A194588 -id:A194588 - cp's OEIS frontend

A194589 a(n) = A194588(n) - A005043(n); complementary Riordan numbers.

0, 0, 1, 1, 5, 11, 34, 92, 265, 751, 2156, 6194, 17874, 51702, 149941, 435749, 1268761, 3700391, 10808548, 31613474, 92577784, 271407896, 796484503, 2339561795, 6877992334, 20236257626, 59581937299, 175546527727, 517538571125, 1526679067331, 4505996000730

Offset: 0

Peter Luschny, Aug 30 2011

nonn

The inverse binomial transform of a(n) is A194590(n).

Peter Luschny, The lost Catalan numbers.

Cf. A005043, A189912, A194588, A100071, A005717.

# First method, describes the derivation:
A056040 := n -> n!/iquo(n,2)!^2:
A057977 := n -> A056040(n)/(iquo(n,2)+1);
A001006 := n -> add(binomial(n,k)*A057977(k)*irem(k+1,2),k=0..n):
A005043 := n -> `if`(n=0,1,A001006(n-1)-A005043(n-1)):
A189912 := n -> add(binomial(n,k)*A057977(k),k=0..n):
A194588 := n -> `if`(n=0,1,A189912(n-1)-A194588(n-1)):
A194589 := n -> A194588(n)-A005043(n):
# Second method, more efficient:
A100071 := n -> A056040(n)*(n/2)^(n-1 mod 2):
A194589 := proc(n) local k;
(n mod 2)+(1/2)*add((-1)^k*binomial(n,k)*A100071(k+1),k=1..n) end:
# Alternatively:
a := n -> `if`(n<3,iquo(n,2),hypergeom([1-n/2,-n,3/2-n/2],[1,2-n],4)): seq(simplify(a(n)), n=0..30); # Peter Luschny, Mar 07 2017

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Mod[n, 2] + (1/2)*Sum[(-1)^k*Binomial[n, k]*2^-Mod[k, 2]*(k+1)^Mod[k, 2]*sf[k+1], {k, 1, n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Jul 30 2013, from 2nd method *)
Table[If[n < 3, Quotient[n, 2], HypergeometricPFQ[{1 - n/2, -n, 3/2 - n/2}, {1, 2-n}, 4]], {n,0,30}] (* Peter Luschny, Mar 07 2017 *)

a(n):=sum(binomial(n+2,k)*binomial(n-k,k),k,0,(n)/2); /* Vladimir Kruchinin, Sep 28 2015 */

a(n) = sum(k=0, n/2, binomial(n+2,k)*binomial(n-k,k));
vector(30, n, a(n-3)) \\ Altug Alkan, Sep 28 2015

a(n) = sum_{k=0..n} C(n,k)*A194590(k).
a(n) = (n mod 2)+(1/2)*sum_{k=1..n} (-1)^k*C(n,k)*(k+1)$*((k+1)/2)^(k mod 2). Here n$ denotes the swinging factorial A056040(n).
a(n) = PSUMSIGN([0,0,1,2,6,16,45,..] = PSUMSIGN([0,0,A005717]) where PSUMSIGN is from Sloane's "Transformations of integer sequences". - Peter Luschny, Jan 17 2012
A(x) = B'(x)*(1/x^2-1/(B(x)*x)), where B(x)/x is g.f. of A005043. - Vladimir Kruchinin, Sep 28 2015
a(n) = Sum_{k=0..n/2} C(n+2,k)*C(n-k,k). - Vladimir Kruchinin, Sep 28 2015
a(n) = hypergeom([1-n/2,-n,3/2-n/2],[1,2-n],4) for n>=3. - Peter Luschny, Mar 07 2017
a(n) ~ 3^(n + 1/2) / (8*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 17 2024

