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A227506 Schroeder triangle sums: a(2*n-1) = A010683(2*n-2) and a(2*n) = A010683(2*n-1) - A001003(2*n-1). 2
1, 1, 7, 17, 121, 353, 2591, 8257, 61921, 207905, 1582791, 5501073, 42344121, 150827073, 1170747519, 4247388417, 33186295681, 122125206977, 959260792775, 3570473750929, 28167068630713, 105820555054241, 837838806587167, 3172136074486337 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The terms of this sequence equal the Fi1 sums, see A180662, of the Schroeder triangle A033877 (with offset 1 and n for columns and k for rows).

LINKS

Table of n, a(n) for n=1..24.

FORMULA

a(n) = Sum_{k=1..floor((n+1)/2)} A033877(2*k-1,n).

a(2*n-1) = A010683(2*n-2) and a(2*n) = A010683(2*n-1) - A001003(2*n-1).

G.f.: (1-4*x+x^2 - sqrt(1-6*x+x^2) + x*sqrt(1+6*x+x^2))/(8*x).

MAPLE

A227506 := proc(n) local k, T; T := proc(n, k) option remember; if n=1 then return(1) fi; if k<n then return(0) fi; T(n, k-1)+T(n-1, k-1)+T(n-1, k) end; add(T(2*k-1, n), k = 1..iquo(n+1, 2)) end: seq(A227506(n), n = 1..24); # Peter Luschny, Jul 17 2013

A227506 := proc(n): if type(n, odd) then A010683(n-1) else A010683(n-1) - A001003(n-1) fi: end: A010683 := proc(n): if n = 0 then 1 else (2/n)*add(binomial(n, k)* binomial(n+k+1, k-1), k=1..n) fi: end: A001003 := proc(n): if n = 0 then 1 else add(binomial(n, j)*binomial(n+j, n-1), j=0..n)/(2*n) fi: end: seq(A227506(n), n=1..24);

MATHEMATICA

T[n_, k_] := T[n, k] = Which[n == 1, 1, k < n, 0, True, T[n, k - 1] + T[n - 1, k - 1] + T[n - 1, k]];

a[n_] := Sum[T[2 k - 1, n], {k, 1, (n + 1)/2}];

Array[a, 24] (* Jean-Fran├žois Alcover, Jul 11 2019, from Sage *)

PROG

(Sage)

def A227506(n):

    @CachedFunction

    def T(n, k):

        if n==1: return 1

        if k <n: return 0

        return T(n, k-1)+T(n-1, k-1)+T(n-1, k)

    return add(T(2*k-1, n) for k in (1..(n+1)//2))

[A227506(n) for n in (1..24)]  # Peter Luschny, Jul 16 2013

CROSSREFS

Cf. A033877, A010683, A001003.

Sequence in context: A165246 A266382 A118108 * A244279 A325584 A214149

Adjacent sequences:  A227503 A227504 A227505 * A227507 A227508 A227509

KEYWORD

nonn,easy

AUTHOR

Johannes W. Meijer, Jul 15 2013

STATUS

approved

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Last modified October 29 12:00 EDT 2020. Contains 338066 sequences. (Running on oeis4.)