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A194589 a(n) = A194588(n) - A005043(n); complementary Riordan numbers. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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-Fran├žois 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

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Last modified March 19 13:25 EDT 2019. Contains 321330 sequences. (Running on oeis4.)