A217540 Scambler statistic on Dyck paths. Triangle T(n, k) read by rows, n >= 0, -n <= k <= n, T(n, k) is the number of Dyck paths of semilength n and k = number of returns + number of hills - number of peaks.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 2, 0, 1, 0, 0, 1, 3, 4, 2, 3, 0, 1, 0, 0, 1, 6, 10, 9, 8, 3, 4, 0, 1, 0, 0, 1, 10, 25, 30, 26, 17, 13, 4, 5, 0, 1, 0, 0, 1, 15, 56, 90, 90, 70, 49, 27, 19, 5, 6, 0, 1, 0, 0, 1, 21, 112, 245, 301, 266, 197, 128, 80, 39, 26, 6, 7, 0, 1

Offset: 0

Peter Luschny, Oct 21 2012

			[n\k] -8,-7,-6, -5,  -4,  -3,  -2,  -1,  0,   1,  2,  3,  4, 5, 6, 7, 8
-----------------------------------------------------------------------
[ 0 ]                                    1,
[ 1 ]                               0,   0,   1,
[ 2 ]                          0,   0,   1,   0,  1,
[ 3 ]                     0,   0,   1,   1,   2,  0,  1,
[ 4 ]                0,   0,   1,   3,   4,   2,  3,  0,  1,
[ 5 ]           0,   0,   1,   6,  10,   9,   8,  3,  4,  0, 1,
[ 6 ]       0,  0,   1,  10,  25,  30,  26,  17, 13,  4,  5, 0, 1,
[ 7 ]    0, 0,  1,  15,  56,  90,  90,  70,  49, 27, 19,  5, 6, 0, 1,
[ 8 ] 0, 0, 1, 21, 112, 245, 301, 266, 197, 128, 80, 39, 26, 6, 7, 0, 1
.
T(5, -2) = 6 counting the Dyck words
[1101011000] (()()(())) [1101100100] (()(())()) [1101101000] (()(()()))
[1110010100] ((())()()) [1110100100] ((()())()) [1110101000] ((()()())) .

Alois P. Heinz, Rows n = 0..100, flattened
Peter Luschny, The Scambler_statistic_on_Dyck_words.

b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, z, expand(`if`(y=0, z, 1)*(b(x-1, y+1, true)
      +b(x-1, y-1, false)*`if`(t and y<>1, 1/z, 1)))))
    end:
T:= n-> `if`(n=0, 1, (p-> seq(coeff(p, z, i), i=-n..n))
                     (b(2*n-1, 1, true))):
seq(T(n), n=0..10);  # Alois P. Heinz, Jun 10 2014

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, z, Expand[If[y == 0, z, 1]*(b[x-1, y+1, True]+b[x-1, y-1, False]*If[t && y != 1, 1/z, 1])]]]; T[n_] := If[n == 0, 1, Function[p, Table[Coefficient[p, z, i], {i, -n, n}]][b[2*n-1, 1, True]]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Oct 24 2016, after Alois P. Heinz *)

def A217540(n, k):
    def characteristic(d):
        count = 1
        h = d.heights()
        for i in (1..len(d)-1):
            if d[i-1]==1 and d[i]==0: count -= 1
            if h[i]==0: count +=1
            else:
                if h[i-1]==0 and h[i+1]==0: count += 1
        return count
    if n == 0: return 1
    count = 0
    for d in DyckWords(n):
        if k == characteristic(d): count += 1
    return count
for n in (0..6): [A217540(n, k) for k in (-n..n)]

T(n,-1) = A014531(n-2) = [0,0,0],1,3,10,30,90,...
T(n, 0) = A113682(n-2) = [1,0],1,1,4,9,26,70,197,...
T(n, 1) = A194588(n-1) = [0],1,0,2,2,8,17,49,128,...
Sum(k>=0,T(n,k))     = A189912(n-1) = [1],1,2,4,10,25, 66,177,..
Sum(k< 0,T(n,k))     = A217539(n)   =   0,0,0,1, 4,17, 66,252,..
Sum(-n<=k<=n,T(n,k)) = A000108(n)   =   1,1,2,5,14,42,132,429,..

cp's OEIS Frontend

A194589 a(n) = A194588(n) - A005043(n); complementary Riordan numbers.

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