The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A344500 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A344500

a(n) = Sum_{k=0..n} binomial(n, k)*CT(n, k) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*CT(n, k), where CT(n, k) is the Catalan triangle A053121.
(history; published version)

#12 by Vaclav Kotesovec at Sun Apr 21 04:40:45 EDT 2024

STATUS

editing

approved

#11 by Vaclav Kotesovec at Sun Apr 21 04:40:34 EDT 2024

FORMULA

From Vaclav Kotesovec, Apr 21 2024: (Start)

Recurrence: 2*(n+1)*(2*n + 1)*(13*n^3 - 17*n^2 - 12*n + 10)*a(n) = (143*n^5 - 44*n^4 - 525*n^3 + 320*n^2 + 94*n - 60)*a(n-1) + 4*(n-1)*(26*n^4 + 5*n^3 - 108*n^2 + 7*n + 22)*a(n-2) + 16*(n-2)*(n-1)*(13*n^3 + 22*n^2 - 7*n - 6)*a(n-3).

a(n) ~ sqrt((247 - 9131*13^(2/3)/(1788163 + 409728*sqrt(78))^(1/3) + (13*(1788163 + 409728*sqrt(78)))^(1/3))/13) * ((11 + (1/3)*(172017 - 16848*sqrt(78))^(1/3) + (6371 + 624*sqrt(78))^(1/3))^n / (sqrt(Pi*n) * 2^(2*n + 3) * 3^(n + 1/2))). (End)

#10 by Vaclav Kotesovec at Sun Apr 21 04:23:14 EDT 2024

MATHEMATICA

Table[Sum[(1 + (-1)^(n+j))/2 * Binomial[n, j] * Binomial[n+1, (n-j)/2] * (j+1)/(n+1), {j, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Apr 21 2024 *)

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approved

editing

#9 by Peter Luschny at Sat May 22 15:49:32 EDT 2021

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approved

#8 by Peter Luschny at Sat May 22 15:49:29 EDT 2021

CROSSREFS

Cf. A053121, A344501.

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approved

editing

#7 by Peter Luschny at Sat May 22 14:53:25 EDT 2021

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proposed

approved

#6 by Peter Luschny at Sat May 22 12:57:03 EDT 2021

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proposed

#5 by Peter Luschny at Sat May 22 12:56:55 EDT 2021

NAME

a(n) = Sum_{k=0..n} binomial(n, k)*CT(n, k) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*CT(n, k), where CT(n, k) is the Catalan triangle A053121.

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proposed

editing

#4 by Peter Luschny at Sat May 22 12:53:49 EDT 2021

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editing

proposed

#3 by Peter Luschny at Sat May 22 12:43:44 EDT 2021

FORMULA

a(n) = Sum_{j=0..n} even(n + j)*binomial(n, j)*binomial(n + 1, (n - j)/2)*(j + 1)/(n + 1), where even(k) = 1 if k is even and otherwise 0.

Discussion

Sat May 22

12:52

Peter Luschny: The identity given in the name seems to be rare. I know it only from the unitary Hermite polynomials. Does anyone know a characterization of such triangles?