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

0,5

Comments

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

Links

Table of n, a(n) for n=0..30.

Peter Luschny, The lost Catalan numbers.

Formula

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

Maple

# 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

Mathematica

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 *)

Prog

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

Crossrefs

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

Sequence in context: A077917 A127864 A055936 * A189918 A318415 A164560

Adjacent sequences: A194586 A194587 A194588 * A194590 A194591 A194592

Keyword

nonn

Author

Peter Luschny, Aug 30 2011

Status

approved