A033297 -id:A033297 - cp's OEIS frontend

A032357 Convolution of Catalan numbers and powers of -1.

1, 0, 2, 3, 11, 31, 101, 328, 1102, 3760, 13036, 45750, 162262, 580638, 2093802, 7601043, 27756627, 101888163, 375750537, 1391512653, 5172607767, 19293659253, 72188904387, 270870709263, 1019033438061, 3842912963391, 14524440108761

Offset: 0

Wolfdieter Lang

Absolute value of the alternating sum of Catalan Numbers. - Alexander Adamchuk, Jul 03 2006
Sums of two consecutive terms are a(n-1) + a(n) = 1, 2, 5, 14, 42, ... = A000108(n) (Catalan Numbers). The prime p divides a((p-3)/2) for p = 11, 19, 29, 31, 41, 59, 61, 71, ... = A045468 (Primes congruent to {1, 4} mod 5). Prime p divides a(2*p+1) for p = 5, 11, 19, 29, 31, 41, 59, 61, 71, ... = A038872 (Primes congruent to {0, 1, 4} mod 5). Also odd primes where 5 is a square mod p. - Alexander Adamchuk, Jul 03 2006
Hankel transform is F(2*n+1), where F = A000045. - Paul Barry, Jul 22 2008
Equals INVERTi transform of A000958. - Gary W. Adamson, Apr 10 2009
Inverse binomial transform of A002212. - Philippe Deléham, Sep 17 2009
Number of singleton and plus-decomposable (2143, 2413, 3142)-avoiding permutations with no +bonds (ascents by 1), with offset 1. Equivalently, number of (2143, 2413, 3142)-avoiding permutations that start with 1 or end with n (top entry). E.g., 132 and 213 for n = 3; 1324, 1432, 3214 for n = 4. - Alexander Burstein, May 22 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Shalosh B. Ekhad and Mingjia Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, 2017.

Cf. A000045, A000108, A000958, A001622, A002212, A014137, A014138, A033297, A038872, A045468, A064739.

rec:= (n+1)*a(n) +3*(-n+1)*a(n-1) +2*(-2*n+1)*a(n-2)=0:
A:= gfun:-rectoproc({rec,a(0)=1,a(1)=0},a(n),remember):
seq(A(n),n=0..50); # Robert Israel, May 22 2015

Table[Sum[(-1)^(k+n)*CatalanNumber[k],{k,0,n}],{n,0,60}] (* Alexander Adamchuk, Jul 03 2006 *)
Round@Table[(-1)^n/GoldenRatio + CatalanNumber[n + 1] Hypergeometric2F1[1, n + 3/2, n + 3, -4], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Oct 02 2016 *)
Table[(CatalanNumber[n] (2 + (n + 1) Hypergeometric2F1[1, -n, 1/2, 5/4]) - (-1)^n)/2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 03 2016 *)

def A032357():
    f, c, n = 1, 1, 1
    while True:
        yield f
        n += 1
        c = c * (4*n - 6) // n
        f = c - f
a = A032357()
print([next(a) for  in range(27)]) # _Peter Luschny, Nov 30 2016

G.f.: c(x)/(1 + x), where c(x) is the g.f. for the Catalan numbers A000108.
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k), where C(k) = A000108(k).
a(n) = ((-1)^(n+1) - binomial(2*(n+1), n+1)*Sum_{k=0..n+1} (-5)^k*binomial(n+1, k)/binomial(2*k, k))/2.
a(n) = C(2*n, n)/(n+1) - a(n-1) = A000108(n) - a(n-1) with a(0) = 1. - Labos Elemer, Apr 26 2003
Conjecture: (n+1)*a(n) + 3*(-n+1)*a(n-1) + 2*(-2*n+1)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012
Conjecture is true since the g.f. satisfies (x - 3*x^2 - 4*x^3)*g'(x) + (1 - 6*x^2)*g(x) = 1. - Robert Israel, May 22 2015
a(n) = (-1)^n/A001622 + A000108(n+1)*hypergeom([1, n + 3/2], [n + 3], -4). - Vladimir Reshetnikov, Oct 02 2016
a(n) ~ 2^(2*n + 2) / (5*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 03 2016
a(n) = (A000108(n) * (2 + (n + 1)*hypergeom([1,-n], [1/2], 5/4)) - (-1)^n)/2. - Vladimir Reshetnikov, Oct 03 2016

More terms from Christian G. Bower, Apr 15 1998

More terms from Alexander Adamchuk, Jul 03 2006

A167477 Expansion of (1-3x+5x^2-x^3)/(1-3x+x^2)^2.

