,  0,  0,  0,  0, -1,  1,
   2,  0,  0,  0,  0,  0, -1,  1
  ... (End)

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Elena Barcucci, Antonio Bernini, Stefano Bilotta, and Renzo Pinzani, Restricting Dyck Paths and 312-avoiding Permutations, arXiv:2307.02837 [math.CO], 2023. Mentions this sequence.
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
D. Foata and G.-N. Han, The doubloon polynomial triangle, Ram. J. 23 (2010), 107-126.
Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965. [Mentions application to design of antenna arrays. Annotated scan.]

Same first 3 rows as in A054525.

Cf. A008482, A037012, A080232, A112466, A112467, A174293, A174294, A174295, A174296, A174297.

[n eq 0 select 1 else (2*k-n)*Binomial(n,k)/n: k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 04 2019

seq(seq( `if`(n=0, 1, (2*k-n)*binomial(n,k)/n), k=0..n), n=0..10); # G. C. Greubel, Dec 04 2019

T[n_, k_]= If[n==0, 1, ((2*k-n)/n)*Binomial[n, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Roger L. Bagula, Feb 16 2009; modified by G. C. Greubel, Dec 04 2019 *)

T(n, k) = if(n==0, 1, (2*k-n)*binomial(n,k)/n ); \\ G. C. Greubel, Dec 04 2019

def T(n, k):
    if (n==0): return 1
    else: return (2*k-n)*binomial(n,k)/n
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 04 2019

Number triangle T(n, k) = binomial(n, n-k) - 2*binomial(n-1, n-k-1).
Sum_{k=0..n} T(n, k)*x^k = (x-1)*(x+1)^(n-1). - Philippe Deléham, Oct 03 2005
T(n,k) = ((2*k-n)/n)*binomial(n, k), with T(0,0)=1. - Roger L. Bagula, Feb 16 2009; modified by G. C. Greubel, Dec 04 2019
T(n,k) = T(n-1,k-1) + T(n-1,k) with T(0,0)=1, T(1,0)=-1, T(n,k)=0 for k>n or for n<0. - Philippe Deléham, Nov 01 2011
G.f.: (1-2x)/(1-(1+y)*x). - Philippe Deléham, Dec 15 2011
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A133494(n), A081294(n), A005053(n), A067411(n), A199661(n), A083233(n) for x = 1, 2, 3, 4, 5, 6, 7, respectively. - Philippe Deléham, Dec 15 2011
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-1 - x + x^2/2! + x^3/3!) = -1 - 2*x - 2*x^2/2! + 5*x^4/4! + 14*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014
Sum_{k=0..n} T(n,k) = 0^n = A000007(n). - G. C. Greubel, Dec 04 2019

A174296 Row sums of A174294.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2

Offset: 0

Mats Granvik, Mar 15 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010693, A112468, A112467, A174294, A174295, A174297.

[n lt 2 select (n+1) else 2 + (n mod 2): n in [0..110]]; // G. C. Greubel, Nov 25 2021

Table[If[n<2, n+1, (5-(-1)^n)/2], {n,0,110}] (* G. C. Greubel, Nov 25 2021 *)

[1,2]+[(5-(-1)^n)/2 for n in (2..110)] # G. C. Greubel, Nov 25 2021

a(A004280(n)) = 3 for n > 2.
From G. C. Greubel, Nov 25 2021: (Start)
a(n) = a(n-2) for n > 3, with a(0) = 1, a(1) = 2, a(2) = 2, a(3) = 3.
a(n) = (5 - (-1)^n)/2 for n > 1, with a(0) = 1, a(1) = 2.
a(n) = (n+1)*[n<2] + A010693(n)*[n>1].
G.f.: (1_+ 2*x + x^2 + x^3)/(1 - x^2).
E.g.f.: (1/2)*( -exp(-x) - 2*(1+x) + 5*exp(x) ). (End)

A174294 Triangle T(n,k), read by rows, T(n,k) = (T(n-1,k-1) + T(n-2,k-1)) - (T(n-1,k) + T(n-2,k)), with T(n, 0) = T(n, k) = 1 and T(n, 1) = (n mod 2).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, -1, 0, 1, 1, 0, 0, 2, -2, 0, 1, 1, 1, 0, 0, 3, -3, 0, 1, 1, 0, 1, -2, 1, 4, -4, 0, 1, 1, 1, 0, 3, -6, 3, 5, -5, 0, 1, 1, 0, 0, 0, 6, -12, 6, 6, -6, 0, 1, 1, 1, 1, -3, 3, 9, -20, 10, 7, -7, 0, 1, 1, 0, 0, 4, -12, 12, 11, -30, 15, 8, -8, 0, 1

Offset: 0

Mats Granvik, Mar 15 2010

			Table begins:
  n\k|...0...1...2...3...4...5...6...7...8...9..10
  ---|--------------------------------------------
  0..|...1
  1..|...1...1
  2..|...1...0...1
  3..|...1...1...0...1
  4..|...1...0...0...0...1
  5..|...1...1...1..-1...0...1
  6..|...1...0...0...2..-2...0...1
  7..|...1...1...0...0...3..-3...0...1
  8..|...1...0...1..-2...1...4..-4...0...1
  9..|...1...1...0...3..-6...3...5..-5...0...1
  10.|...1...0...0...0...6.-12...6...6..-6...0...1

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A112467, A112468, A174294, A174295, A174296, A174297.

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, If[k==1, Mod[n, 2], T[n-1, k-1] +T[n-2, k-1] -T[n-1, k] -T[n-2, k] ]]];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 25 2021 *)

@CachedFunction
def T(n,k): # A174294
    if (k<0 or k>n): return 0
    elif (k==0 or k==n): return 1
    elif (k==1): return n%2
    else: return T(n-1, k-1) + T(n-2, k-1) - T(n-1, k) - T(n-2, k)
flatten([[T(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 25 2021

T(n,k) = (T(n-1,k-1) + T(n-2,k-1)) - (T(n-1,k) + T(n-2,k)), with T(n, 0) = T(n, k) = 1 and T(n, 1) = (n mod 2).

cp's OEIS Frontend

A174297 First column of A174295.

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

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A112467 Riordan array ((1-2x)/(1-x), x/(1-x)).

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A174296 Row sums of A174294.

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A174294 Triangle T(n,k), read by rows, T(n,k) = (T(n-1,k-1) + T(n-2,k-1)) - (T(n-1,k) + T(n-2,k)), with T(n, 0) = T(n, k) = 1 and T(n, 1) = (n mod 2).

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