A155050 -id:A155050 - cp's OEIS frontend

A155052 Image of symmetric Catalan based triangle A155050 under ((1+x)/(1-x),x)^{-1}.

1, -1, 1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 2, -1, 1, -1, 1, -2, 2, -1, 1, 1, -1, 2, 1, 2, -1, 1, -1, 1, -2, -1, 1, 2, -1, 1, 1, -1, 2, 1, 8, 1, 2, -1, 1, -1, 1, -2, -1, -8, 8, 1, 2, -1, 1, 1, -1, 2, 1, 8, 20, 8, 1, 2, -1, 1

Offset: 0

Paul Barry, Jan 19 2009

Row sums are aerated Catalan numbers (A000108).

			Triangle begins
1,
-1, 1,
1, -1, 1,
-1, 1, -1, 1,
1, -1, 2, -1, 1,
-1, 1, -2, 2, -1, 1,
1, -1, 2, 1, 2, -1, 1,
-1, 1, -2, -1, 1, 2, -1, 1,
1, -1, 2, 1, 8, 1, 2, -1, 1,
-1, 1, -2, -1, -8, 8, 1, 2, -1, 1,
1, -1, 2, 1, 8, 20, 8, 1, 2, -1, 1

A155051 Expansion of c(x^2)*(1+x)/(1-x), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 13, 18, 32, 46, 88, 130, 262, 394, 823, 1252, 2682, 4112, 8974, 13836, 30632, 47428, 106214, 165000, 373012, 581024, 1323924, 2066824, 4741264, 7415704, 17110549, 26805394, 62163064, 97520734, 227165524

Offset: 0

Paul Barry, Jan 19 2009

Row sums of A155050.

Conjecture: A000975(n) = A264784(a(n-1)) for n > 0. - Reinhard Zumkeller, Dec 04 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A155050, A000975, A264784.

A155051[n_] := 2*Sum[CatalanNumber[k/2]*(1 + (-1)^k)/2, {k, 0, n}] -
CatalanNumber[n/2]*(1 + (-1)^n)/2; Table[A155051[n], {n, 0, 50}] (* G. C. Greubel, Sep 30 2017 *)

a(n) = 2*Sum_{k=0..n,} ( C(k/2)*(1+(-1)^k)/2 ) - C(n/2)*(1+(-1)^n)/2, C(n) = A000108;
a(n) = (C(n/2) + 2*Sum_{k=0..(n/2-1), C(k)})*(1+(-1)^n)/2 + Sum_{k=0..n/2, C(k)}*(1-(-1)^n), C(n) = A000108.
Conjecture: (n+2)*a(n) -2*a(n-1) +(-5*n+4)*a(n-2) +8*a(n-3) +4*(n-3)*a(n-4)=0. - R. J. Mathar, Feb 05 2015
Conjecture: -(n+2)*(n-3)*a(n) +(n^2-n-10)*a(n-1) +4*(n^2-4*n+5)*a(n-2) -4*(n-2)^2*a(n-3)=0. - R. J. Mathar, Feb 05 2015

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A155052 Image of symmetric Catalan based triangle A155050 under ((1+x)/(1-x),x)^{-1}.

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A155051 Expansion of c(x^2)*(1+x)/(1-x), c(x) the g.f. of A000108.

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