Original entry on oeis.org

1, 3, 12, 44, 149, 479, 1487, 4503, 13386, 39226, 113641, 326173, 928957, 2628459, 7395624, 20708264, 57739517, 160391483, 444068171, 1225831551, 3374848806, 9268963318, 25401364177, 69472849849, 189661024249, 516904018899

Offset: 0

Paul Barry, Nov 04 2009

Hankel transform of A033297 (when this starts 1,1,4,...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

LinearRecurrence[{6, -11, 6, -1}, {1, 3, 12, 44}, 100] (* G. C. Greubel, Jun 13 2016 *)

a(n) = 4*A001871(n-1) - A001871(n) + A001519(n+2). - R. J. Mathar, Jun 28 2011

A168377 Riordan array (1/(1 + x), x*c(x)), where c(x) is the o.g.f. of Catalan numbers A000108.

Original entry on oeis.org

1, -1, 1, 1, 0, 1, -1, 2, 1, 1, 1, 3, 4, 2, 1, -1, 11, 10, 7, 3, 1, 1, 31, 32, 21, 11, 4, 1, -1, 101, 100, 69, 37, 16, 5, 1, 1, 328, 329, 228, 128, 59, 22, 6, 1, -1, 1102, 1101, 773, 444, 216, 88, 29, 7, 1, 1, 3760, 3761, 2659, 1558, 785, 341, 125, 37, 8, 1

Offset: 0

Philippe Deléham, Nov 24 2009

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
   1;
  -1,   1;
   1,   0,   1;
  -1,   2,   1,  1;
   1,   3,   4,  2,  1;
  -1,  11,  10,  7,  3,  1;
   1,  31,  32, 21, 11,  4, 1;
  -1, 101, 100, 69, 37, 16, 5, 1;
  ...
From _Philippe Deléham_, Sep 14 2014: (Start)
Production matrix begins:
  -1, 1
   0, 1, 1
   0, 1, 1, 1
   0, 1, 1, 1, 1
   0, 1, 1, 1, 1, 1
   0, 1, 1, 1, 1, 1, 1
   0, 1, 1, 1, 1, 1, 1, 1
   0, 1, 1, 1, 1, 1, 1, 1, 1
   ... (End)

Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production matrices and Riordan arrays, arXiv:math/0702638 [math.CO], 2007.
Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production matrices and Riordan arrays, Annals of Combinatorics, 13 (2009), 65-85.
L. W. Shapiro, S. Getu, W.-J. Woan, and L. C. Woodson, The Riordan group, Discrete Applied Mathematics, 34(1-3) (1991), 229-239.
Wikipedia, Riordan array.

Cf. A000012, A000108, A000124, A023443, A032357, A033297, A033999, A091491, A096470, A106566, A127540.

A000108(n) = binomial(2*n, n)/(n+1);
A032357(n) = sum(k=0, n, (-1)^(n-k)*A000108(k));
T(n, k) = if ((k==0), (-1)^n, if ((n<0) || (k<0), 0, if (k==1, A032357(n-1), if (n > k-1, T(n, k-1) - T(n-1, k-2), 0))));
for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ Petros Hadjicostas, Aug 08 2020

T(n,0) = (-1)^n and T(n,n) = 1.
Sum_{0 <= k <= n} T(n,k) = A032357(n).
From Petros Hadjicostas, Aug 08 2020: (Start)
T(n,k) = T(n,k-1) - T(n-1,k-2) for 2 <= k <= n with initial conditions T(n,0) = (-1)^n (n >= 0) and T(n,1) = A032357(n-1) (n >= 1).
T(n,2) = A033297(n).
T(n,n-1) = n - 2 for n >= 1.
|T(n,k)| = |A096470(n,n-k)| for 0 <= k <= n.
Bivariate o.g.f.: 1/((1 + x)*(1 - x*y*c(x))), where c(x) is the o.g.f. of A000108.
Bivariate o.g.f.: (1 - y + x*y*c(x))/((1 + x)*(1 - y + x*y^2)).
Bivariate o.g.f. of |T(n,k)|: (o.g.f. of T(n,k)) + 2*x/(1 - x^2). (End)

A096470 Triangle T(n,k), read by rows, formed by setting all entries in the zeroth column and in the main diagonal ((n,n) entries) to 1 and defining the rest of the entries by the recursion T(n,k) = T(n-1,k) - T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, 2, 1, 1, -2, 4, -3, 1, 1, -3, 7, -10, 11, 1, 1, -4, 11, -21, 32, -31, 1, 1, -5, 16, -37, 69, -100, 101, 1, 1, -6, 22, -59, 128, -228, 329, -328, 1, 1, -7, 29, -88, 216, -444, 773, -1101, 1102, 1, 1, -8, 37, -125, 341, -785, 1558, -2659, 3761, -3760, 1, 1, -9, 46, -171, 512, -1297, 2855, -5514, 9275, -13035, 13036, 1

Offset: 0

Gerald McGarvey, Aug 12 2004

If A(x,y) is the bivariate o.g.f. of a triangular array T(n,k) and B(x,y) is the bivariate o.g.f. of its mirror image T(n,n-k), then B(x,y) = A(x*y, y^(-1)) and A(x,y) = B(x*y, y^(-1)). - Petros Hadjicostas, Aug 08 2020

			From _Petros Hadjicostas_, Aug 08 2020: (Start)
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  0,  1;
  1, -1,  2,   1;
  1, -2,  4,  -3,   1;
  1, -3,  7, -10,  11,    1;
  1, -4, 11, -21,  32,  -31,   1;
  1, -5, 16, -37,  69, -100, 101,    1;
  1, -6, 22, -59, 128, -228, 329, -328, 1;
  ... (End)

Cf. A000108, A000124, A032357, A033297, A168377.

T(n, k) = if ((k==0) || (n==k), 1, if ((n<0) || (k<0), 0, if (n>k, T(n-1, k) - T(n, k-1), 0)));
for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ Petros Hadjicostas, Aug 08 2020

T(n,k) = T(n-1,k) - T(n,k-1) for 1 <= k <= n-1 with T(n,0) = 1 = T(n,n) for n >= 0.
The 2nd column is T(n,2) = A000124(n-2) for n >= 2 (Hogben's central polygonal numbers).
The "first subdiagonal" (unsigned) is |T(n,n-1)| = A032357(n-1) for n >= 1 (Convolution of Catalan numbers and powers of -1).
The "2nd subdiagonal" (unsigned) is |T(n,n-2)| = A033297(n) = Sum_{i=0..n-2} (-1)^i*C(n-1-i) for n >= 2, where C(n) are the Catalan numbers (A000108).
From Petros Hadjicostas, Aug 08 2020: (Start)
|T(n,k)| = |A168377(n,n-k)| for 0 <= k <= n.
Bivariate o.g.f.: (1 + y + x*y*c(-x*y))/((1 - x*y)*(1 - x + y)), where c(x) = 2/(1 + sqrt(1 - 4*x)) = o.g.f. of A000108.
Bivariate o.g.f. of |T(n,k)|: (1 - y - x*y*c(x*y))/((1 + x*y)*(1 - x - y)) + 2*x*y/(1 - x^2*y^2).
Bivariate o.g.f. of mirror image T(n,n-k): (1 + y + x*y*c(-x))/((1 - x)*(1 + y - x*y^2)).
Bivariate o.g.f. of |T(n,n-k)|: (1 - y + x*y*c(x))/((1 + x)*(1 - y + x*y^2)) + 2*x/(1 - x^2). (End)

Offset changed to 0 by Petros Hadjicostas, Aug 08 2020

A143949 Triangle read by rows: T(n,k) is the number of n-Dyck paths containing k odd-length descents to ground level (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 4, 4, 5, 0, 1, 10, 17, 7, 7, 0, 1, 32, 46, 34, 10, 9, 0, 1, 100, 155, 94, 55, 13, 11, 0, 1, 329, 502, 335, 154, 80, 16, 13, 0, 1, 1101, 1701, 1110, 580, 226, 109, 19, 15, 0, 1, 3761, 5820, 3865, 1960, 898, 310, 142, 22, 17, 0, 1, 13035, 20251

Offset: 0

Emeric Deutsch, Oct 05 2008

Row sums are the Catalan numbers (A000108).
T(0,n)=A033297(n).
Sum(k*T(n,k),k=0..n)=A000957(n+2) (the Fine numbers).
The case of even-length descents to ground level is considered in A111301.

			T(4,2) = 5 because we have U(D)U(D)UUDD, U(D)UUDDU(D), U(D)UUU(DDD), UUDDU(D)U(D) and UUU(DDD)U(D) (the odd-length descents to ground level are shown between parentheses).
Triangle starts:
1;
0,1;
1,0,1;
1,3,0,1;
4,4,5,0,1;
10,17,7,7,0,1;

Cf. A000108, A033297, A000957, A111301.

C:=((1-sqrt(1-4*z))*1/2)/z: G:=1/(1-z*(t+z*C)/(1-z^2*C^2)): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(expand(coeff(Gser,z,n))) end do: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) end do; # yields sequence in triangular form

G.f.: G(s,z) = 1/[1-z(t+zC)/(1-z^2*C^2)], where C = [1-sqrt(1-4z)]/(2z) is the Catalan function.

The trivariate g.f. H(t,s,z), where t (s) marks odd-length (even-length) descents to ground level and z marks semilength, is H=1/[1-z(t+szC)/(1-z^2*C^2)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.

A237596 Convolution triangle of A000958(n+1).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 8, 7, 3, 1, 24, 22, 12, 4, 1, 75, 73, 43, 18, 5, 1, 243, 246, 156, 72, 25, 6, 1, 808, 844, 564, 283, 110, 33, 7, 1, 2742, 2936, 2046, 1092, 465, 158, 42, 8, 1, 9458, 10334, 7449, 4178, 1906, 714, 217, 52, 9, 1, 33062, 36736, 27231, 15904, 7670, 3096, 1043, 288, 63, 10, 1

Offset: 0

Philippe Deléham, Feb 09 2014

Riordan array (f(x)/x, f(x)) where f(x) is the g.f. of A000958.
Reversal of A236918.
Row sums are A109262(n+1).
Diagonal sums are A033297(n+2).

			Triangle begins:
    1;
    1,   1;
    3,   2,   1;
    8,   7,   3,   1;
   24,  22,  12,   4,   1;
   75,  73,  43,  18,   5,  1;
  243, 246, 156,  72,  25,  6, 1;
  808, 844, 564, 283, 110, 33, 7, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832.

Columns give A000958, A114495.

Cf. A033297 (diagonal sums), A109262 (row sums), A236918 (row reversal).

# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, n -> A000958(n)); # Peter Luschny, Oct 19 2022

P[n_, x_]:= P[n, x]= If[n==0, 1, Sum[(j/(2*n-j))*Binomial[2*n-j, n-j]*Fibonacci[j, x], {j,0,n}]];
T[n_, k_] := Coefficient[P[n+1, x], x, k];
Table[T[n, k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 14 2022 *)

def f(n,x): return sum( binomial(n-j-1, j)*x^(n-2*j-1) for j in (0..(n-1)//2) )
def p(n,x):
    if (n==0): return 1
    else: return sum( (j/(2*n-j))*binomial(2*n-j, n-j)*f(j, x) for j in (0..n) )
def A237596(n,k): return ( p(n+1,x) ).series(x, n+1).list()[k]
flatten([[A237596(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 14 2022

G.f. for the column k-1: ((1-sqrt(1-4*x))^k/(1+sqrt(1-4*x) + 2*x)^k)/x.
Sum_{k=0..n} T(n,k) = A109262(n+1).
From G. C. Greubel, Jun 14 2022: (Start)
T(n, k) = coefficient of [x^k]( p(n+1, x) ), where p(n, x) = Sum_{j=0..n} (j/(2*n-j))*binomial(2*n-j, n-j)*Fibonacci(j, x) with p(0, x) = 1 and Fibonacci(n, x) are the Fibonacci polynomials.
T(n, k) = A236918(n, n-k). (End)

A278072 Riordan array(1/(1+x), (1-sqrt(1-4x))/(2x)).

Original entry on oeis.org

1, -1, 1, 1, 1, 1, -1, 4, 3, 1, 1, 10, 11, 5, 1, -1, 32, 37, 22, 7, 1, 1, 100, 128, 88, 37, 9, 1, -1, 329, 444, 341, 171, 56, 11, 1, 1, 1101, 1558, 1297, 739, 294, 79, 13, 1, -1, 3761, 5514, 4891, 3069, 1406, 465, 106, 15, 1, 1, 13035, 19680, 18365, 12435, 6346, 2442, 692, 137, 17, 1

Offset: 0

Peter Luschny, Nov 22 2016

			[   1]
[  -1,   1]
[   1,   1,   1]
[  -1,   4,   3,   1]
[   1,  10,  11,   5,   1]
[  -1,  32,  37,  22,   7,  1]
[   1, 100, 128,  88,  37,  9,  1]
[  -1, 329, 444, 341, 171, 56, 11, 1]

Cf. A033297, A084849.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/(1+#)&, (1-Sqrt[1-4#])/(2#)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

# uses[riordan_array from A256893]
riordan_array(1/(1+x), (1-sqrt(1-4*x))/(2*x), 8)

cp's OEIS Frontend

A032357 Convolution of Catalan numbers and powers of -1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

Extensions

A167477 Expansion of (1-3x+5x^2-x^3)/(1-3x+x^2)^2.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A168377 Riordan array (1/(1 + x), x*c(x)), where c(x) is the o.g.f. of Catalan numbers A000108.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A096470 Triangle T(n,k), read by rows, formed by setting all entries in the zeroth column and in the main diagonal ((n,n) entries) to 1 and defining the rest of the entries by the recursion T(n,k) = T(n-1,k) - T(n,k-1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A143949 Triangle read by rows: T(n,k) is the number of n-Dyck paths containing k odd-length descents to ground level (0<=k<=n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A237596 Convolution triangle of A000958(n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

A278072 Riordan array(1/(1+x), (1-sqrt(1-4*x))/(2*x)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Sage

A278072 Riordan array(1/(1+x), (1-sqrt(1-4x))/(2x)).